Integrand size = 28, antiderivative size = 683 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {3 f (e+f x)^2}{8 a d^2}+\frac {(e+f x)^4}{8 a f}+\frac {b^2 (e+f x)^4}{4 a^3 f}-\frac {6 b f^2 (e+f x) \cosh (c+d x)}{a^2 d^3}-\frac {b (e+f x)^3 \cosh (c+d x)}{a^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 a d^2}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {6 b \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {6 b f^3 \sinh (c+d x)}{a^2 d^4}+\frac {3 b f (e+f x)^2 \sinh (c+d x)}{a^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d} \] Output:
3/8*f*(f*x+e)^2/a/d^2+1/8*(f*x+e)^4/a/f+1/4*b^2*(f*x+e)^4/a^3/f-6*b*f^2*(f *x+e)*cosh(d*x+c)/a^2/d^3-b*(f*x+e)^3*cosh(d*x+c)/a^2/d-3/8*f^3*cosh(d*x+c )^2/a/d^4-3/4*f*(f*x+e)^2*cosh(d*x+c)^2/a/d^2-b*(a^2+b^2)^(1/2)*(f*x+e)^3* ln(1+a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^3/d+b*(a^2+b^2)^(1/2)*(f*x+e)^3*l n(1+a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^3/d-3*b*(a^2+b^2)^(1/2)*f*(f*x+e)^ 2*polylog(2,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^3/d^2+3*b*(a^2+b^2)^(1/2) *f*(f*x+e)^2*polylog(2,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^3/d^2+6*b*(a^2 +b^2)^(1/2)*f^2*(f*x+e)*polylog(3,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^3/d ^3-6*b*(a^2+b^2)^(1/2)*f^2*(f*x+e)*polylog(3,-a*exp(d*x+c)/(b+(a^2+b^2)^(1 /2)))/a^3/d^3-6*b*(a^2+b^2)^(1/2)*f^3*polylog(4,-a*exp(d*x+c)/(b-(a^2+b^2) ^(1/2)))/a^3/d^4+6*b*(a^2+b^2)^(1/2)*f^3*polylog(4,-a*exp(d*x+c)/(b+(a^2+b ^2)^(1/2)))/a^3/d^4+6*b*f^3*sinh(d*x+c)/a^2/d^4+3*b*f*(f*x+e)^2*sinh(d*x+c )/a^2/d^2+3/4*f^2*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/a/d^3+1/2*(f*x+e)^3*cosh (d*x+c)*sinh(d*x+c)/a/d
Leaf count is larger than twice the leaf count of optimal. \(2218\) vs. \(2(683)=1366\).
Time = 9.49 (sec) , antiderivative size = 2218, normalized size of antiderivative = 3.25 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^3*Cosh[c + d*x]^2)/(a + b*Csch[c + d*x]),x]
Output:
(e^3*(c/d + x - (2*b*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/( Sqrt[-a^2 - b^2]*d))*Csch[c + d*x]*(b + a*Sinh[c + d*x]))/(4*a*(a + b*Csch [c + d*x])) + (3*e^2*f*Csch[c + d*x]*(x^2 - (2*b*(d*x*(Log[1 + (a*E^(c + d *x))/(b - Sqrt[a^2 + b^2])] - Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2] )]) + PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - PolyLog[2, -((a *E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^2))*(b + a*Sinh [c + d*x]))/(8*a*(a + b*Csch[c + d*x])) + (e*f^2*Csch[c + d*x]*(x^3 - (3*b *(d^2*x^2*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] - d^2*x^2*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (a*E^(c + d*x)) /(-b + Sqrt[a^2 + b^2])] - 2*d*x*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^ 2 + b^2]))] - 2*PolyLog[3, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] + 2*Pol yLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3)) *(b + a*Sinh[c + d*x]))/(4*a*(a + b*Csch[c + d*x])) + (f^3*Csch[c + d*x]*( x^4 - (4*b*(d^3*x^3*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] - d^3*x ^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 3*d^2*x^2*PolyLog[2, ( a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 3*d^2*x^2*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 6*d*x*PolyLog[3, (a*E^(c + d*x))/(-b + Sqr t[a^2 + b^2])] + 6*d*x*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])) ] + 6*PolyLog[4, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 6*PolyLog[4, -( (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^4))*(b + a...
Result contains complex when optimal does not.
Time = 3.59 (sec) , antiderivative size = 641, normalized size of antiderivative = 0.94, number of steps used = 31, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6128, 6113, 3042, 3792, 17, 3042, 3791, 17, 6099, 17, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117, 3803, 25, 2694, 27, 2620, 3011, 7163, 2720, 7143}
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.
