Integrand size = 34, antiderivative size = 699 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {f \arctan (\sinh (c+d x))}{a d^2}-\frac {b^2 f \arctan (\sinh (c+d x))}{a^3 d^2}+\frac {b^4 f \arctan (\sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 f x \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 f x \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}-\frac {3 f x \text {arctanh}(\cosh (c+d x))}{2 a d}+\frac {b^2 f x \text {arctanh}(\cosh (c+d x))}{a^3 d}+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 a d}-\frac {b^2 (e+f x) \text {arctanh}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x) \coth (2 c+2 d x)}{a^2 d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 f \log (\cosh (c+d x))}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b f \log (\sinh (2 c+2 d x))}{a^2 d^2}+\frac {3 f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}-\frac {3 f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {b^5 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac {b^5 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x) \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x) \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x) \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d} \]
2*b*(f*x+e)*coth(2*d*x+2*c)/a^2/d-1/2*(f*x+e)*csch(d*x+c)^2*sech(d*x+c)/a/ d+3/2*f*polylog(2,-exp(d*x+c))/a/d^2-3/2*f*polylog(2,exp(d*x+c))/a/d^2+f*a rctan(sinh(d*x+c))/a/d^2-3/2*f*x*arctanh(cosh(d*x+c))/a/d+3/2*(f*x+e)*arct anh(cosh(d*x+c))/a/d-3/2*(f*x+e)*sech(d*x+c)/a/d-2*b^2*f*x*arctanh(exp(d*x +c))/a^3/d+b^4*f*arctan(sinh(d*x+c))/a^3/(a^2+b^2)/d^2+b^2*f*x*arctanh(cos h(d*x+c))/a^3/d+b^3*f*ln(cosh(d*x+c))/a^2/(a^2+b^2)/d^2-b^5*(f*x+e)*ln(1+b *exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(3/2)/d+b^5*(f*x+e)*ln(1+b* exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(3/2)/d-b^5*f*polylog(2,-b*e xp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(3/2)/d^2+b^5*f*polylog(2,-b* exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(3/2)/d^2-b^4*(f*x+e)*sech(d *x+c)/a^3/(a^2+b^2)/d-b^3*(f*x+e)*tanh(d*x+c)/a^2/(a^2+b^2)/d-b^2*f*polylo g(2,-exp(d*x+c))/a^3/d^2+b^2*f*polylog(2,exp(d*x+c))/a^3/d^2-b^2*f*arctan( sinh(d*x+c))/a^3/d^2-b^2*(f*x+e)*arctanh(cosh(d*x+c))/a^3/d-b*f*ln(sinh(2* d*x+2*c))/a^2/d^2+b^2*(f*x+e)*sech(d*x+c)/a^3/d+3*f*x*arctanh(exp(d*x+c))/ a/d-1/2*f*csch(d*x+c)/a/d^2
Result contains complex when optimal does not.
Time = 9.44 (sec) , antiderivative size = 798, normalized size of antiderivative = 1.14 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{(a-i b) d^2}+\frac {f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{(a+i b) d^2}+\frac {\left (2 b d e \cosh \left (\frac {1}{2} (c+d x)\right )-a f \cosh \left (\frac {1}{2} (c+d x)\right )-2 b c f \cosh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right )}{4 a^2 d^2}+\frac {(-d e+c f-f (c+d x)) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {i f \log (\cosh (c+d x))}{2 (a-i b) d^2}-\frac {i f \log (\cosh (c+d x))}{2 (a+i b) d^2}+\frac {-2 a b f (c+d x)-\left (2 a b f+3 a^2 (d e+d f x)-2 b^2 (d e+d f x)\right ) \log \left (1-e^{-c-d x}\right )+\left (-2 a b f+3 a^2 (d e+d f x)-2 b^2 (d e+d f x)\right ) \log \left (1+e^{-c-d x}\right )-\left (3 a^2-2 b^2\right ) f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+\left (3 a^2-2 b^2\right ) f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )}{2 a^3 d^2}-\frac {b^5 \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac {(-d e+c f-f (c+d x)) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (2 b d e \sinh \left (\frac {1}{2} (c+d x)\right )+a f \sinh \left (\frac {1}{2} (c+d x)\right )-2 b c f \sinh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 a^2 d^2}+\frac {\text {sech}(c+d x) (-a d e+a c f-a f (c+d x)+b d e \sinh (c+d x)-b c f \sinh (c+d x)+b f (c+d x) \sinh (c+d x))}{\left (a^2+b^2\right ) d^2} \]
