Optimal. Leaf size=132 \[ -\frac {2 b (a-b)^{3/2} (a+b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^5}-\frac {\sinh ^3(x) (4 b-3 a \cosh (x))}{12 a^2}+\frac {\sinh (x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right )}{8 a^4}+\frac {x \left (3 a^4-12 a^2 b^2+8 b^4\right )}{8 a^5} \]
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Rubi [A] time = 0.37, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3872, 2865, 2735, 2659, 205} \[ \frac {x \left (-12 a^2 b^2+3 a^4+8 b^4\right )}{8 a^5}+\frac {\sinh (x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right )}{8 a^4}-\frac {2 b (a-b)^{3/2} (a+b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^5}-\frac {\sinh ^3(x) (4 b-3 a \cosh (x))}{12 a^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2659
Rule 2735
Rule 2865
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sinh ^4(x)}{a+b \text {sech}(x)} \, dx &=-\int \frac {\cosh (x) \sinh ^4(x)}{-b-a \cosh (x)} \, dx\\ &=-\frac {(4 b-3 a \cosh (x)) \sinh ^3(x)}{12 a^2}+\frac {\int \frac {\left (-a b+\left (3 a^2-4 b^2\right ) \cosh (x)\right ) \sinh ^2(x)}{-b-a \cosh (x)} \, dx}{4 a^2}\\ &=\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 a^4}-\frac {(4 b-3 a \cosh (x)) \sinh ^3(x)}{12 a^2}-\frac {\int \frac {-a b \left (5 a^2-4 b^2\right )+\left (3 a^4-12 a^2 b^2+8 b^4\right ) \cosh (x)}{-b-a \cosh (x)} \, dx}{8 a^4}\\ &=\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 a^4}-\frac {(4 b-3 a \cosh (x)) \sinh ^3(x)}{12 a^2}+\frac {\left (b \left (a^2-b^2\right )^2\right ) \int \frac {1}{-b-a \cosh (x)} \, dx}{a^5}\\ &=\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 a^4}-\frac {(4 b-3 a \cosh (x)) \sinh ^3(x)}{12 a^2}+\frac {\left (2 b \left (a^2-b^2\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a-b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^5}\\ &=\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac {2 (a-b)^{3/2} b (a+b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^5}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 a^4}-\frac {(4 b-3 a \cosh (x)) \sinh ^3(x)}{12 a^2}\\ \end {align*}
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Mathematica [A] time = 0.74, size = 219, normalized size = 1.66 \[ \frac {36 a^4 x+3 a^4 \sinh (4 x)-8 a^3 b \sinh (3 x)-144 a^2 b^2 x+24 a b \left (5 a^2-4 b^2\right ) \sinh (x)-24 a^2 \left (a^2-b^2\right ) \sinh (2 x)+\frac {192 b^5 \tan ^{-1}\left (\frac {(b-a) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {384 a^2 b^3 \tan ^{-1}\left (\frac {(b-a) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {192 a^4 b \tan ^{-1}\left (\frac {(b-a) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+96 b^4 x}{96 a^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 1812, normalized size = 13.73 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 197, normalized size = 1.49 \[ \frac {3 \, a^{3} e^{\left (4 \, x\right )} - 8 \, a^{2} b e^{\left (3 \, x\right )} - 24 \, a^{3} e^{\left (2 \, x\right )} + 24 \, a b^{2} e^{\left (2 \, x\right )} + 120 \, a^{2} b e^{x} - 96 \, b^{3} e^{x}}{192 \, a^{4}} + \frac {{\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} x}{8 \, a^{5}} + \frac {{\left (8 \, a^{3} b e^{x} - 3 \, a^{4} - 24 \, {\left (5 \, a^{3} b - 4 \, a b^{3}\right )} e^{\left (3 \, x\right )} + 24 \, {\left (a^{4} - a^{2} b^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-4 \, x\right )}}{192 \, a^{5}} - \frac {2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 488, normalized size = 3.70 \[ \frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 a}-\frac {1}{8 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {3}{8 a \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 a}+\frac {1}{8 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3}{8 a \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {2 b \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {4 b^{3} \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{3} \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {2 b^{5} \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5} \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) b^{4}}{a^{5}}-\frac {b}{a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {b^{3}}{a^{4} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {b}{3 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b^{3}}{a^{4} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {b}{a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) b^{2}}{2 a^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) b^{4}}{a^{5}}+\frac {b}{3 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {b}{2 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {b^{2}}{2 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) b^{2}}{2 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.01, size = 275, normalized size = 2.08 \[ \frac {{\mathrm {e}}^{4\,x}}{64\,a}-\frac {{\mathrm {e}}^{-4\,x}}{64\,a}+\frac {x\,\left (3\,a^4-12\,a^2\,b^2+8\,b^4\right )}{8\,a^5}-\frac {{\mathrm {e}}^{-x}\,\left (5\,a^2\,b-4\,b^3\right )}{8\,a^4}+\frac {{\mathrm {e}}^{-2\,x}\,\left (a^2-b^2\right )}{8\,a^3}-\frac {{\mathrm {e}}^{2\,x}\,\left (a^2-b^2\right )}{8\,a^3}+\frac {b\,{\mathrm {e}}^{-3\,x}}{24\,a^2}-\frac {b\,{\mathrm {e}}^{3\,x}}{24\,a^2}+\frac {{\mathrm {e}}^x\,\left (5\,a^2\,b-4\,b^3\right )}{8\,a^4}+\frac {b\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (a^4\,b-2\,a^2\,b^3+b^5\right )}{a^6}-\frac {2\,b\,{\left (a+b\right )}^{3/2}\,\left (a+b\,{\mathrm {e}}^x\right )\,{\left (b-a\right )}^{3/2}}{a^6}\right )\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}{a^5}-\frac {b\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (a^4\,b-2\,a^2\,b^3+b^5\right )}{a^6}+\frac {2\,b\,{\left (a+b\right )}^{3/2}\,\left (a+b\,{\mathrm {e}}^x\right )\,{\left (b-a\right )}^{3/2}}{a^6}\right )\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}{a^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{4}{\relax (x )}}{a + b \operatorname {sech}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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