\(\displaystyle \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) |
\(\Big \downarrow \) 6128 |
\(\displaystyle \int \frac {(e+f x)^3 \sinh (c+d x) \cosh ^2(c+d x)}{a \sinh (c+d x)+b}dx\) |
\(\Big \downarrow \) 6113 |
\(\displaystyle \frac {\int (e+f x)^3 \cosh ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{a}\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle \frac {\frac {3 f^2 \int (e+f x) \cosh ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^3dx-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle \frac {\frac {3 f^2 \int (e+f x) \cosh ^2(c+d x)dx}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {3 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle \frac {\frac {3 f^2 \left (\frac {1}{2} \int (e+f x)dx-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{b+a \sinh (c+d x)}dx}{a}\) |
\(\Big \downarrow \) 6099 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{b+a \sinh (c+d x)}dx}{a^2}-\frac {b \int (e+f x)^3dx}{a^2}+\frac {\int (e+f x)^3 \sinh (c+d x)dx}{a}\right )}{a}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{b+a \sinh (c+d x)}dx}{a^2}+\frac {\int (e+f x)^3 \sinh (c+d x)dx}{a}-\frac {b (e+f x)^4}{4 a^2 f}\right )}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{b-i a \sin (i c+i d x)}dx}{a^2}+\frac {\int -i (e+f x)^3 \sin (i c+i d x)dx}{a}-\frac {b (e+f x)^4}{4 a^2 f}\right )}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \int (e+f x)^3 \sin (i c+i d x)dx}{a}-\frac {b (e+f x)^4}{4 a^2 f}\right )}{a}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \cosh (c+d x)dx}{d}\right )}{a}-\frac {b (e+f x)^4}{4 a^2 f}\right )}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{a}-\frac {b (e+f x)^4}{4 a^2 f}\right )}{a}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{a}-\frac {b (e+f x)^4}{4 a^2 f}\right )}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{a}-\frac {b (e+f x)^4}{4 a^2 f}\right )}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{a}-\frac {b (e+f x)^4}{4 a^2 f}\right )}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{a}-\frac {b (e+f x)^4}{4 a^2 f}\right )}{a}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )}{a}-\frac {b (e+f x)^4}{4 a^2 f}\right )}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )}{d}\right )}{a}-\frac {b (e+f x)^4}{4 a^2 f}\right )}{a}\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{b-i a \sin (i c+i d x)}dx}{a^2}-\frac {b (e+f x)^4}{4 a^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}\) |
\(\Big \downarrow \) 3803 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (\frac {2 \left (a^2+b^2\right ) \int -\frac {e^{c+d x} (e+f x)^3}{-e^{2 (c+d x)} a+a-2 b e^{c+d x}}dx}{a^2}-\frac {b (e+f x)^4}{4 a^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^3}{-e^{2 (c+d x)} a+a-2 b e^{c+d x}}dx}{a^2}-\frac {b (e+f x)^4}{4 a^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}\) |
\(\Big \downarrow \) 2694 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \int -\frac {e^{c+d x} (e+f x)^3}{2 \left (e^{c+d x} a+b-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {a \int -\frac {e^{c+d x} (e+f x)^3}{2 \left (e^{c+d x} a+b+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^4}{4 a^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \int \frac {e^{c+d x} (e+f x)^3}{e^{c+d x} a+b+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {a \int \frac {e^{c+d x} (e+f x)^3}{e^{c+d x} a+b-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^4}{4 a^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \left (\frac {(e+f x)^3 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} a}{b+\sqrt {a^2+b^2}}+1\right )dx}{a d}\right )}{2 \sqrt {a^2+b^2}}-\frac {a \left (\frac {(e+f x)^3 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} a}{b-\sqrt {a^2+b^2}}+1\right )dx}{a d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^4}{4 a^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \left (\frac {(e+f x)^3 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}-\frac {a \left (\frac {(e+f x)^3 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^4}{4 a^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \left (\frac {(e+f x)^3 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}-\frac {a \left (\frac {(e+f x)^3 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^4}{4 a^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \left (\frac {(e+f x)^3 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}-\frac {a \left (\frac {(e+f x)^3 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^4}{4 a^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {a \left (\frac {(e+f x)^3 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}-\frac {a \left (\frac {(e+f x)^3 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{d}\right )}{a d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2}-\frac {b (e+f x)^4}{4 a^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{a}\right )}{a}\) |
Input:
Int[((e + f*x)^3*Cosh[c + d*x]^2)/(a + b*Csch[c + d*x]),x]
Output:
((e + f*x)^4/(8*f) - (3*f*(e + f*x)^2*Cosh[c + d*x]^2)/(4*d^2) + ((e + f*x )^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + (3*f^2*((e + f*x)^2/(4*f) - (f*Co sh[c + d*x]^2)/(4*d^2) + ((e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)))/( 2*d^2))/a - (b*(-1/4*(b*(e + f*x)^4)/(a^2*f) - (2*(a^2 + b^2)*(-1/2*(a*((( e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a*d) - (3*f*(- (((e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/d) + ( 2*f*(((e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/d - (f*PolyLog[4, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/d^2))/d))/(a*d))) /Sqrt[a^2 + b^2] + (a*(((e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a*d) - (3*f*(-(((e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b + S qrt[a^2 + b^2]))])/d) + (2*f*(((e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/d - (f*PolyLog[4, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b ^2]))])/d^2))/d))/(a*d)))/(2*Sqrt[a^2 + b^2])))/a^2 - (I*((I*(e + f*x)^3*C osh[c + d*x])/d - ((3*I)*f*(((e + f*x)^2*Sinh[c + d*x])/d + ((2*I)*f*((I*( e + f*x)*Cosh[c + d*x])/d - (I*f*Sinh[c + d*x])/d^2))/d))/d))/a))/a
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]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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]
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]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( -I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(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]
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]
Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Int[(e + f*x)^m*Sinh[c + d*x]*(F [c + d*x]^n/(b + a*Sinh[c + d*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && H yperbolicQ[F] && IntegersQ[m, n]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )^{2}}{a +\operatorname {csch}\left (d x +c \right ) b}d x\]
Input:
int((f*x+e)^3*cosh(d*x+c)^2/(a+csch(d*x+c)*b),x)
Output:
int((f*x+e)^3*cosh(d*x+c)^2/(a+csch(d*x+c)*b),x)
Leaf count of result is larger than twice the leaf count of optimal. 3847 vs. \(2 (627) = 1254\).
