Integrand size = 32, antiderivative size = 454 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {(e+f x)^2}{2 b f}-\frac {a f \arctan (\sinh (c+d x))}{b^2 d^2}+\frac {a^3 f \arctan (\sinh (c+d x))}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac {a^2 f \log (\cosh (c+d x))}{b^3 d^2}+\frac {f \log (\cosh (c+d x))}{b d^2}+\frac {a^4 f \log (\cosh (c+d x))}{b^3 \left (a^2+b^2\right ) d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {a (e+f x) \text {sech}(c+d x)}{b^2 d}-\frac {a^3 (e+f x) \text {sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \tanh (c+d x)}{b^3 d}-\frac {(e+f x) \tanh (c+d x)}{b d}-\frac {a^4 (e+f x) \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d} \] Output:
1/2*(f*x+e)^2/b/f-a*f*arctan(sinh(d*x+c))/b^2/d^2+a^3*f*arctan(sinh(d*x+c) )/b^2/(a^2+b^2)/d^2-a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/( a^2+b^2)^(3/2)/d+a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2 +b^2)^(3/2)/d-a^2*f*ln(cosh(d*x+c))/b^3/d^2+f*ln(cosh(d*x+c))/b/d^2+a^4*f* ln(cosh(d*x+c))/b^3/(a^2+b^2)/d^2-a^3*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^ 2)^(1/2)))/b/(a^2+b^2)^(3/2)/d^2+a^3*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2 )^(1/2)))/b/(a^2+b^2)^(3/2)/d^2+a*(f*x+e)*sech(d*x+c)/b^2/d-a^3*(f*x+e)*se ch(d*x+c)/b^2/(a^2+b^2)/d+a^2*(f*x+e)*tanh(d*x+c)/b^3/d-(f*x+e)*tanh(d*x+c )/b/d-a^4*(f*x+e)*tanh(d*x+c)/b^3/(a^2+b^2)/d
Result contains complex when optimal does not.
Time = 5.45 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.79 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {(c+d x) (c f-d (2 e+f x))}{b}+\frac {2 f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a-i b}+\frac {2 f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a+i b}+\frac {f \log (\cosh (c+d x))}{i a-b}-\frac {f \log (\cosh (c+d x))}{i a+b}+\frac {2 a^3 \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{b \left (a^2+b^2\right )^{3/2}}+\frac {2 d (e+f x) \text {sech}(c+d x) (-a+b \sinh (c+d x))}{a^2+b^2}}{2 d^2} \] Input:
Integrate[((e + f*x)*Sinh[c + d*x]*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x]
Output:
-1/2*(((c + d*x)*(c*f - d*(2*e + f*x)))/b + (2*f*ArcTan[Tanh[(c + d*x)/2]] )/(a - I*b) + (2*f*ArcTan[Tanh[(c + d*x)/2]])/(a + I*b) + (f*Log[Cosh[c + d*x]])/(I*a - b) - (f*Log[Cosh[c + d*x]])/(I*a + b) + (2*a^3*(-2*d*e*ArcTa nh[(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*Pol yLog[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]))]))/(b*(a^2 + b^2)^(3/2)) + (2*d*(e + f*x)*Se ch[c + d*x]*(-a + b*Sinh[c + d*x]))/(a^2 + b^2))/d^2
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.
| \(\displaystyle \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6115 |
| \(\displaystyle \frac {\int (e+f x) \tanh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int -\left ((e+f x) \tan (i c+i d x)^2\right )dx}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\int (e+f x) \tan (i c+i d x)^2dx}{b}\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {i f \int i \tanh (c+d x)dx}{d}-\int (e+f x)dx+\frac {(e+f x) \tanh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {i f \int i \tanh (c+d x)dx}{d}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {-\frac {f \int \tanh (c+d x)dx}{d}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {-\frac {f \int -i \tan (i c+i d x)dx}{d}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {i f \int \tan (i c+i d x)dx}{d}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 6101 |
| \(\displaystyle -\frac {a \left (\frac {\int (e+f x) \text {sech}(c+d x) \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 5974 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \int \text {sech}(c+d x)dx}{d}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x) \text {sech}(c+d x)}{d}+\frac {f \int \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 6117 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}-\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}-\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {i f \int -i \tanh (c+d x)dx}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \int \tanh (c+d x)dx}{d}}{b}-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}-\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \int -i \tan (i c+i d x)dx}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}-\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x) \tanh (c+d x)}{d}+\frac {i f \int \tan (i c+i d x)dx}{d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {b^2 \int \frac {e+f x}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}-\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {2 b^2 \int -\frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \int \frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \arctan (\sinh (c+d x))}{d^2}-\frac {(e+f x) \text {sech}(c+d x)}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x) \tanh (c+d x)}{d}-\frac {f \log (\cosh (c+d x))}{d^2}}{b}-\frac {a \left (\frac {\int \left (a (e+f x) \text {sech}^2(c+d x)-b (e+f x) \text {sech}(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {-\frac {f \log (\cosh (c+d x))}{d^2}+\frac {(e+f x) \tanh (c+d x)}{d}-\frac {(e+f x)^2}{2 f}}{b}\) |
Input:
Int[((e + f*x)*Sinh[c + d*x]*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(1896\) vs. \(2(432)=864\).
