Integrand size = 34, antiderivative size = 1122 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^2 f x}{2 a^3 d}+\frac {3 b f x \arctan \left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^5 (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {3 b f x \arctan (\sinh (c+d x))}{2 a^2 d}+\frac {3 b (e+f x) \arctan (\sinh (c+d x))}{2 a^2 d}-\frac {2 b^2 f x \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {4 (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}+\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}+\frac {3 b (e+f x) \text {csch}(c+d x)}{2 a^2 d}-\frac {f \text {csch}(2 c+2 d x)}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{a d}-\frac {b^6 (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 )^2 d}-\frac {b^6 (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 )^2 d}+\frac {b^6 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b^2 f x \log (\tanh (c+d x))}{a^3 d}+\frac {b^2 (e+f x) \log (\tanh (c+d x))}{a^3 d}-\frac {3 i b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a^2 d^2}+\frac {i b^5 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac {i b^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {3 i b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 a^2 d^2}-\frac {i b^5 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {i b^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b^6 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 )^2 d^2}-\frac {b^6 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 )^2 d^2}+\frac {b^6 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right )^2 d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {b f \text {sech}(c+d x)}{2 a^2 d^2}-\frac {b^3 f \text {sech}(c+d x)}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b^4 (e+f x) \text {sech}^2(c+d x)}{2 a^3 \left (a^2+b^2\right ) d}-\frac {b (e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a^2 d}-\frac {b^2 f \tanh (c+d x)}{2 a^3 d^2}+\frac {b^4 f \tanh (c+d x)}{2 a^3 \left (a^2+b^2\right ) d^2}-\frac {b^3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 a^2 \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x) \tanh ^2(c+d x)}{2 a^3 d} \]
-f*polylog(2,exp(2*d*x+2*c))/a/d^2+4*(f*x+e)*arctanh(exp(2*d*x+2*c))/a/d+f *polylog(2,-exp(2*d*x+2*c))/a/d^2+1/2*b^2*f*polylog(2,exp(2*d*x+2*c))/a^3/ d^2-2*b^5*(f*x+e)*arctan(exp(d*x+c))/a^2/(a^2+b^2)^2/d-2*b^2*f*x*arctanh(e xp(2*d*x+2*c))/a^3/d+1/2*b^6*f*polylog(2,-exp(2*d*x+2*c))/a^3/(a^2+b^2)^2/ d^2-1/2*b^3*f*sech(d*x+c)/a^2/(a^2+b^2)/d^2-1/2*b^4*(f*x+e)*sech(d*x+c)^2/ a^3/(a^2+b^2)/d-1/2*b*(f*x+e)*csch(d*x+c)*sech(d*x+c)^2/a^2/d+1/2*b^4*f*ta nh(d*x+c)/a^3/(a^2+b^2)/d^2-3/2*I*b*f*polylog(2,-I*exp(d*x+c))/a^2/d^2+1/2 *b^2*f*x/a^3/d-2*(f*x+e)*coth(2*d*x+2*c)*csch(2*d*x+2*c)/a/d+1/2*b*f*sech( d*x+c)/a^2/d^2-1/2*b^2*f*tanh(d*x+c)/a^3/d^2-1/2*b^2*(f*x+e)*tanh(d*x+c)^2 /a^3/d+1/2*I*b^3*f*polylog(2,-I*exp(d*x+c))/a^2/(a^2+b^2)/d^2+b*f*arctanh( cosh(d*x+c))/a^2/d^2+3/2*b*(f*x+e)*csch(d*x+c)/a^2/d-f*csch(2*d*x+2*c)/a/d ^2-3/2*b*f*x*arctan(sinh(d*x+c))/a^2/d+3/2*b*(f*x+e)*arctan(sinh(d*x+c))/a ^2/d+b^6*(f*x+e)*ln(1+exp(2*d*x+2*c))/a^3/(a^2+b^2)^2/d-b^6*(f*x+e)*ln(1+b *exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^2/d-b^6*(f*x+e)*ln(1+b*exp( d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^2/d-b^2*f*x*ln(tanh(d*x+c))/a^3/ d-b^6*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^2/d^2-b ^6*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^2/d^2+b^2* (f*x+e)*ln(tanh(d*x+c))/a^3/d+3*b*f*x*arctan(exp(d*x+c))/a^2/d-b^3*(f*x+e) *arctan(exp(d*x+c))/a^2/(a^2+b^2)/d-1/2*b^3*(f*x+e)*sech(d*x+c)*tanh(d*x+c )/a^2/(a^2+b^2)/d-I*b^5*f*polylog(2,I*exp(d*x+c))/a^2/(a^2+b^2)^2/d^2-1...
