Optimal. Leaf size=115 \[ -\frac {2 a x}{b^3}-\frac {2 a^2 \left (2 a^2+3 b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^3 \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \cosh (x) \sinh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))} \]
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Rubi [A]
time = 0.20, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2871, 3102,
2814, 2739, 632, 212} \begin {gather*} \frac {\left (2 a^2+b^2\right ) \cosh (x)}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \sinh (x) \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {2 a^2 \left (2 a^2+3 b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^3 \left (a^2+b^2\right )^{3/2}}-\frac {2 a x}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 2739
Rule 2814
Rule 2871
Rule 3102
Rubi steps
\begin {align*} \int \frac {\sinh ^3(x)}{(a+b \sinh (x))^2} \, dx &=-\frac {a^2 \cosh (x) \sinh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\int \frac {a^2-a b \sinh (x)+\left (2 a^2+b^2\right ) \sinh ^2(x)}{a+b \sinh (x)} \, dx}{b \left (a^2+b^2\right )}\\ &=\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \cosh (x) \sinh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {i \int \frac {-i a^2 b+2 i a \left (a^2+b^2\right ) \sinh (x)}{a+b \sinh (x)} \, dx}{b^2 \left (a^2+b^2\right )}\\ &=-\frac {2 a x}{b^3}+\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \cosh (x) \sinh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\left (a^2 \left (2 a^2+3 b^2\right )\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{b^3 \left (a^2+b^2\right )}\\ &=-\frac {2 a x}{b^3}+\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \cosh (x) \sinh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\left (2 a^2 \left (2 a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )}\\ &=-\frac {2 a x}{b^3}+\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \cosh (x) \sinh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\left (4 a^2 \left (2 a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )}\\ &=-\frac {2 a x}{b^3}-\frac {2 a^2 \left (2 a^2+3 b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^3 \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \cosh (x) \sinh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 95, normalized size = 0.83 \begin {gather*} \frac {-2 a x-\frac {2 a^2 \left (2 a^2+3 b^2\right ) \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+\cosh (x) \left (b+\frac {a^3 b}{\left (a^2+b^2\right ) (a+b \sinh (x))}\right )}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.51, size = 161, normalized size = 1.40
method | result | size |
default | \(-\frac {1}{b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {2 a \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{3}}-\frac {4 a^{2} \left (\frac {\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{2 a^{2}+2 b^{2}}+\frac {a b}{2 a^{2}+2 b^{2}}}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (2 a^{2}+3 b^{2}\right ) \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{b^{3}}+\frac {1}{b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {2 a \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{3}}\) | \(161\) |
risch | \(-\frac {2 a x}{b^{3}}+\frac {{\mathrm e}^{x}}{2 b^{2}}+\frac {{\mathrm e}^{-x}}{2 b^{2}}-\frac {2 a^{3} \left (a \,{\mathrm e}^{x}-b \right )}{b^{3} \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}-b \right )}+\frac {2 a^{4} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b^{3}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b}-\frac {2 a^{4} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b^{3}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b}\) | \(315\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 208, normalized size = 1.81 \begin {gather*} \frac {{\left (2 \, a^{2} + 3 \, b^{2}\right )} a^{2} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} b^{3} + b^{5}\right )} \sqrt {a^{2} + b^{2}}} + \frac {a^{2} b^{2} + b^{4} + 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} e^{\left (-x\right )} + {\left (4 \, a^{4} - a^{2} b^{2} - b^{4}\right )} e^{\left (-2 \, x\right )}}{2 \, {\left ({\left (a^{2} b^{4} + b^{6}\right )} e^{\left (-x\right )} + 2 \, {\left (a^{3} b^{3} + a b^{5}\right )} e^{\left (-2 \, x\right )} - {\left (a^{2} b^{4} + b^{6}\right )} e^{\left (-3 \, x\right )}\right )}} - \frac {2 \, a x}{b^{3}} + \frac {e^{\left (-x\right )}}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1053 vs.