Time = 1.25 (sec) , antiderivative size = 1897, normalized size of antiderivative = 4.18
Input:
int((f*x+e)*sinh(d*x+c)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVE RBOSE)
Output:
-2/d^2*a^3/(a^2+b^2)^(3/2)*c*b*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c) +2*a)/(a^2+b^2)^(1/2))+2/b/(a^2+b^2)^(3/2)/d^2*a^5*f/(2*a^2+2*b^2)*dilog(( b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-2/b/(a^2+b^2)^(3/2)/d ^2*a^5*f/(2*a^2+2*b^2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^ 2)^(1/2)))+2*b/(a^2+b^2)^(1/2)/d^2*a*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp( d*x+c)+2*a)/(a^2+b^2)^(1/2))+2*b^3/(a^2+b^2)^(3/2)/d^2*a*f/(2*a^2+2*b^2)*a rctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2*b/(a^2+b^2)^(3/2)/d^2*a ^3*f/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/b/( a^2+b^2)^(3/2)/d*a^5*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2 +b^2)^(1/2))+2*b/(a^2+b^2)^(3/2)/d^2*a^3*f/(2*a^2+2*b^2)*dilog((b*exp(d*x+ c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-2*b/(a^2+b^2)^(3/2)/d^2*a^3*f/( 2*a^2+2*b^2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2))) +2*b^3/(a^2+b^2)/d^2*f/(2*a^2+2*b^2)*ln(1+exp(2*d*x+2*c))-b^3/(a^2+b^2)/d^ 2*f/(2*a^2+2*b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+b/(a^2+b^2)^2/d^2* f*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)*a^2-4/(a^2+b^2)/d^2*a^3*f/(2*a^2+2 *b^2)*arctan(exp(d*x+c))+2*b/(a^2+b^2)/d^2*a^2*f/(2*a^2+2*b^2)*ln(1+exp(2* d*x+2*c))-2*b/(a^2+b^2)/d^2*a^2*f/(2*a^2+2*b^2)*ln(b*exp(2*d*x+2*c)+2*a*ex p(d*x+c)-b)-4*b^2/(a^2+b^2)/d^2*f/(2*a^2+2*b^2)*a*arctan(exp(d*x+c))-2*b/( a^2+b^2)^(5/2)/d^2*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^3 -2*b^3/(a^2+b^2)^(5/2)/d^2*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)...
Leaf count of result is larger than twice the leaf count of optimal. 1571 vs. \(2 (430) = 860\).
Time = 0.14 (sec) , antiderivative size = 1571, normalized size of antiderivative = 3.46 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)*sinh(d*x+c)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm ="fricas")
Output:
1/2*((a^4 + 2*a^2*b^2 + b^4)*d^2*f*x^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^2*e*x + 4*(a^2*b^2 + b^4)*d*e + ((a^4 + 2*a^2*b^2 + b^4)*d^2*f*x^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*e - 2*(a^2*b^2 + b^4)*d*f)*x)*cosh(d*x + c)^2 + ((a^4 + 2*a^2*b^2 + b^4)*d^2*f*x^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*e - 2*(a^2* b^2 + b^4)*d*f)*x)*sinh(d*x + c)^2 - 2*(a^3*b*f*cosh(d*x + c)^2 + 2*a^3*b* f*cosh(d*x + c)*sinh(d*x + c) + a^3*b*f*sinh(d*x + c)^2 + a^3*b*f)*sqrt((a ^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 2*(a^3*b*f*cosh(d* x + c)^2 + 2*a^3*b*f*cosh(d*x + c)*sinh(d*x + c) + a^3*b*f*sinh(d*x + c)^2 + a^3*b*f)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 2*(a^3*b*d*e - a^3*b*c*f + (a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a ^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d*e - a^3*b*c*f )*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh( d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 2*(a^3*b*d*e - a^3*b*c*f + ( a^3*b*d*e - a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*e - a^3*b*c*f)*cosh(d* x + c)*sinh(d*x + c) + (a^3*b*d*e - a^3*b*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^ 2)/b^2) + 2*a) - 2*(a^3*b*d*f*x + a^3*b*c*f + (a^3*b*d*f*x + a^3*b*c*f)*co sh(d*x + c)^2 + 2*(a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c)...
