Optimal. Leaf size=108 \[ -\frac {a \left (2 a^2-b^2\right ) x}{2 b^4}-\frac {2 a^4 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^4 \sqrt {a^2+b^2}}-\frac {\left (2-\frac {3 a^2}{b^2}\right ) \cosh (x)}{3 b}-\frac {a \cosh (x) \sinh (x)}{2 b^2}+\frac {\cosh (x) \sinh ^2(x)}{3 b} \]
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Rubi [A]
time = 0.23, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2872, 3128,
3102, 2814, 2739, 632, 212} \begin {gather*} -\frac {\left (2-\frac {3 a^2}{b^2}\right ) \cosh (x)}{3 b}-\frac {a x \left (2 a^2-b^2\right )}{2 b^4}-\frac {2 a^4 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^4 \sqrt {a^2+b^2}}-\frac {a \sinh (x) \cosh (x)}{2 b^2}+\frac {\sinh ^2(x) \cosh (x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 2739
Rule 2814
Rule 2872
Rule 3102
Rule 3128
Rubi steps
\begin {align*} \int \frac {\sinh ^4(x)}{a+b \sinh (x)} \, dx &=\frac {\cosh (x) \sinh ^2(x)}{3 b}-\frac {\int \frac {\sinh (x) \left (2 a+2 b \sinh (x)+3 a \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{3 b}\\ &=-\frac {a \cosh (x) \sinh (x)}{2 b^2}+\frac {\cosh (x) \sinh ^2(x)}{3 b}-\frac {\int \frac {-3 a^2+a b \sinh (x)-2 \left (3 a^2-2 b^2\right ) \sinh ^2(x)}{a+b \sinh (x)} \, dx}{6 b^2}\\ &=\frac {\left (3 a^2-2 b^2\right ) \cosh (x)}{3 b^3}-\frac {a \cosh (x) \sinh (x)}{2 b^2}+\frac {\cosh (x) \sinh ^2(x)}{3 b}-\frac {i \int \frac {3 i a^2 b-3 i a \left (2 a^2-b^2\right ) \sinh (x)}{a+b \sinh (x)} \, dx}{6 b^3}\\ &=-\frac {a \left (2 a^2-b^2\right ) x}{2 b^4}+\frac {\left (3 a^2-2 b^2\right ) \cosh (x)}{3 b^3}-\frac {a \cosh (x) \sinh (x)}{2 b^2}+\frac {\cosh (x) \sinh ^2(x)}{3 b}+\frac {a^4 \int \frac {1}{a+b \sinh (x)} \, dx}{b^4}\\ &=-\frac {a \left (2 a^2-b^2\right ) x}{2 b^4}+\frac {\left (3 a^2-2 b^2\right ) \cosh (x)}{3 b^3}-\frac {a \cosh (x) \sinh (x)}{2 b^2}+\frac {\cosh (x) \sinh ^2(x)}{3 b}+\frac {\left (2 a^4\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^4}\\ &=-\frac {a \left (2 a^2-b^2\right ) x}{2 b^4}+\frac {\left (3 a^2-2 b^2\right ) \cosh (x)}{3 b^3}-\frac {a \cosh (x) \sinh (x)}{2 b^2}+\frac {\cosh (x) \sinh ^2(x)}{3 b}-\frac {\left (4 a^4\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^4}\\ &=-\frac {a \left (2 a^2-b^2\right ) x}{2 b^4}-\frac {2 a^4 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^4 \sqrt {a^2+b^2}}+\frac {\left (3 a^2-2 b^2\right ) \cosh (x)}{3 b^3}-\frac {a \cosh (x) \sinh (x)}{2 b^2}+\frac {\cosh (x) \sinh ^2(x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 105, normalized size = 0.97 \begin {gather*} \frac {3 b \left (4 a^2-3 b^2\right ) \cosh (x)+b^3 \cosh (3 x)+3 a \left (-4 a^2 x+2 b^2 x+\frac {8 a^3 \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-b^2 \sinh (2 x)\right )}{12 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs.
\(2(94)=188\).
time = 0.37, size = 202, normalized size = 1.87
method | result | size |
risch | \(-\frac {a^{3} x}{b^{4}}+\frac {a x}{2 b^{2}}+\frac {{\mathrm e}^{3 x}}{24 b}-\frac {a \,{\mathrm e}^{2 x}}{8 b^{2}}+\frac {{\mathrm e}^{x} a^{2}}{2 b^{3}}-\frac {3 \,{\mathrm e}^{x}}{8 b}+\frac {{\mathrm e}^{-x} a^{2}}{2 b^{3}}-\frac {3 \,{\mathrm e}^{-x}}{8 b}+\frac {a \,{\mathrm e}^{-2 x}}{8 b^{2}}+\frac {{\mathrm e}^{-3 x}}{24 b}+\frac {a^{4} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, b^{4}}-\frac {a^{4} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, b^{4}}\) | \(201\) |
default | \(-\frac {1}{3 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a +b}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b -b^{2}}{2 b^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \left (2 a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b^{4}}+\frac {1}{3 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-a +b}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {-2 a^{2}+a b +b^{2}}{2 b^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \left (2 a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b^{4}}+\frac {2 a^{4} \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{4} \sqrt {a^{2}+b^{2}}}\) | \(202\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 158, normalized size = 1.46 \begin {gather*} \frac {a^{4} \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 )}{\sqrt {a^{2} + b^{2}} b^{4}} - \frac {{\left (3 \, a b e^{\left (-x\right )} - b^{2} - 3 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )} e^{\left (-2 \, x\right )}\right )} e^{\left (3 \, x\right )}}{24 \, b^{3}} + \frac {3 \, a b e^{\left (-2 \, x\right )} + b^{2} e^{\left (-3 \, x\right )} + 3 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )} e^{\left (-x\right )}}{24 \, b^{3}} - \frac {{\left (2 \, a^{3} - a b^{2}\right )} x}{2 \, b^{4}} \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. 799 vs.