(f*ArcTan[Tanh[(c + d*x)/2]])/((a - I*b)*d^2) + (f*ArcTan[Tanh[(c + d*x)/2 ]])/((a + I*b)*d^2) + ((2*b*d*e*Cosh[(c + d*x)/2] - a*f*Cosh[(c + d*x)/2] - 2*b*c*f*Cosh[(c + d*x)/2] + 2*b*f*(c + d*x)*Cosh[(c + d*x)/2])*Csch[(c + d*x)/2])/(4*a^2*d^2) + ((-(d*e) + c*f - f*(c + d*x))*Csch[(c + d*x)/2]^2) /(8*a*d^2) + ((I/2)*f*Log[Cosh[c + d*x]])/((a - I*b)*d^2) - ((I/2)*f*Log[C osh[c + d*x]])/((a + I*b)*d^2) + (-2*a*b*f*(c + d*x) - (2*a*b*f + 3*a^2*(d *e + d*f*x) - 2*b^2*(d*e + d*f*x))*Log[1 - E^(-c - d*x)] + (-2*a*b*f + 3*a ^2*(d*e + d*f*x) - 2*b^2*(d*e + d*f*x))*Log[1 + E^(-c - d*x)] - (3*a^2 - 2 *b^2)*f*PolyLog[2, -E^(-c - d*x)] + (3*a^2 - 2*b^2)*f*PolyLog[2, E^(-c - d *x)])/(2*a^3*d^2) - (b^5*(-2*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^ 2]] + 2*c*f*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + f*(c + d*x)*Log [1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a ^2 + b^2])] - f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(a^ 3*(a^2 + b^2)^(3/2)*d^2) + ((-(d*e) + c*f - f*(c + d*x))*Sech[(c + d*x)/2] ^2)/(8*a*d^2) + (Sech[(c + d*x)/2]*(2*b*d*e*Sinh[(c + d*x)/2] + a*f*Sinh[( c + d*x)/2] - 2*b*c*f*Sinh[(c + d*x)/2] + 2*b*f*(c + d*x)*Sinh[(c + d*x)/2 ]))/(4*a^2*d^2) + (Sech[c + d*x]*(-(a*d*e) + a*c*f - a*f*(c + d*x) + b*d*e *Sinh[c + d*x] - b*c*f*Sinh[c + d*x] + b*f*(c + d*x)*Sinh[c + d*x]))/((a^2 + b^2)*d^2)
Result contains complex when optimal does not.
Time = 4.39 (sec) , antiderivative size = 611, normalized size of antiderivative = 0.87, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, Rules used = {6123, 5985, 2009, 6123, 5984, 3042, 25, 4672, 26, 3042, 26, 3956, 6123, 5985, 2009, 6107, 3042, 3803, 25, 2694, 27, 2620, 2715, 2838, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6123 |
| \(\displaystyle \frac {\int (e+f x) \text {csch}^3(c+d x) \text {sech}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5985 |
| \(\displaystyle \frac {-f \int \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{2 d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \text {sech}(c+d x)}{2 d}\right )dx+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 6123 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {csch}^2(c+d x) \text {sech}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (\frac {4 \int (e+f x) \text {csch}^2(2 c+2 d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {4 \int -\left ((e+f x) \csc (2 i c+2 i d x)^2\right )dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \int (e+f x) \csc (2 i c+2 i d x)^2dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {i f \int -i \coth (2 c+2 d x)dx}{2 d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \int \coth (2 c+2 d x)dx}{2 d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \int -i \tan \left (2 i c+2 i d x+\frac {\pi }{2}\right )dx}{2 d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}+\frac {i f \int \tan \left (\frac {1}{2} (4 i c+\pi )+2 i d x\right )dx}{2 d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6123 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {\int (e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5985 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-f \int \left (\frac {\text {sech}(c+d x)}{d}-\frac {\text {arctanh}(\cosh (c+d x))}{d}\right )dx-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {b^2 \int \frac {e+f x}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {2 b^2 \int -\frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \int \frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int \left (a (e+f x) \text {sech}^2(c+d x)-b (e+f x) \text {sech}(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-f \left (-\frac {\arctan (\sinh (c+d x))}{d^2}-\frac {3 x \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {3 x \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 d^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 d^2}+\frac {\text {csch}(c+d x)}{2 d^2}\right )+\frac {3 (e+f x) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {\arctan (\sinh (c+d x))}{d^2}+\frac {2 x \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {x \text {arctanh}(\cosh (c+d x))}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {(e+f x) \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x) \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {-\frac {a f \log (\cosh (c+d x))}{d^2}+\frac {a (e+f x) \tanh (c+d x)}{d}-\frac {b f \arctan (\sinh (c+d x))}{d^2}+\frac {b (e+f x) \text {sech}(c+d x)}{d}}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \log (-i \sinh (2 c+2 d x))}{4 d^2}\right )}{a}\right )}{a}\) |
