Integrand size = 20, antiderivative size = 823 \[ \int \frac {(c+d x)^3}{(a+b \cosh (e+f x))^2} \, dx=-\frac {(c+d x)^3}{\left (a^2-b^2\right ) f}+\frac {3 d (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {a (c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {3 d (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {a (c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac {3 a d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac {3 a d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac {6 d^3 \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^4}-\frac {6 a d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}-\frac {6 d^3 \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^4}+\frac {6 a d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac {6 a d^3 \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^4}-\frac {6 a d^3 \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^4}-\frac {b (c+d x)^3 \sinh (e+f x)}{\left (a^2-b^2\right ) f (a+b \cosh (e+f x))} \] Output:
-(d*x+c)^3/(a^2-b^2)/f+3*d*(d*x+c)^2*ln(1+b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)) )/(a^2-b^2)/f^2+a*(d*x+c)^3*ln(1+b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^ 2)^(3/2)/f+3*d*(d*x+c)^2*ln(1+b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)/ f^2-a*(d*x+c)^3*ln(1+b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2)/f+6 *d^2*(d*x+c)*polylog(2,-b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)/f^3+3* a*d*(d*x+c)^2*polylog(2,-b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2) /f^2+6*d^2*(d*x+c)*polylog(2,-b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)/ f^3-3*a*d*(d*x+c)^2*polylog(2,-b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2) ^(3/2)/f^2-6*d^3*polylog(3,-b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)/f^ 4-6*a*d^2*(d*x+c)*polylog(3,-b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^( 3/2)/f^3-6*d^3*polylog(3,-b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)/f^4+ 6*a*d^2*(d*x+c)*polylog(3,-b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/ 2)/f^3+6*a*d^3*polylog(4,-b*exp(f*x+e)/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2 )/f^4-6*a*d^3*polylog(4,-b*exp(f*x+e)/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2) /f^4-b*(d*x+c)^3*sinh(f*x+e)/(a^2-b^2)/f/(a+b*cosh(f*x+e))
Leaf count is larger than twice the leaf count of optimal. \(11178\) vs. \(2(823)=1646\).
Time = 13.54 (sec) , antiderivative size = 11178, normalized size of antiderivative = 13.58 \[ \int \frac {(c+d x)^3}{(a+b \cosh (e+f x))^2} \, dx=\text {Result too large to show} \] Input:
Integrate[(c + d*x)^3/(a + b*Cosh[e + f*x])^2,x]
Output:
Result too large to show
Time = 4.09 (sec) , antiderivative size = 746, normalized size of antiderivative = 0.91, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.850, Rules used = {3042, 3805, 26, 3042, 3801, 2694, 27, 2620, 3011, 6096, 2620, 3011, 2720, 7143, 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.
\(\displaystyle \int \frac {(c+d x)^3}{(a+b \cosh (e+f x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c+d x)^3}{\left (a+b \sin \left (i e+i f x+\frac {\pi }{2}\right )\right )^2}dx\) |
\(\Big \downarrow \) 3805 |
\(\displaystyle \frac {a \int \frac {(c+d x)^3}{a+b \cosh (e+f x)}dx}{a^2-b^2}+\frac {3 i b d \int -\frac {i (c+d x)^2 \sinh (e+f x)}{a+b \cosh (e+f x)}dx}{f \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {a \int \frac {(c+d x)^3}{a+b \cosh (e+f x)}dx}{a^2-b^2}+\frac {3 b d \int \frac {(c+d x)^2 \sinh (e+f x)}{a+b \cosh (e+f x)}dx}{f \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \int \frac {(c+d x)^3}{a+b \sin \left (i e+i f x+\frac {\pi }{2}\right )}dx}{a^2-b^2}+\frac {3 b d \int \frac {(c+d x)^2 \sinh (e+f x)}{a+b \cosh (e+f x)}dx}{f \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 3801 |
\(\displaystyle \frac {2 a \int \frac {e^{e+f x} (c+d x)^3}{2 e^{e+f x} a+b e^{2 (e+f x)}+b}dx}{a^2-b^2}+\frac {3 b d \int \frac {(c+d x)^2 \sinh (e+f x)}{a+b \cosh (e+f x)}dx}{f \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 2694 |