Time = 10.27 (sec) , antiderivative size = 1552, normalized size of antiderivative = 1.38 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \]
8*(((I/16)*(2*a^6 + 3*a^4*b^2 + b^6)*(d*e - c*f)*(c + d*x))/(a^3*(a^2 + b^ 2)^2*d^2) + ((I/32)*(2*a^6 + 3*a^4*b^2 + b^6)*f*(c + d*x)^2)/(a^3*(a^2 + b ^2)^2*d^2) + (b*f*Log[Cosh[(c + d*x)/2]])/(8*a^2*d^2) - (b*f*Log[Sinh[(c + d*x)/2]])/(8*a^2*d^2) - ((2*a^2 - b^2)*((d^2*f*x^2)/2 + d*e*(c + d*x) - 2 *(d*e - c*f)*(c + d*x) + 2*f*(c + d*x)*Log[1 + E^(-c - d*x)] + 2*(d*e - c* f)*Log[1 + E^(c + d*x)] - 2*f*PolyLog[2, -E^(-c - d*x)]))/(16*a^3*d^2) + ( (1/16 - I/16)*a*(2*a^2 + 3*b^2)*((d^2*f*x^2)/2 + d*e*(c + d*x) - (1 + I)*( d*e - c*f)*(c + d*x) + (1 + I)*f*(c + d*x)*Log[1 - I*E^(-c - d*x)] + (1 + I)*(d*e - c*f)*Log[I - E^(c + d*x)] - (1 + I)*f*PolyLog[2, I*E^(-c - d*x)] ))/((a^2 + b^2)^2*d^2) + ((1/16 + I/16)*(((-1/2*I)*b^6*(d*e - c*f + f*(c + d*x))^2)/f - (1 - I)*(2*a^2 - b^2)*(a^2 + b^2)^2*(d*e - c*f + f*(c + d*x) )*Log[1 - E^(-c - d*x)] + (1 - I)*a^4*(2*a^2 + 3*b^2)*(d*e - c*f + f*(c + d*x))*Log[1 + I*E^(-c - d*x)] - (1 - I)*a^4*(2*a^2 + 3*b^2)*f*PolyLog[2, ( -I)*E^(-c - d*x)] + (1 - I)*(2*a^2 - b^2)*(a^2 + b^2)^2*f*PolyLog[2, E^(-c - d*x)]))/(a^3*(a^2 + b^2)^2*d^2) + ((I/16)*b*(3*a^2 + 5*b^2)*((-2*I)*d*e *ArcTan[E^(c + d*x)] + (2*I)*c*f*ArcTan[E^(c + d*x)] + f*(c + d*x)*Log[1 - I*E^(c + d*x)] - f*(c + d*x)*Log[1 + I*E^(c + d*x)] - f*PolyLog[2, (-I)*E ^(c + d*x)] + f*PolyLog[2, I*E^(c + d*x)]))/((a^2 + b^2)^2*d^2) - (b^6*(-2 *d*e*(c + d*x) + 2*c*f*(c + d*x) - f*(c + d*x)^2 + (4*a*Sqrt[a^2 + b^2]*d* e*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/Sqrt[-(a^2 + b^2)^2] - ...
Time = 5.60 (sec) , antiderivative size = 943, normalized size of antiderivative = 0.84, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.735, Rules used = {6123, 5984, 3042, 26, 4673, 26, 3042, 26, 4670, 2715, 2838, 6123, 5985, 2009, 6123, 5985, 2009, 6107, 6107, 6095, 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.