((3*(e + f*x)*ArcTanh[Cosh[c + d*x]])/(2*d) - f*(-(ArcTan[Sinh[c + d*x]]/d ^2) - (3*x*ArcTanh[E^(c + d*x)])/d + (3*x*ArcTanh[Cosh[c + d*x]])/(2*d) + Csch[c + d*x]/(2*d^2) - (3*PolyLog[2, -E^(c + d*x)])/(2*d^2) + (3*PolyLog[ 2, E^(c + d*x)])/(2*d^2)) - (3*(e + f*x)*Sech[c + d*x])/(2*d) - ((e + f*x) *Csch[c + d*x]^2*Sech[c + d*x])/(2*d))/a - (b*((-4*(((e + f*x)*Coth[2*c + 2*d*x])/(2*d) - (f*Log[(-I)*Sinh[2*c + 2*d*x]])/(4*d^2)))/a - (b*((-(((e + f*x)*ArcTanh[Cosh[c + d*x]])/d) - f*(ArcTan[Sinh[c + d*x]]/d^2 + (2*x*Arc Tanh[E^(c + d*x)])/d - (x*ArcTanh[Cosh[c + d*x]])/d + PolyLog[2, -E^(c + d *x)]/d^2 - PolyLog[2, E^(c + d*x)]/d^2) + ((e + f*x)*Sech[c + d*x])/d)/a - (b*((-2*b^2*(-1/2*(b*(((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + (f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/( b*d^2)))/Sqrt[a^2 + b^2] + (b*(((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqr t[a^2 + b^2])])/(b*d) + (f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^ 2]))])/(b*d^2)))/(2*Sqrt[a^2 + b^2])))/(a^2 + b^2) + (-((b*f*ArcTan[Sinh[c + d*x]])/d^2) - (a*f*Log[Cosh[c + d*x]])/d^2 + (b*(e + f*x)*Sech[c + d*x] )/d + (a*(e + f*x)*Tanh[c + d*x])/d)/(a^2 + b^2)))/a))/a))/a
3.5.97.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( -I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csch[2*a + 2*b*x ]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> With[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Simp[(c + d*x)^m u, x] - Simp[d*m Int[(c + d*x)^(m - 1)*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n , p]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b^2/(a^2 + b^2) Int[(e + f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Simp[1/(a^2 + b^2) Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; F reeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0 ]
Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/a Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Simp[b/ a Int[(e + f*x)^m*Sech[c + d*x]^p*(Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2766\) vs. \(2(659)=1318\).
Time = 37.58 (sec) , antiderivative size = 2767, normalized size of antiderivative = 3.96
-3/2/d^2/a/(a^2+b^2)^(5/2)*c*b^5*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b ^2)^(1/2))-2/d^2*a/(a^2+b^2)^(5/2)*c*b^3*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a )/(a^2+b^2)^(1/2))-3/2/d^2*a^3/(a^2+b^2)^(5/2)*c*b*f*arctanh(1/2*(2*b*exp( d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/d^2/(a^2+b^2)^(5/2)*b*f*arctanh(1/2*(2*b*ex p(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^3+2/d^2/(a^2+b^2)^(5/2)*b^3*f*arctanh(1/2 *(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a+3/2/d/a/(a^2+b^2)^(5/2)*e*b^5*arc tanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/(a^2+b^2)^(5/2)/d/a*b^5*f *ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/(a^2+b^2)^ (5/2)/d/a*b^5*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x -1/(a^2+b^2)^(5/2)/d/a^3*b^7*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a ^2+b^2)^(1/2)))*x+1/(a^2+b^2)^(5/2)/d/a^3*b^7*f*ln((b*exp(d*x+c)+(a^2+b^2) ^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/(a^2+b^2)^(5/2)/d^2/a*b^5*f*ln((b*exp(d *x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-(3*d*f*x*a^3*exp(5*d*x+5*c )+a*d*f*x*b^2*exp(5*d*x+5*c)+3*d*e*a^3*exp(5*d*x+5*c)+a*b^2*d*e*exp(5*d*x+ 5*c)-2*b^3*d*f*x*exp(4*d*x+4*c)-2*d*f*x*a^3*exp(3*d*x+3*c)+a^3*f*exp(5*d*x +5*c)+2*a*d*f*x*b^2*exp(3*d*x+3*c)+a*b^2*f*exp(5*d*x+5*c)-2*b^3*d*e*exp(4* d*x+4*c)-2*d*e*a^3*exp(3*d*x+3*c)-4*exp(2*d*x+2*c)*a^2*b*d*f*x+2*a*b^2*d*e *exp(3*d*x+3*c)+3*a^3*d*f*x*exp(d*x+c)-4*a^2*b*d*e*exp(2*d*x+2*c)+exp(d*x+ c)*a*b^2*d*f*x+3*a^3*d*e*exp(d*x+c)+4*a^2*b*d*f*x+exp(d*x+c)*a*b^2*d*e+2*b ^3*d*f*x-a^3*f*exp(d*x+c)+4*a^2*b*d*e-a*b^2*f*exp(d*x+c)+2*b^3*d*e)/d^2...