\(\displaystyle \frac {2 a \left (\frac {b \int \frac {e^{e+f x} (c+d x)^3}{2 \left (a+b e^{e+f x}-\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}-\frac {b \int \frac {e^{e+f x} (c+d x)^3}{2 \left (a+b e^{e+f x}+\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {3 b d \int \frac {(c+d x)^2 \sinh (e+f x)}{a+b \cosh (e+f x)}dx}{f \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 a \left (\frac {b \int \frac {e^{e+f x} (c+d x)^3}{a+b e^{e+f x}-\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}-\frac {b \int \frac {e^{e+f x} (c+d x)^3}{a+b e^{e+f x}+\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {3 b d \int \frac {(c+d x)^2 \sinh (e+f x)}{a+b \cosh (e+f x)}dx}{f \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {2 a \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \int (c+d x)^2 \log \left (\frac {e^{e+f x} b}{a-\sqrt {a^2-b^2}}+1\right )dx}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \int (c+d x)^2 \log \left (\frac {e^{e+f x} b}{a+\sqrt {a^2-b^2}}+1\right )dx}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {3 b d \int \frac {(c+d x)^2 \sinh (e+f x)}{a+b \cosh (e+f x)}dx}{f \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {2 a \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {3 b d \int \frac {(c+d x)^2 \sinh (e+f x)}{a+b \cosh (e+f x)}dx}{f \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 6096 |
\(\displaystyle \frac {2 a \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {3 b d \left (\int \frac {e^{e+f x} (c+d x)^2}{a+b e^{e+f x}-\sqrt {a^2-b^2}}dx+\int \frac {e^{e+f x} (c+d x)^2}{a+b e^{e+f x}+\sqrt {a^2-b^2}}dx-\frac {(c+d x)^3}{3 b d}\right )}{f \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {2 a \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {3 b d \left (-\frac {2 d \int (c+d x) \log \left (\frac {e^{e+f x} b}{a-\sqrt {a^2-b^2}}+1\right )dx}{b f}-\frac {2 d \int (c+d x) \log \left (\frac {e^{e+f x} b}{a+\sqrt {a^2-b^2}}+1\right )dx}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {(c+d x)^3}{3 b d}\right )}{f \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {3 b d \left (-\frac {2 d \left (\frac {d \int \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}-\frac {2 d \left (\frac {d \int \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {(c+d x)^3}{3 b d}\right )}{f \left (a^2-b^2\right )}+\frac {2 a \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {3 b d \left (-\frac {2 d \left (\frac {d \int e^{-e-f x} \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )de^{e+f x}}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}-\frac {2 d \left (\frac {d \int e^{-e-f x} \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )de^{e+f x}}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {(c+d x)^3}{3 b d}\right )}{f \left (a^2-b^2\right )}+\frac {2 a \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {2 a \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )dx}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {3 b d \left (-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {(c+d x)^3}{3 b d}\right )}{f \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {2 a \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )dx}{f}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )dx}{f}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {3 b d \left (-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {(c+d x)^3}{3 b d}\right )}{f \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {2 a \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-e-f x} \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )de^{e+f x}}{f^2}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-e-f x} \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )de^{e+f x}}{f^2}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {3 b d \left (-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {(c+d x)^3}{3 b d}\right )}{f \left (a^2-b^2\right )}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {3 b d \left (-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {(c+d x)^3}{3 b d}\right )}{f \left (a^2-b^2\right )}+\frac {2 a \left (\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}-\frac {d \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f^2}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2-b^2}+a}+1\right )}{b f}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}-\frac {d \operatorname {PolyLog}\left (4,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f^2}\right )}{f}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2-b^2}}\right )}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {b (c+d x)^3 \sinh (e+f x)}{f \left (a^2-b^2\right ) (a+b \cosh (e+f x))}\) |
Input:
Int[(c + d*x)^3/(a + b*Cosh[e + f*x])^2,x]
Output:
(3*b*d*(-1/3*(c + d*x)^3/(b*d) + ((c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])])/(b*f) + ((c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a + Sqrt [a^2 - b^2])])/(b*f) - (2*d*(-(((c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/f) + (d*PolyLog[3, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/f^2))/(b*f) - (2*d*(-(((c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/f) + (d*PolyLog[3, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/f^2))/(b*f)))/((a^2 - b^2)*f) + (2*a*((b*(((c + d*x)^3*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])])/(b*f) - (3*d*(-(((c + d*x)^2*PolyL og[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/f) + (2*d*(((c + d*x)*Pol yLog[3, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/f - (d*PolyLog[4, -((b* E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/f^2))/f))/(b*f)))/(2*Sqrt[a^2 - b^2] ) - (b*(((c + d*x)^3*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 - b^2])])/(b*f) - (3*d*(-(((c + d*x)^2*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]) )])/f) + (2*d*(((c + d*x)*PolyLog[3, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2 ]))])/f - (d*PolyLog[4, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/f^2))/f ))/(b*f)))/(2*Sqrt[a^2 - b^2])))/(a^2 - b^2) - (b*(c + d*x)^3*Sinh[e + f*x ])/((a^2 - b^2)*f*(a + b*Cosh[e + f*x]))
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[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Comple x[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Simp[2 Int[((c + d*x)^m*(E^((-I)*e + f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*((-I) *e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c , d, e, f, fz}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f *x]))), x] + (Simp[a/(a^2 - b^2) Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] , x] - Simp[b*d*(m/(f*(a^2 - b^2))) Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_ .)*(x_)]*(b_.) + (a_)), 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[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]
\[\int \frac {\left (d x +c \right )^{3}}{\left (a +b \cosh \left (f x +e \right )\right )^{2}}d x\]
Input:
int((d*x+c)^3/(a+b*cosh(f*x+e))^2,x)
Output:
int((d*x+c)^3/(a+b*cosh(f*x+e))^2,x)
Leaf count of result is larger than twice the leaf count of optimal. 7116 vs. \(2 (761) = 1522\).
Time = 0.27 (sec) , antiderivative size = 7116, normalized size of antiderivative = 8.65 \[ \int \frac {(c+d x)^3}{(a+b \cosh (e+f x))^2} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^3/(a+b*cosh(f*x+e))^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(c+d x)^3}{(a+b \cosh (e+f x))^2} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**3/(a+b*cosh(f*x+e))**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x)^3}{(a+b \cosh (e+f x))^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^3/(a+b*cosh(f*x+e))^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
\[ \int \frac {(c+d x)^3}{(a+b \cosh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left (b \cosh \left (f x + e\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((d*x+c)^3/(a+b*cosh(f*x+e))^2,x, algorithm="giac")
Output:
integrate((d*x + c)^3/(b*cosh(f*x + e) + a)^2, x)
Timed out. \[ \int \frac {(c+d x)^3}{(a+b \cosh (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{{\left (a+b\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2} \,d x \] Input:
int((c + d*x)^3/(a + b*cosh(e + f*x))^2,x)
Output:
int((c + d*x)^3/(a + b*cosh(e + f*x))^2, x)
\[ \int \frac {(c+d x)^3}{(a+b \cosh (e+f x))^2} \, dx=\text {too large to display} \] Input:
int((d*x+c)^3/(a+b*cosh(f*x+e))^2,x)
Output:
(12*e**(2*e + 2*f*x)*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a**3*b*c**2*d*f**2 + 12*e**(2*e + 2*f*x)*sqrt( - a**2 + b* *2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a**3*b*c*d**2*f + 6*e* *(2*e + 2*f*x)*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a**3*b*d**3 - 4*e**(2*e + 2*f*x)*sqrt( - a**2 + b**2)*atan((e**( e + f*x)*b + a)/sqrt( - a**2 + b**2))*a*b**3*c**3*f**3 - 12*e**(2*e + 2*f* x)*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a* b**3*c**2*d*f**2 - 12*e**(2*e + 2*f*x)*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a*b**3*c*d**2*f - 6*e**(2*e + 2*f*x)*sqr t( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a*b**3*d **3 + 24*e**(e + f*x)*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a**4*c**2*d*f**2 + 24*e**(e + f*x)*sqrt( - a**2 + b**2)*a tan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a**4*c*d**2*f + 12*e**(e + f*x)*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))* a**4*d**3 - 8*e**(e + f*x)*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/ sqrt( - a**2 + b**2))*a**2*b**2*c**3*f**3 - 24*e**(e + f*x)*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a**2*b**2*c**2*d*f* *2 - 24*e**(e + f*x)*sqrt( - a**2 + b**2)*atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a**2*b**2*c*d**2*f - 12*e**(e + f*x)*sqrt( - a**2 + b**2)* atan((e**(e + f*x)*b + a)/sqrt( - a**2 + b**2))*a**2*b**2*d**3 + 12*sqr...