Time = 0.19 (sec) , antiderivative size = 3847, normalized size of antiderivative = 5.63 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \cosh ^{2}{\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**3*cosh(d*x+c)**2/(a+b*csch(d*x+c)),x)
Output:
Integral((e + f*x)**3*cosh(c + d*x)**2/(a + b*csch(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="maxima")
Output:
-1/8*e^3*((4*b*e^(-d*x - c) - a)*e^(2*d*x + 2*c)/(a^2*d) - 4*(a^2 + 2*b^2) *(d*x + c)/(a^3*d) + (4*b*e^(-d*x - c) + a*e^(-2*d*x - 2*c))/(a^2*d) + 8*( a^2*b + b^3)*log((a*e^(-d*x - c) - b - sqrt(a^2 + b^2))/(a*e^(-d*x - c) - b + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3*d)) + 1/32*(4*(a^2*d^4*f^3*e^(2 *c) + 2*b^2*d^4*f^3*e^(2*c))*x^4 + 16*(a^2*d^4*e*f^2*e^(2*c) + 2*b^2*d^4*e *f^2*e^(2*c))*x^3 + 24*(a^2*d^4*e^2*f*e^(2*c) + 2*b^2*d^4*e^2*f*e^(2*c))*x ^2 + (4*a^2*d^3*f^3*x^3*e^(4*c) + 6*(2*d^3*e*f^2 - d^2*f^3)*a^2*x^2*e^(4*c ) + 6*(2*d^3*e^2*f - 2*d^2*e*f^2 + d*f^3)*a^2*x*e^(4*c) - 3*(2*d^2*e^2*f - 2*d*e*f^2 + f^3)*a^2*e^(4*c))*e^(2*d*x) - 16*(a*b*d^3*f^3*x^3*e^(3*c) + 3 *(d^3*e*f^2 - d^2*f^3)*a*b*x^2*e^(3*c) + 3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d* f^3)*a*b*x*e^(3*c) - 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a*b*e^(3*c))*e^(d*x ) - 16*(a*b*d^3*f^3*x^3*e^c + 3*(d^3*e*f^2 + d^2*f^3)*a*b*x^2*e^c + 3*(d^3 *e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*a*b*x*e^c + 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f ^3)*a*b*e^c)*e^(-d*x) - (4*a^2*d^3*f^3*x^3 + 6*(2*d^3*e*f^2 + d^2*f^3)*a^2 *x^2 + 6*(2*d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*a^2*x + 3*(2*d^2*e^2*f + 2*d* e*f^2 + f^3)*a^2)*e^(-2*d*x))*e^(-2*c)/(a^3*d^4) - integrate(2*((a^2*b*f^3 *e^c + b^3*f^3*e^c)*x^3 + 3*(a^2*b*e*f^2*e^c + b^3*e*f^2*e^c)*x^2 + 3*(a^2 *b*e^2*f*e^c + b^3*e^2*f*e^c)*x)*e^(d*x)/(a^4*e^(2*d*x + 2*c) + 2*a^3*b*e^ (d*x + c) - a^4), x)
\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="giac")
Output:
integrate((f*x + e)^3*cosh(d*x + c)^2/(b*csch(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \] Input:
int((cosh(c + d*x)^2*(e + f*x)^3)/(a + b/sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)^2*(e + f*x)^3)/(a + b/sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )^{2}}{a +b \,\mathrm {csch}\left (d x +c \right )}d x \] Input:
int((f*x+e)^3*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x)
Output:
int((f*x+e)^3*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x)