\(\int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [340]

Optimal result

Integrand size = 32, antiderivative size = 321 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 (e+f x)^2}{2 b^3 f}+\frac {(e+f x)^2}{4 b f}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}-\frac {f \cosh ^2(c+d x)}{4 b d^2}-\frac {a \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {a \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d} \] Output:

1/2*a^2*(f*x+e)^2/b^3/f+1/4*(f*x+e)^2/b/f-a*(f*x+e)*cosh(d*x+c)/b^2/d-1/4* 
f*cosh(d*x+c)^2/b/d^2-a*(a^2+b^2)^(1/2)*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+ 
b^2)^(1/2)))/b^3/d+a*(a^2+b^2)^(1/2)*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2 
)^(1/2)))/b^3/d-a*(a^2+b^2)^(1/2)*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^( 
1/2)))/b^3/d^2+a*(a^2+b^2)^(1/2)*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1 
/2)))/b^3/d^2+a*f*sinh(d*x+c)/b^2/d^2+1/2*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/ 
b/d
 

Mathematica [A] (warning: unable to verify)

Time = 1.79 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.81 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 b^2 e \left (\frac {c}{d}+x-\frac {2 a \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2} d}\right )+b^2 f \left (x^2-\frac {2 a \left (d x \left (\log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-\log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^2}\right )+\frac {2 e \left (\left (4 a^2+b^2\right ) (c+d x)-\frac {2 a \left (4 a^2+3 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-4 a b \cosh (c+d x)+b^2 \sinh (2 (c+d x))\right )}{d}+\frac {f \left (\left (4 a^2+b^2\right ) (-c+d x) (c+d x)-8 a b d x \cosh (c+d x)-b^2 \cosh (2 (c+d x))-\frac {2 a \left (4 a^2+3 b^2\right ) \left (2 c \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+(c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-(c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+\operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}+8 a b \sinh (c+d x)+2 b^2 d x \sinh (2 (c+d x))\right )}{d^2}}{8 b^3} \] Input:

Integrate[((e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), 
x]
 

Output:

(2*b^2*e*(c/d + x - (2*a*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2] 
])/(Sqrt[-a^2 - b^2]*d)) + b^2*f*(x^2 - (2*a*(d*x*(Log[1 + (b*E^(c + d*x)) 
/(a - Sqrt[a^2 + b^2])] - Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])]) 
+ PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - PolyLog[2, -((b*E^( 
c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^2)) + (2*e*((4*a^2 
+ b^2)*(c + d*x) - (2*a*(4*a^2 + 3*b^2)*ArcTan[(b - a*Tanh[(c + d*x)/2])/S 
qrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 4*a*b*Cosh[c + d*x] + b^2*Sinh[2*(c + 
 d*x)]))/d + (f*((4*a^2 + b^2)*(-c + d*x)*(c + d*x) - 8*a*b*d*x*Cosh[c + d 
*x] - b^2*Cosh[2*(c + d*x)] - (2*a*(4*a^2 + 3*b^2)*(2*c*ArcTanh[(a + b*E^( 
c + d*x))/Sqrt[a^2 + b^2]] + (c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a 
^2 + b^2])] - (c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + P 
olyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - PolyLog[2, -((b*E^(c + 
 d*x))/(a + Sqrt[a^2 + b^2]))]))/Sqrt[a^2 + b^2] + 8*a*b*Sinh[c + d*x] + 2 
*b^2*d*x*Sinh[2*(c + d*x)]))/d^2)/(8*b^3)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.61 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.01, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6113, 3042, 3791, 17, 6099, 17, 3042, 26, 3777, 3042, 3117, 3803, 25, 2694, 27, 2620, 2715, 2838}

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.