\[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \sinh {\left (c + d x \right )} \tanh ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)*sinh(d*x+c)*tanh(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)*sinh(c + d*x)*tanh(c + d*x)**2/(a + b*sinh(c + d*x)), x )
\[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \sinh \left (d x + c\right ) \tanh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)*sinh(d*x+c)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm ="maxima")
Output:
-(a^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqr t(a^2 + b^2)))/((a^2*b + b^3)*sqrt(a^2 + b^2)*d) - 2*(a*e^(-d*x - c) - b)/ ((a^2 + b^2 + (a^2 + b^2)*e^(-2*d*x - 2*c))*d) - (d*x + c)/(b*d))*e - 1/2* (4*a^3*integrate(-x*e^(d*x + c)/(a^2*b^2 + b^4 - (a^2*b^2*e^(2*c) + b^4*e^ (2*c))*e^(2*d*x) - 2*(a^3*b*e^c + a*b^3*e^c)*e^(d*x)), x) - ((a^2*d*e^(2*c ) + b^2*d*e^(2*c))*x^2*e^(2*d*x) + 4*a*b*x*e^(d*x + c) + 4*b^2*x + (a^2*d + b^2*d)*x^2)/(a^2*b*d + b^3*d + (a^2*b*d*e^(2*c) + b^3*d*e^(2*c))*e^(2*d* x)) + 4*b*x/((a^2 + b^2)*d) + 4*a*arctan(e^(d*x + c))/((a^2 + b^2)*d^2) - 2*b*log(e^(2*d*x + 2*c) + 1)/((a^2 + b^2)*d^2))*f
Timed out. \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)*sinh(d*x+c)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm ="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {sinh}\left (c+d\,x\right )\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((sinh(c + d*x)*tanh(c + d*x)^2*(e + f*x))/(a + b*sinh(c + d*x)),x)
Output:
int((sinh(c + d*x)*tanh(c + d*x)^2*(e + f*x))/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x) \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
int((f*x+e)*sinh(d*x+c)*tanh(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
( - 4*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**5*b*f - 8*e**(2*c + 2*d*x)*at an(e**(c + d*x))*a**3*b**3*f - 4*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a*b** 5*f - 4*atan(e**(c + d*x))*a**5*b*f - 8*atan(e**(c + d*x))*a**3*b**3*f - 4 *atan(e**(c + d*x))*a*b**5*f - 4*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan(( e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**3*b**2*d*e*i - 4*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**3*b**2*d*e*i - 16*e**(5*c + 2*d*x)*int((e**(3*d*x)*x)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5 *d*x)*a + e**(4*c + 4*d*x)*b + 4*e**(3*c + 3*d*x)*a - e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**7*d**2*f - 32*e**(5*c + 2*d*x)*int((e**(3*d*x )*x)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a + e**(4*c + 4*d*x)*b + 4*e **(3*c + 3*d*x)*a - e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**5*b** 2*d**2*f - 16*e**(5*c + 2*d*x)*int((e**(3*d*x)*x)/(e**(6*c + 6*d*x)*b + 2* e**(5*c + 5*d*x)*a + e**(4*c + 4*d*x)*b + 4*e**(3*c + 3*d*x)*a - e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**3*b**4*d**2*f - 2*e**(2*c + 2*d*x) *log(e**(2*c + 2*d*x) + 1)*a**6*f - 2*e**(2*c + 2*d*x)*log(e**(2*c + 2*d*x ) + 1)*a**4*b**2*f + 2*e**(2*c + 2*d*x)*log(e**(2*c + 2*d*x) + 1)*a**2*b** 4*f + 2*e**(2*c + 2*d*x)*log(e**(2*c + 2*d*x) + 1)*b**6*f + 4*e**(2*c + 2* d*x)*a**6*d*f*x + 2*e**(2*c + 2*d*x)*a**4*b**2*d**2*e*x + e**(2*c + 2*d*x) *a**4*b**2*d**2*f*x**2 + 4*e**(2*c + 2*d*x)*a**4*b**2*d*f*x + 4*e**(2*c + 2*d*x)*a**2*b**4*d**2*e*x + 2*e**(2*c + 2*d*x)*a**2*b**4*d**2*f*x**2 - ...