Optimal. Leaf size=215 \[ \frac {a^3 b x}{\left (a^2-b^2\right )^3}+\frac {a b^3 x}{\left (a^2-b^2\right )^3}-\frac {a b x}{\left (a^2-b^2\right )^2}+\frac {a b \left (a^2+b^2\right ) x}{\left (a^2-b^2\right )^3}+\frac {b^2 \cosh ^2(x)}{2 \left (a^2-b^2\right )^2}-\frac {3 a^2 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac {b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac {a b \cosh (x) \sinh (x)}{\left (a^2-b^2\right )^2}+\frac {a^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac {a b^2}{\left (a^2-b^2\right )^2 (a+b \tanh (x))} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.36, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {3190, 3179,
2715, 8, 3177, 3212, 3188, 2644, 30, 3165, 3564, 3612, 3611} \begin {gather*} \frac {a b x \left (a^2+b^2\right )}{\left (a^2-b^2\right )^3}-\frac {a b x}{\left (a^2-b^2\right )^2}+\frac {a^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac {b^2 \cosh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac {a b^2}{\left (a^2-b^2\right )^2 (a+b \tanh (x))}-\frac {a b \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^2}-\frac {3 a^2 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac {b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}+\frac {a b^3 x}{\left (a^2-b^2\right )^3}+\frac {a^3 b x}{\left (a^2-b^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 30
Rule 2644
Rule 2715
Rule 3165
Rule 3177
Rule 3179
Rule 3188
Rule 3190
Rule 3212
Rule 3564
Rule 3611
Rule 3612
Rubi steps
\begin {align*} \int \frac {\cosh ^3(x) \sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx &=\frac {a \int \frac {\cosh ^2(x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}-\frac {b \int \frac {\cosh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}+\frac {(a b) \int \frac {\cosh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx}{a^2-b^2}\\ &=\frac {b^2 \cosh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac {a^2 \int \cosh (x) \sinh (x) \, dx}{\left (a^2-b^2\right )^2}-2 \frac {(a b) \int \cosh ^2(x) \, dx}{\left (a^2-b^2\right )^2}+\frac {\left (a^2 b\right ) \int \frac {\cosh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}+\frac {b^3 \int \frac {\cosh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}+\frac {(a b) \int \frac {1}{(a+b \tanh (x))^2} \, dx}{a^2-b^2}\\ &=\frac {a^3 b x}{\left (a^2-b^2\right )^3}+\frac {a b^3 x}{\left (a^2-b^2\right )^3}+\frac {b^2 \cosh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac {a b^2}{\left (a^2-b^2\right )^2 (a+b \tanh (x))}-\frac {\left (i a^2 b^2\right ) \int \frac {-i b \cosh (x)-i a \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}-\frac {\left (i b^4\right ) \int \frac {-i b \cosh (x)-i a \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}-\frac {a^2 \text {Subst}(\int x \, dx,x,i \sinh (x))}{\left (a^2-b^2\right )^2}-2 \left (\frac {a b \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^2}+\frac {(a b) \int 1 \, dx}{2 \left (a^2-b^2\right )^2}\right )+\frac {(a b) \int \frac {a-b \tanh (x)}{a+b \tanh (x)} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac {a^3 b x}{\left (a^2-b^2\right )^3}+\frac {a b^3 x}{\left (a^2-b^2\right )^3}+\frac {a b \left (a^2+b^2\right ) x}{\left (a^2-b^2\right )^3}+\frac {b^2 \cosh ^2(x)}{2 \left (a^2-b^2\right )^2}-\frac {a^2 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac {b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}+\frac {a^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}-2 \left (\frac {a b x}{2 \left (a^2-b^2\right )^2}+\frac {a b \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^2}\right )+\frac {a b^2}{\left (a^2-b^2\right )^2 (a+b \tanh (x))}-\frac {\left (2 i a^2 b^2\right ) \int \frac {-i b-i a \tanh (x)}{a+b \tanh (x)} \, dx}{\left (a^2-b^2\right )^3}\\ &=\frac {a^3 b x}{\left (a^2-b^2\right )^3}+\frac {a b^3 x}{\left (a^2-b^2\right )^3}+\frac {a b \left (a^2+b^2\right ) x}{\left (a^2-b^2\right )^3}+\frac {b^2 \cosh ^2(x)}{2 \left (a^2-b^2\right )^2}-\frac {3 a^2 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac {b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}+\frac {a^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}-2 \left (\frac {a b x}{2 \left (a^2-b^2\right )^2}+\frac {a b \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^2}\right )+\frac {a b^2}{\left (a^2-b^2\right )^2 (a+b \tanh (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.90, size = 183, normalized size = 0.85 \begin {gather*} \frac {a \cosh (x) \left (\left (a^4-b^4\right ) \cosh (2 x)-4 b \left (-a \left (a^2+3 b^2\right ) x+b \left (3 a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))+a \left (a^2-b^2\right ) \cosh (x) \sinh (x)\right )\right )+b \sinh (x) \left (\left (a^4-b^4\right ) \cosh (2 x)+4 b \left (-a^2 b+b^3+a^3 x+3 a b^2 x-b \left (3 a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))\right )-2 a b \left (a^2-b^2\right ) \sinh (2 x)\right )}{4 (a-b)^3 (a+b)^3 (a \cosh (x)+b \sinh (x))} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.55, size = 178, normalized size = 0.83
method | result | size |
default | \(\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{\left (a +b \right )^{3}}-\frac {2 b^{2} \left (\frac {b \left (a^{2}-b^{2}\right ) \tanh \left (\frac {x}{2}\right )}{a +2 b \tanh \left (\frac {x}{2}\right )+a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}+\frac {\left (3 a^{2}+b^{2}\right ) \ln \left (a +2 b \tanh \left (\frac {x}{2}\right )+a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}{2}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}+\frac {1}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{\left (a -b \right )^{3}}\) | \(178\) |