\(\displaystyle \int \frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\)

\(\Big \downarrow \) 6113

\(\displaystyle \frac {\int (e+f x) \cosh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{b}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {\frac {1}{2} \int (e+f x)dx-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\)

\(\Big \downarrow \) 6099

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)dx}{b^2}+\frac {\int (e+f x) \sinh (c+d x)dx}{b}\right )}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x) \sinh (c+d x)dx}{b}-\frac {a (e+f x)^2}{2 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{b^2}+\frac {\int -i (e+f x) \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^2}{2 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \int (e+f x) \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^2}{2 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{b}-\frac {a (e+f x)^2}{2 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b}-\frac {a (e+f x)^2}{2 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 3117

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {e+f x}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3803

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (\frac {2 \left (a^2+b^2\right ) \int -\frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 2694

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{b}\)

Input:

Int[((e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(a + b*Sinh[c + d*x]),x]
 

Output:

((e + f*x)^2/(4*f) - (f*Cosh[c + d*x]^2)/(4*d^2) + ((e + f*x)*Cosh[c + d*x 
]*Sinh[c + d*x])/(2*d))/b - (a*(-1/2*(a*(e + f*x)^2)/(b^2*f) - (2*(a^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 + Sqrt[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])))/b^2 - (I*((I*(e + f*x)*Cosh[c + d*x])/d - (I 
*f*Sinh[c + d*x])/d^2))/b))/b
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

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]
 

rule 2694

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]
 

rule 2715

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]
 

rule 2838

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3791

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> 
 Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x 
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n)   Int[(c + d* 
x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 
 1]
 

rule 3803

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]
 

rule 6099

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. 
)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-a/b^2   Int[(e + f*x)^m*Cos 
h[c + d*x]^(n - 2), x], x] + (Simp[1/b   Int[(e + f*x)^m*Cosh[c + d*x]^(n - 
 2)*Sinh[c + d*x], x], x] + Simp[(a^2 + b^2)/b^2   Int[(e + f*x)^m*(Cosh[c 
+ d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, 
 x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
 

rule 6113

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + 
 (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S 
imp[1/b   Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x] - S 
imp[a/b   Int[(e + f*x)^m*Cosh[c + d*x]^p*(Sinh[c + d*x]^(n - 1)/(a + b*Sin 
h[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[ 
n, 0] && IGtQ[p, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1011\) vs. \(2(293)=586\).

Time = 5.04 (sec) , antiderivative size = 1012, normalized size of antiderivative = 3.15

method	result	size
risch	\(\text {Expression too large to display}\)	\(1012\)

Input:

int((f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVE 
RBOSE)
 

Output:

1/2/b^3*a^2*f*x^2+1/4/b*f*x^2+1/b^3*a^2*e*x+1/2/b*e*x+1/16*(2*d*f*x+2*d*e- 
f)/b/d^2*exp(2*d*x+2*c)-1/2*a*(d*f*x+d*e-f)/b^2/d^2*exp(d*x+c)-1/2*a*(d*f* 
x+d*e+f)/b^2/d^2*exp(-d*x-c)-1/16*(2*d*f*x+2*d*e+f)/b/d^2*exp(-2*d*x-2*c)+ 
2/d*a^3/b^3*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^( 
1/2))+2/d*a/b*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2) 
^(1/2))-1/d*a^3/b^3*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a) 
/(-a+(a^2+b^2)^(1/2)))*x+1/d*a^3/b^3*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a 
^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/d^2*a^3/b^3*f/(a^2+b^2)^(1/2)*ln 
((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/d^2*a^3/b^3*f 
/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))* 
c-1/d^2*a^3/b^3*f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/ 
(-a+(a^2+b^2)^(1/2)))+1/d^2*a^3/b^3*f/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+ 
(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/d*a/b*f/(a^2+b^2)^(1/2)*ln((-b*e 
xp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/d*a/b*f/(a^2+b^2)^( 
1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/d^2*a/b* 
f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2) 
))*c+1/d^2*a/b*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a 
^2+b^2)^(1/2)))*c-1/d^2*a/b*f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^ 
2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/d^2*a/b*f/(a^2+b^2)^(1/2)*dilog((b*exp 
(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-2/d^2*a^3/b^3*f*c/(a^2+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1284 vs. \(2 (291) = 582\).