Leaf count of result is larger than twice the leaf count of optimal. 11126 vs. \(2 (653) = 1306\).
Time = 0.55 (sec) , antiderivative size = 11126, normalized size of antiderivative = 15.92 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {csch}\left (d x + c\right )^{3} \operatorname {sech}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-1/2*(2*b^5*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/((a^5 + a^3*b^2)*sqrt(a^2 + b^2)*d) + 2*(4*a^2*b*e^(- 2*d*x - 2*c) + 2*b^3*e^(-4*d*x - 4*c) - 4*a^2*b - 2*b^3 + (3*a^3 + a*b^2)* e^(-d*x - c) - 2*(a^3 - a*b^2)*e^(-3*d*x - 3*c) + (3*a^3 + a*b^2)*e^(-5*d* x - 5*c))/((a^4 + a^2*b^2 - (a^4 + a^2*b^2)*e^(-2*d*x - 2*c) - (a^4 + a^2* b^2)*e^(-4*d*x - 4*c) + (a^4 + a^2*b^2)*e^(-6*d*x - 6*c))*d) - (3*a^2 - 2* b^2)*log(e^(-d*x - c) + 1)/(a^3*d) + (3*a^2 - 2*b^2)*log(e^(-d*x - c) - 1) /(a^3*d))*e - (32*b^5*integrate(-1/16*x*e^(d*x + c)/(a^5*b + a^3*b^3 - (a^ 5*b*e^(2*c) + a^3*b^3*e^(2*c))*e^(2*d*x) - 2*(a^6*e^c + a^4*b^2*e^c)*e^(d* x)), x) + 96*a^2*d*integrate(1/64*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 64*b ^2*d*integrate(1/64*x/(a^3*d*e^(d*x + c) + a^3*d), x) + 96*a^2*d*integrate (1/64*x/(a^3*d*e^(d*x + c) - a^3*d), x) - 64*b^2*d*integrate(1/64*x/(a^3*d *e^(d*x + c) - a^3*d), x) - a*b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c) + 1 )/(a^3*d^2)) - a*b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c) - 1)/(a^3*d^2)) - (2*b^3*d*x*e^(4*d*x + 4*c) + 4*a^2*b*d*x*e^(2*d*x + 2*c) + 2*(a^3*d*e^(3 *c) - a*b^2*d*e^(3*c))*x*e^(3*d*x) - 2*(2*a^2*b*d + b^3*d)*x - (a^3*e^(5*c ) + a*b^2*e^(5*c) + (3*a^3*d*e^(5*c) + a*b^2*d*e^(5*c))*x)*e^(5*d*x) + (a^ 3*e^c + a*b^2*e^c - (3*a^3*d*e^c + a*b^2*d*e^c)*x)*e^(d*x))/(a^4*d^2 + a^2 *b^2*d^2 + (a^4*d^2*e^(6*c) + a^2*b^2*d^2*e^(6*c))*e^(6*d*x) - (a^4*d^2*e^ (4*c) + a^2*b^2*d^2*e^(4*c))*e^(4*d*x) - (a^4*d^2*e^(2*c) + a^2*b^2*d^2...
Timed out. \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {e+f\,x}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]