\(\displaystyle \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
\(\Big \downarrow \) 6123 |
\(\displaystyle \frac {\int (e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
\(\Big \downarrow \) 5984 |
\(\displaystyle \frac {8 \int (e+f x) \text {csch}^3(2 c+2 d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {8 \int -i (e+f x) \csc (2 i c+2 i d x)^3dx}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \int (e+f x) \csc (2 i c+2 i d x)^3dx}{a}\) |
\(\Big \downarrow \) 4673 |
\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (\frac {1}{2} \int -i (e+f x) \text {csch}(2 c+2 d x)dx-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (-\frac {1}{2} i \int (e+f x) \text {csch}(2 c+2 d x)dx-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (-\frac {1}{2} i \int i (e+f x) \csc (2 i c+2 i d x)dx-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (\frac {1}{2} \int (e+f x) \csc (2 i c+2 i d x)dx-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i f \int \log \left (1-e^{2 c+2 d x}\right )dx}{2 d}-\frac {i f \int \log \left (1+e^{2 c+2 d x}\right )dx}{2 d}+\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i f \int e^{-2 c-2 d x} \log \left (1-e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {i f \int e^{-2 c-2 d x} \log \left (1+e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}+\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 6123 |
\(\displaystyle -\frac {b \left (\frac {\int (e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 5985 |
\(\displaystyle -\frac {b \left (\frac {-f \int \left (\frac {\text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}-\frac {3 \arctan (\sinh (c+d x))}{2 d}-\frac {3 \text {csch}(c+d x)}{2 d}\right )dx-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 6123 |
\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {\int (e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 5985 |
\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \int \left (\frac {\log (\tanh (c+d x))}{d}-\frac {\tanh ^2(c+d x)}{2 d}\right )dx-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 6107 |
\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 6107 |
\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 6095 |
\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \left (\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \left (-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {b^2 \left (\frac {b^2 \left (-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {b^2 \left (\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {b^2 \left (\frac {\int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\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}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int \left (a (e+f x) \text {sech}^3(c+d x)-b (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {8 i \left (-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}+\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )\right )}{a}-\frac {b \left (\frac {\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )}{a}-\frac {b \left (\frac {-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}-f \left (\frac {2 \text {arctanh}\left (e^{2 c+2 d x}\right ) x}{d}+\frac {\log (\tanh (c+d x)) x}{d}-\frac {x}{2 d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}\right )}{a}-\frac {b \left (\frac {\left (\frac {\left (-\frac {(e+f x)^2}{2 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)}{b d}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}\right ) b^2}{a^2+b^2}+\frac {\frac {b (e+f x)^2}{2 f}+\frac {2 a \arctan \left (e^{c+d x}\right ) (e+f x)}{d}-\frac {b \log \left (1+e^{2 (c+d x)}\right ) (e+f x)}{d}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d^2}}{a^2+b^2}\right ) b^2}{a^2+b^2}+\frac {\frac {b (e+f x) \text {sech}^2(c+d x)}{2 d}+\frac {a f \text {sech}(c+d x)}{2 d^2}+\frac {a (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}+\frac {a (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}-\frac {b f \tanh (c+d x)}{2 d^2}}{a^2+b^2}\right )}{a}\right )}{a}\right )}{a}\) |
((-8*I)*(((-1/8*I)*f*Csch[2*c + 2*d*x])/d^2 - ((I/4)*(e + f*x)*Coth[2*c + 2*d*x]*Csch[2*c + 2*d*x])/d + ((I*(e + f*x)*ArcTanh[E^(2*c + 2*d*x)])/d + ((I/4)*f*PolyLog[2, -E^(2*c + 2*d*x)])/d^2 - ((I/4)*f*PolyLog[2, E^(2*c + 2*d*x)])/d^2)/2))/a - (b*(((-3*(e + f*x)*ArcTan[Sinh[c + d*x]])/(2*d) - (3 *(e + f*x)*Csch[c + d*x])/(2*d) + ((e + f*x)*Csch[c + d*x]*Sech[c + d*x]^2 )/(2*d) - f*((3*x*ArcTan[E^(c + d*x)])/d - (3*x*ArcTan[Sinh[c + d*x]])/(2* d) + ArcTanh[Cosh[c + d*x]]/d^2 - (((3*I)/2)*PolyLog[2, (-I)*E^(c + d*x)]) /d^2 + (((3*I)/2)*PolyLog[2, I*E^(c + d*x)])/d^2 + Sech[c + d*x]/(2*d^2))) /a - (b*((((e + f*x)*Log[Tanh[c + d*x]])/d - ((e + f*x)*Tanh[c + d*x]^2)/( 2*d) - f*(-1/2*x/d + (2*x*ArcTanh[E^(2*c + 2*d*x)])/d + (x*Log[Tanh[c + d* x]])/d + PolyLog[2, -E^(2*c + 2*d*x)]/(2*d^2) - PolyLog[2, E^(2*c + 2*d*x) ]/(2*d^2) + Tanh[c + d*x]/(2*d^2)))/a - (b*((b^2*((b^2*(-1/2*(e + f*x)^2/( b*f) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + (f*Poly Log[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^2) + (f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^2)))/(a^2 + b^2) + ((b*(e + f*x)^2)/(2*f) + (2*a*(e + f*x)*ArcTan[E^(c + d*x)])/d - (b*(e + f*x)*Log [1 + E^(2*(c + d*x))])/d - (I*a*f*PolyLog[2, (-I)*E^(c + d*x)])/d^2 + (I*a *f*PolyLog[2, I*E^(c + d*x)])/d^2 - (b*f*PolyLog[2, -E^(2*(c + d*x))])/(2* d^2))/(a^2 + b^2)))/(a^2 + b^2) + ((a*(e + f*x)*ArcTan[E^(c + d*x)])/d ...
3.5.100.3.1 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[(((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[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[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S imp[b^2*((n - 2)/(n - 1)) Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
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[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3562 vs. \(2 (1045 ) = 2090\).