\(2 (111) = 222\).
time = 0.50, size = 1053, normalized size = 9.16 \begin {gather*} -\frac {a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} - {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \cosh \left (x\right )^{4} - {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \sinh \left (x\right )^{4} - 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5} - 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} x\right )} \cosh \left (x\right )^{3} - 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5} - 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} x + 2 \, {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (a^{6} + a^{4} b^{2} + 2 \, {\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} x\right )} \cosh \left (x\right )^{2} + 2 \, {\left (2 \, a^{6} + 2 \, a^{4} b^{2} - 3 \, {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \cosh \left (x\right )^{2} + 4 \, {\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} x - 3 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5} - 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} x\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 2 \, {\left ({\left (2 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \cosh \left (x\right )^{3} + {\left (2 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \sinh \left (x\right )^{3} + 2 \, {\left (2 \, a^{5} + 3 \, a^{3} b^{2}\right )} \cosh \left (x\right )^{2} + {\left (4 \, a^{5} + 6 \, a^{3} b^{2} + 3 \, {\left (2 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - {\left (2 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \cosh \left (x\right ) - {\left (2 \, a^{4} b + 3 \, a^{2} b^{3} - 3 \, {\left (2 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \cosh \left (x\right )^{2} - 4 \, {\left (2 \, a^{5} + 3 \, a^{3} b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) - 2 \, {\left (3 \, a^{5} b + 4 \, a^{3} b^{3} + a b^{5} + 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} x\right )} \cosh \left (x\right ) - 2 \, {\left (3 \, a^{5} b + 4 \, a^{3} b^{3} + a b^{5} + 2 \, {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \cosh \left (x\right )^{3} + 3 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5} - 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} x\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} x - 4 \, {\left (a^{6} + a^{4} b^{2} + 2 \, {\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} x\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{2 \, {\left ({\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} \cosh \left (x\right )^{3} + {\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} \sinh \left (x\right )^{3} + 2 \, {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} \cosh \left (x\right )^{2} + {\left (2 \, a^{5} b^{3} + 4 \, a^{3} b^{5} + 2 \, a b^{7} + 3 \, {\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - {\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} \cosh \left (x\right ) - {\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8} - 3 \, {\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} \cosh \left (x\right )^{2} - 4 \, {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a 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.
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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.
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Giac [A]
time = 0.42, size = 184, normalized size = 1.60 \begin {gather*} \frac {{\left (2 \, a^{4} + 3 \, a^{2} b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} b^{3} + b^{5}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, a x}{b^{3}} + \frac {e^{x}}{2 \, b^{2}} - \frac {{\left (a^{2} b^{2} + b^{4} + {\left (4 \, a^{4} - a^{2} b^{2} - b^{4}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} e^{x}\right )} e^{\left (-x\right )}}{2 \, {\left (a^{2} + b^{2}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.78, size = 274, normalized size = 2.38 \begin {gather*} \frac {{\mathrm {e}}^{-x}}{2\,b^2}+\frac {\frac {2\,a^3}{b\,\left (a^2\,b+b^3\right )}-\frac {2\,a^4\,{\mathrm {e}}^x}{b^2\,\left (a^2\,b+b^3\right )}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}+\frac {{\mathrm {e}}^x}{2\,b^2}-\frac {2\,a\,x}{b^3}-\frac {a^2\,\ln \left (-\frac {2\,{\mathrm {e}}^x\,\left (2\,a^4+3\,a^2\,b^2\right )}{a^2\,b^4+b^6}-\frac {2\,a^2\,\left (b-a\,{\mathrm {e}}^x\right )\,\left (2\,a^2+3\,b^2\right )}{b^4\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (2\,a^2+3\,b^2\right )}{b^3\,{\left (a^2+b^2\right )}^{3/2}}+\frac {a^2\,\ln \left (\frac {2\,a^2\,\left (b-a\,{\mathrm {e}}^x\right )\,\left (2\,a^2+3\,b^2\right )}{b^4\,{\left (a^2+b^2\right )}^{3/2}}-\frac {2\,{\mathrm {e}}^x\,\left (2\,a^4+3\,a^2\,b^2\right )}{a^2\,b^4+b^6}\right )\,\left (2\,a^2+3\,b^2\right )}{b^3\,{\left (a^2+b^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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