Time = 0.13 (sec) , antiderivative size = 1284, normalized size of antiderivative = 4.00 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm 
="fricas")
 

Output:

-1/16*(2*b^2*d*f*x - (2*b^2*d*f*x + 2*b^2*d*e - b^2*f)*cosh(d*x + c)^4 - ( 
2*b^2*d*f*x + 2*b^2*d*e - b^2*f)*sinh(d*x + c)^4 + 2*b^2*d*e + 8*(a*b*d*f* 
x + a*b*d*e - a*b*f)*cosh(d*x + c)^3 + 4*(2*a*b*d*f*x + 2*a*b*d*e - 2*a*b* 
f - (2*b^2*d*f*x + 2*b^2*d*e - b^2*f)*cosh(d*x + c))*sinh(d*x + c)^3 + b^2 
*f - 4*((2*a^2 + b^2)*d^2*f*x^2 + 2*(2*a^2 + b^2)*d^2*e*x)*cosh(d*x + c)^2 
 - 2*(2*(2*a^2 + b^2)*d^2*f*x^2 + 4*(2*a^2 + b^2)*d^2*e*x + 3*(2*b^2*d*f*x 
 + 2*b^2*d*e - b^2*f)*cosh(d*x + c)^2 - 12*(a*b*d*f*x + a*b*d*e - a*b*f)*c 
osh(d*x + c))*sinh(d*x + c)^2 + 16*(a*b*f*cosh(d*x + c)^2 + 2*a*b*f*cosh(d 
*x + c)*sinh(d*x + c) + a*b*f*sinh(d*x + c)^2)*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) - 16*(a*b*f*cosh(d*x + c)^2 + 2*a*b*f*co 
sh(d*x + c)*sinh(d*x + c) + a*b*f*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*d 
ilog((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) - 16*((a*b*d*e - a*b*c*f)*cosh(d*x + 
 c)^2 + 2*(a*b*d*e - a*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d*e - a*b 
*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*s 
inh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 16*((a*b*d*e - a*b*c*f)* 
cosh(d*x + c)^2 + 2*(a*b*d*e - a*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b 
*d*e - a*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) + 16*((a*b*d*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)*cosh(d*x+c)**2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm 
="maxima")
 

Output:

-1/16*(32*(a^3*e^c + a*b^2*e^c)*integrate(x*e^(d*x)/(b^4*e^(2*d*x + 2*c) + 
 2*a*b^3*e^(d*x + c) - b^4), x) - (4*(2*a^2*d^2*e^(2*c) + b^2*d^2*e^(2*c)) 
*x^2 + (2*b^2*d*x*e^(4*c) - b^2*e^(4*c))*e^(2*d*x) - 8*(a*b*d*x*e^(3*c) - 
a*b*e^(3*c))*e^(d*x) - 8*(a*b*d*x*e^c + a*b*e^c)*e^(-d*x) - (2*b^2*d*x + b 
^2)*e^(-2*d*x))*e^(-2*c)/(b^3*d^2))*f - 1/8*e*((4*a*e^(-d*x - c) - b)*e^(2 
*d*x + 2*c)/(b^2*d) + 8*sqrt(a^2 + b^2)*a*log((b*e^(-d*x - c) - a - sqrt(a 
^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(b^3*d) - 4*(2*a^2 + b^ 
2)*(d*x + c)/(b^3*d) + (4*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c))/(b^2*d))
 

Giac [F]

\[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm 
="giac")
 

Output:

integrate((f*x + e)*cosh(d*x + c)^2*sinh(d*x + c)/(b*sinh(d*x + c) + a), x 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:

int((cosh(c + d*x)^2*sinh(c + d*x)*(e + f*x))/(a + b*sinh(c + d*x)),x)
 

Output:

int((cosh(c + d*x)^2*sinh(c + d*x)*(e + f*x))/(a + b*sinh(c + d*x)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-32 e^{2 d x +2 c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{2} d e i +2 e^{4 d x +4 c} b^{4} d e -16 a^{2} b^{2} f -2 b^{4} d e -e^{4 d x +4 c} b^{4} f -32 a^{2} b^{2} d f x -8 e^{3 d x +3 c} a \,b^{3} d f x +16 e^{2 d x +2 c} a^{2} b^{2} d^{2} e x +8 e^{2 d x +2 c} a^{2} b^{2} d^{2} f \,x^{2}+32 e^{d x +c} a^{3} b d f x +24 e^{d x +c} a \,b^{3} d f x +64 e^{2 d x +2 c} \left (\int \frac {x}{e^{4 d x +4 c} b +2 e^{3 d x +3 c} a -e^{2 d x +2 c} b}d x \right ) a^{4} b \,d^{2} f +64 e^{2 d x +2 c} \left (\int \frac {x}{e^{4 d x +4 c} b +2 e^{3 d x +3 c} a -e^{2 d x +2 c} b}d x \right ) a^{2} b^{3} d^{2} f -160 e^{2 d x +c} \left (\int \frac {x}{e^{3 d x +2 c} b +2 e^{2 d x +c} a -e^{d x} b}d x \right ) a^{3} b^{2} d^{2} f -32 e^{2 d x +c} \left (\int \frac {x}{e^{3 d x +2 c} b +2 e^{2 d x +c} a -e^{d x} b}d x \right ) a \,b^{4} d^{2} f +8 e^{3 d x +3 c} a \,b^{3} f +32 e^{d x +c} a^{3} b f +24 e^{d x +c} a \,b^{3} f -32 a^{4} d f x -2 b^{4} d f x -16 a^{4} f -b^{4} f +2 e^{4 d x +4 c} b^{4} d f x -8 e^{3 d x +3 c} a \,b^{3} d e +8 e^{2 d x +2 c} b^{4} d^{2} e x +4 e^{2 d x +2 c} b^{4} d^{2} f \,x^{2}-8 e^{d x +c} a \,b^{3} d e -128 e^{2 d x +c} \left (\int \frac {x}{e^{3 d x +2 c} b +2 e^{2 d x +c} a -e^{d x} b}d x \right ) a^{5} d^{2} f}{16 e^{2 d x +2 c} b^{5} d^{2}} \] Input:

int((f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)
 

Output:

( - 32*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sq 
rt(a**2 + b**2))*a*b**2*d*e*i + 2*e**(4*c + 4*d*x)*b**4*d*e + 2*e**(4*c + 
4*d*x)*b**4*d*f*x - e**(4*c + 4*d*x)*b**4*f - 8*e**(3*c + 3*d*x)*a*b**3*d* 
e - 8*e**(3*c + 3*d*x)*a*b**3*d*f*x + 8*e**(3*c + 3*d*x)*a*b**3*f + 64*e** 
(2*c + 2*d*x)*int(x/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a - e**(2*c + 
 2*d*x)*b),x)*a**4*b*d**2*f + 64*e**(2*c + 2*d*x)*int(x/(e**(4*c + 4*d*x)* 
b + 2*e**(3*c + 3*d*x)*a - e**(2*c + 2*d*x)*b),x)*a**2*b**3*d**2*f + 16*e* 
*(2*c + 2*d*x)*a**2*b**2*d**2*e*x + 8*e**(2*c + 2*d*x)*a**2*b**2*d**2*f*x* 
*2 + 8*e**(2*c + 2*d*x)*b**4*d**2*e*x + 4*e**(2*c + 2*d*x)*b**4*d**2*f*x** 
2 - 128*e**(c + 2*d*x)*int(x/(e**(2*c + 3*d*x)*b + 2*e**(c + 2*d*x)*a - e* 
*(d*x)*b),x)*a**5*d**2*f - 160*e**(c + 2*d*x)*int(x/(e**(2*c + 3*d*x)*b + 
2*e**(c + 2*d*x)*a - e**(d*x)*b),x)*a**3*b**2*d**2*f - 32*e**(c + 2*d*x)*i 
nt(x/(e**(2*c + 3*d*x)*b + 2*e**(c + 2*d*x)*a - e**(d*x)*b),x)*a*b**4*d**2 
*f + 32*e**(c + d*x)*a**3*b*d*f*x + 32*e**(c + d*x)*a**3*b*f - 8*e**(c + d 
*x)*a*b**3*d*e + 24*e**(c + d*x)*a*b**3*d*f*x + 24*e**(c + d*x)*a*b**3*f - 
 32*a**4*d*f*x - 16*a**4*f - 32*a**2*b**2*d*f*x - 16*a**2*b**2*f - 2*b**4* 
d*e - 2*b**4*d*f*x - b**4*f)/(16*e**(2*c + 2*d*x)*b**5*d**2)