Time = 89.78 (sec) , antiderivative size = 3563, normalized size of antiderivative = 3.18
10*I/(a^2+b^2)/d^2*b^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c+1/(a^2+b^2)^(5 /2)/d^2*a^2*b^2*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+12/(a^ 2+b^2)/d*a*b^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x+12/(a^2+b^2)/d*a*b^2*f /(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x-12/(a^2+b^2)/d^2*a*c*b^2*f/(4*a^2+4*b^ 2)*ln(1+exp(2*d*x+2*c))+1/2/(a^2+b^2)^(5/2)/d^2*a^2*c*b^2*f*arctanh(1/2*(2 *b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+12/(a^2+b^2)/d^2*a*b^2*f/(4*a^2+4*b^2) *ln(1+I*exp(d*x+c))*c+12/(a^2+b^2)/d^2*a*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d* x+c))*c+6*I/(a^2+b^2)/d^2*a^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))*b-6*I/ (a^2+b^2)/d^2*a^2*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))*b+10*I/(a^2+b^2)/d *b^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x-10*I/(a^2+b^2)/d*b^3*f/(4*a^2+4* b^2)*ln(1+I*exp(d*x+c))*x-10*I/(a^2+b^2)/d^2*b^3*f/(4*a^2+4*b^2)*ln(1+I*ex p(d*x+c))*c-12/(a^2+b^2)/d^2*a^2*c*b*f/(4*a^2+4*b^2)*arctan(exp(d*x+c))-6* I/(a^2+b^2)/d^2*a^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*c+6*I/(a^2+b^2)/d ^2*a^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*b*c+12/(a^2+b^2)/d^2*a*b^2*f/(4* a^2+4*b^2)*dilog(1-I*exp(d*x+c))+1/(a^2+b^2)^(5/2)/d^2/a^2*b^6*f*arctanh(1 /2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+12/(a^2+b^2)/d*a^2*b*e/(4*a^2+4*b ^2)*arctan(exp(d*x+c))+1/(a^2+b^2)^2/d^2/a^3*c*b^6*f*ln(b*exp(2*d*x+2*c)+2 *a*exp(d*x+c)-b)+8/(a^2+b^2)/d*a^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x-10 *I/(a^2+b^2)/d^2*b^3*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))+10*I/(a^2+b^2)/ d^2*b^3*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))-1/d^2/a^2*f*b^4/(a^2+b^2)...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 16848 vs. \(2 (1020) = 2040\).
Time = 0.67 (sec) , antiderivative size = 16848, normalized size of antiderivative = 15.02 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {csch}\left (d x + c\right )^{3} \operatorname {sech}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-(b^6*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^7 + 2*a^5*b^2 + a^3*b^4)*d) + (3*a^2*b + 5*b^3)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b ^4)*d) - (2*a^3 + 3*a*b^2)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b ^4)*d) + (4*a*b^2*e^(-4*d*x - 4*c) - (3*a^2*b + 2*b^3)*e^(-d*x - c) + 2*(2 *a^3 + a*b^2)*e^(-2*d*x - 2*c) + (a^2*b - 2*b^3)*e^(-3*d*x - 3*c) - (a^2*b - 2*b^3)*e^(-5*d*x - 5*c) + 2*(2*a^3 + a*b^2)*e^(-6*d*x - 6*c) + (3*a^2*b + 2*b^3)*e^(-7*d*x - 7*c))/((a^4 + a^2*b^2 - 2*(a^4 + a^2*b^2)*e^(-4*d*x - 4*c) + (a^4 + a^2*b^2)*e^(-8*d*x - 8*c))*d) + (2*a^2 - b^2)*log(e^(-d*x - c) + 1)/(a^3*d) + (2*a^2 - b^2)*log(e^(-d*x - c) - 1)/(a^3*d))*e + (128* a^2*d*integrate(1/64*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 64*b^2*d*integrat e(1/64*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 128*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)) + (a*b^2 + ( a^2*b*e^(7*c) + (3*a^2*b*d*e^(7*c) + 2*b^3*d*e^(7*c))*x)*e^(7*d*x) - (2*a^ 3*e^(6*c) + a*b^2*e^(6*c) + 2*(2*a^3*d*e^(6*c) + a*b^2*d*e^(6*c))*x)*e^(6* d*x) - (a^2*b*e^(5*c) + (a^2*b*d*e^(5*c) - 2*b^3*d*e^(5*c))*x)*e^(5*d*x) - (4*a*b^2*d*x*e^(4*c) + a*b^2*e^(4*c))*e^(4*d*x) - (a^2*b*e^(3*c) - (a^2*b *d*e^(3*c) - 2*b^3*d*e^(3*c))*x)*e^(3*d*x) + (2*a^3*e^(2*c) + a*b^2*e^(2*c ) - 2*(2*a^3*d*e^(2*c) + a*b^2*d*e^(2*c))*x)*e^(2*d*x) + (a^2*b*e^c - (...
Timed out. \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {e+f\,x}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]