risch | \(\frac {x b}{\left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}+\frac {{\mathrm e}^{2 x}}{8 a^{2}+16 a b +8 b^{2}}+\frac {{\mathrm e}^{-2 x}}{8 a^{2}-16 a b +8 b^{2}}+\frac {6 a^{2} x \,b^{2}}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}+\frac {2 b^{4} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}+\frac {2 a \,b^{3}}{\left (a -b \right )^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left ({\mathrm e}^{2 x} a +b \,{\mathrm e}^{2 x}+a -b \right )}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right ) a^{2}}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}-\frac {b^{4} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}\) | \(284\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 240, normalized size = 1.12 \begin {gather*} \frac {b x}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {{\left (3 \, a^{2} b^{2} + b^{4}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4} + {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 20 \, a b^{3} + b^{4}\right )} e^{\left (-2 \, x\right )}}{8 \, {\left ({\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} e^{\left (-2 \, x\right )} + {\left (a^{6} - 2 \, a^{5} b - a^{4} b^{2} + 4 \, a^{3} b^{3} - a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right )} e^{\left (-4 \, x\right )}\right )}} + \frac {e^{\left (-2 \, x\right )}}{8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1661 vs.
\(2 (211) = 422\).
time = 0.41, size = 1661, normalized size = 7.73 \begin {gather*} \frac {{\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )^{6} + 6 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{5} + {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \sinh \left (x\right )^{6} + a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5} + {\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5} + 8 \, {\left (a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right )} x\right )} \cosh \left (x\right )^{4} + {\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5} + 15 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )^{2} + 8 \, {\left (a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right )} x\right )} \sinh \left (x\right )^{4} + 4 \, {\left (5 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )^{3} + {\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5} + 8 \, {\left (a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right )} x\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} + 14 \, a^{2} b^{3} - 19 \, a b^{4} - b^{5} + 8 \, {\left (a^{4} b + 2 \, a^{3} b^{2} - 2 \, a b^{4} - b^{5}\right )} x\right )} \cosh \left (x\right )^{2} + {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} + 14 \, a^{2} b^{3} - 19 \, a b^{4} - b^{5} + 15 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )^{4} + 6 \, {\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5} + 8 \, {\left (a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right )} x\right )} \cosh \left (x\right )^{2} + 8 \, {\left (a^{4} b + 2 \, a^{3} b^{2} - 2 \, a b^{4} - b^{5}\right )} x\right )} \sinh \left (x\right )^{2} - 8 \, {\left ({\left (3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \sinh \left (x\right )^{4} + {\left (3 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )^{2} + {\left (3 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4} - b^{5} + 6 \, {\left (3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (2 \, {\left (3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right )^{3} + {\left (3 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (3 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )^{5} + 2 \, {\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5} + 8 \, {\left (a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right )} x\right )} \cosh \left (x\right )^{3} + {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} + 14 \, a^{2} b^{3} - 19 \, a b^{4} - b^{5} + 8 \, {\left (a^{4} b + 2 \, a^{3} b^{2} - 2 \, a b^{4} - b^{5}\right )} x\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{8 \, {\left ({\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6} - b^{7}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6} - b^{7}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6} - b^{7}\right )} \sinh \left (x\right )^{4} + {\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - 3 \, a^{2} b^{5} - a b^{6} + b^{7}\right )} \cosh \left (x\right )^{2} + {\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - 3 \, a^{2} b^{5} - a b^{6} + b^{7} + 6 \, {\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6} - b^{7}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (2 \, {\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6} - b^{7}\right )} \cosh \left (x\right )^{3} + {\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - 3 \, a^{2} b^{5} - a b^{6} + b^{7}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 240, normalized size = 1.12 \begin {gather*} \frac {b x}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (3 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {e^{\left (2 \, x\right )}}{8 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {2 \, a^{2} b e^{\left (4 \, x\right )} - 4 \, a b^{2} e^{\left (4 \, x\right )} + 2 \, b^{3} e^{\left (4 \, x\right )} - a^{3} e^{\left (2 \, x\right )} - a^{2} b e^{\left (2 \, x\right )} - 11 \, a b^{2} e^{\left (2 \, x\right )} - 3 \, b^{3} e^{\left (2 \, x\right )} - a^{3} - a^{2} b + a b^{2} + b^{3}}{8 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + a e^{\left (2 \, x\right )} - b e^{\left (2 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.83, size = 127, normalized size = 0.59 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}}{8\,{\left (a+b\right )}^2}+\frac {{\mathrm {e}}^{-2\,x}}{8\,{\left (a-b\right )}^2}+\frac {b\,x}{{\left (a-b\right )}^3}-\frac {\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (3\,a^2\,b^2+b^4\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+\frac {2\,a\,b^3}{{\left (a+b\right )}^3\,{\left (a-b\right )}^2\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________