\(2 (96) = 192\).
time = 0.36, size = 799, normalized size = 7.40 \begin {gather*} \frac {{\left (a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )^{6} + {\left (a^{2} b^{3} + b^{5}\right )} \sinh \left (x\right )^{6} - 3 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right )^{5} - 3 \, {\left (a^{3} b^{2} + a b^{4} - 2 \, {\left (a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + a^{2} b^{3} + b^{5} - 12 \, {\left (2 \, a^{5} + a^{3} b^{2} - a b^{4}\right )} x \cosh \left (x\right )^{3} + 3 \, {\left (4 \, a^{4} b + a^{2} b^{3} - 3 \, b^{5}\right )} \cosh \left (x\right )^{4} + 3 \, {\left (4 \, a^{4} b + a^{2} b^{3} - 3 \, b^{5} + 5 \, {\left (a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )^{2} - 5 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + 2 \, {\left (10 \, {\left (a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )^{3} - 15 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right )^{2} - 6 \, {\left (2 \, a^{5} + a^{3} b^{2} - a b^{4}\right )} x + 6 \, {\left (4 \, a^{4} b + a^{2} b^{3} - 3 \, b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (4 \, a^{4} b + a^{2} b^{3} - 3 \, b^{5}\right )} \cosh \left (x\right )^{2} + 3 \, {\left (4 \, a^{4} b + a^{2} b^{3} - 3 \, b^{5} + 5 \, {\left (a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )^{4} - 10 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right )^{3} - 12 \, {\left (2 \, a^{5} + a^{3} b^{2} - a b^{4}\right )} x \cosh \left (x\right ) + 6 \, {\left (4 \, a^{4} b + a^{2} b^{3} - 3 \, b^{5}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 24 \, {\left (a^{4} \cosh \left (x\right )^{3} + 3 \, a^{4} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, a^{4} \cosh \left (x\right ) \sinh \left (x\right )^{2} + a^{4} \sinh \left (x\right )^{3}\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 ) + 3 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right ) + 3 \, {\left (2 \, {\left (a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )^{5} + a^{3} b^{2} + a b^{4} - 5 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right )^{4} - 12 \, {\left (2 \, a^{5} + a^{3} b^{2} - a b^{4}\right )} x \cosh \left (x\right )^{2} + 4 \, {\left (4 \, a^{4} b + a^{2} b^{3} - 3 \, b^{5}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (4 \, a^{4} b + a^{2} b^{3} - 3 \, b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{24 \, {\left ({\left (a^{2} b^{4} + b^{6}\right )} \cosh \left (x\right )^{3} + 3 \, {\left (a^{2} b^{4} + b^{6}\right )} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, {\left (a^{2} b^{4} + b^{6}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (a^{2} b^{4} + b^{6}\right )} \sinh \left (x\right )^{3}\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 = 156, normalized size = 1.44 \begin {gather*} \frac {a^{4} \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 )}{\sqrt {a^{2} + b^{2}} b^{4}} + \frac {b^{2} e^{\left (3 \, x\right )} - 3 \, a b e^{\left (2 \, x\right )} + 12 \, a^{2} e^{x} - 9 \, b^{2} e^{x}}{24 \, b^{3}} - \frac {{\left (2 \, a^{3} - a b^{2}\right )} x}{2 \, b^{4}} + \frac {{\left (3 \, a b^{2} e^{x} + b^{3} + 3 \, {\left (4 \, a^{2} b - 3 \, b^{3}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-3 \, x\right )}}{24 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.76, size = 199, normalized size = 1.84 \begin {gather*} \frac {{\mathrm {e}}^{-3\,x}}{24\,b}+\frac {{\mathrm {e}}^{3\,x}}{24\,b}+\frac {x\,\left (a\,b^2-2\,a^3\right )}{2\,b^4}+\frac {{\mathrm {e}}^x\,\left (4\,a^2-3\,b^2\right )}{8\,b^3}+\frac {a\,{\mathrm {e}}^{-2\,x}}{8\,b^2}-\frac {a\,{\mathrm {e}}^{2\,x}}{8\,b^2}+\frac {{\mathrm {e}}^{-x}\,\left (4\,a^2-3\,b^2\right )}{8\,b^3}-\frac {a^4\,\ln \left (-\frac {2\,a^4\,{\mathrm {e}}^x}{b^5}-\frac {2\,a^4\,\left (b-a\,{\mathrm {e}}^x\right )}{b^5\,\sqrt {a^2+b^2}}\right )}{b^4\,\sqrt {a^2+b^2}}+\frac {a^4\,\ln \left (\frac {2\,a^4\,\left (b-a\,{\mathrm {e}}^x\right )}{b^5\,\sqrt {a^2+b^2}}-\frac {2\,a^4\,{\mathrm {e}}^x}{b^5}\right )}{b^4\,\sqrt {a^2+b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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