Optimal. Leaf size=100 \[ -\frac{4}{3} a b \left (a^2+2 b^2\right ) \cosh (x)-\frac{1}{3} b^2 \left (2 a^2+3 b^2\right ) \sinh (x) \cosh (x)+\frac{1}{3} \text{sech}(x) (a+b \sinh (x))^2 \left (\left (2 a^2+3 b^2\right ) \sinh (x)+a b\right )-\frac{1}{3} \text{sech}^3(x) (b-a \sinh (x)) (a+b \sinh (x))^3+b^4 x \]
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Rubi [A] time = 0.208518, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4391, 2691, 2861, 2734} \[ -\frac{4}{3} a b \left (a^2+2 b^2\right ) \cosh (x)-\frac{1}{3} b^2 \left (2 a^2+3 b^2\right ) \sinh (x) \cosh (x)+\frac{1}{3} \text{sech}(x) (a+b \sinh (x))^2 \left (\left (2 a^2+3 b^2\right ) \sinh (x)+a b\right )-\frac{1}{3} \text{sech}^3(x) (b-a \sinh (x)) (a+b \sinh (x))^3+b^4 x \]
Antiderivative was successfully verified.
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Rule 4391
Rule 2691
Rule 2861
Rule 2734
Rubi steps
\begin{align*} \int (a \text{sech}(x)+b \tanh (x))^4 \, dx &=\int \text{sech}^4(x) (a+b \sinh (x))^4 \, dx\\ &=-\frac{1}{3} \text{sech}^3(x) (b-a \sinh (x)) (a+b \sinh (x))^3-\frac{1}{3} \int \text{sech}^2(x) (a+b \sinh (x))^2 \left (-2 a^2-3 b^2+a b \sinh (x)\right ) \, dx\\ &=-\frac{1}{3} \text{sech}^3(x) (b-a \sinh (x)) (a+b \sinh (x))^3+\frac{1}{3} \text{sech}(x) (a+b \sinh (x))^2 \left (a b+\left (2 a^2+3 b^2\right ) \sinh (x)\right )+\frac{1}{3} \int (a+b \sinh (x)) \left (-2 a b^2-2 b \left (2 a^2+3 b^2\right ) \sinh (x)\right ) \, dx\\ &=b^4 x-\frac{4}{3} a b \left (a^2+2 b^2\right ) \cosh (x)-\frac{1}{3} b^2 \left (2 a^2+3 b^2\right ) \cosh (x) \sinh (x)-\frac{1}{3} \text{sech}^3(x) (b-a \sinh (x)) (a+b \sinh (x))^3+\frac{1}{3} \text{sech}(x) (a+b \sinh (x))^2 \left (a b+\left (2 a^2+3 b^2\right ) \sinh (x)\right )\\ \end{align*}
Mathematica [A] time = 0.176357, size = 79, normalized size = 0.79 \[ \frac{1}{3} \left (2 \left (3 a^2 b^2+a^4-2 b^4\right ) \tanh (x)-4 a b \left (a^2-b^2\right ) \text{sech}^3(x)+\left (-6 a^2 b^2+a^4+b^4\right ) \tanh (x) \text{sech}^2(x)-12 a b^3 \text{sech}(x)+3 b^4 x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 123, normalized size = 1.2 \begin{align*}{a}^{4} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (x\right ) \right ) ^{2}}{3}} \right ) \tanh \left ( x \right ) +4\,{a}^{3}b \left ( 1/3\,{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{2}}{ \left ( \cosh \left ( x \right ) \right ) ^{3}}}+1/3\,{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{2}}{\cosh \left ( x \right ) }}-1/3\,\cosh \left ( x \right ) \right ) +6\,{a}^{2}{b}^{2} \left ( -1/2\,{\frac{\sinh \left ( x \right ) }{ \left ( \cosh \left ( x \right ) \right ) ^{3}}}+1/2\, \left ( 2/3+1/3\, \left ({\rm sech} \left (x\right ) \right ) ^{2} \right ) \tanh \left ( x \right ) \right ) +4\,a{b}^{3} \left ( -1/3\,{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{2}}{ \left ( \cosh \left ( x \right ) \right ) ^{3}}}+2/3\,{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{2}}{\cosh \left ( x \right ) }}-2/3\,\cosh \left ( x \right ) \right ) +{b}^{4} \left ( x-\tanh \left ( x \right ) -{\frac{ \left ( \tanh \left ( x \right ) \right ) ^{3}}{3}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05966, size = 284, normalized size = 2.84 \begin{align*} 2 \, a^{2} b^{2} \tanh \left (x\right )^{3} + \frac{1}{3} \, b^{4}{\left (3 \, x - \frac{4 \,{\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + 2\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1}\right )} - \frac{8}{3} \, a b^{3}{\left (\frac{3 \, e^{\left (-x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} + \frac{2 \, e^{\left (-3 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} + \frac{3 \, e^{\left (-5 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1}\right )} + \frac{4}{3} \, a^{4}{\left (\frac{3 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} + \frac{1}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1}\right )} - \frac{32 \, a^{3} b}{3 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.30987, size = 505, normalized size = 5.05 \begin{align*} -\frac{24 \, a b^{3} \cosh \left (x\right )^{2} + 16 \, a^{3} b + 8 \, a b^{3} -{\left (3 \, b^{4} x - 2 \, a^{4} - 6 \, a^{2} b^{2} + 4 \, b^{4}\right )} \cosh \left (x\right )^{3} - 2 \,{\left (a^{4} + 3 \, a^{2} b^{2} - 2 \, b^{4}\right )} \sinh \left (x\right )^{3} + 3 \,{\left (8 \, a b^{3} -{\left (3 \, b^{4} x - 2 \, a^{4} - 6 \, a^{2} b^{2} + 4 \, b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 3 \,{\left (3 \, b^{4} x - 2 \, a^{4} - 6 \, a^{2} b^{2} + 4 \, b^{4}\right )} \cosh \left (x\right ) - 6 \,{\left (a^{4} - 3 \, a^{2} b^{2} +{\left (a^{4} + 3 \, a^{2} b^{2} - 2 \, b^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )}{3 \,{\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \operatorname{sech}{\left (x \right )} + b \tanh{\left (x \right )}\right )^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13681, size = 149, normalized size = 1.49 \begin{align*} b^{4} x - \frac{4 \,{\left (6 \, a b^{3} e^{\left (5 \, x\right )} + 9 \, a^{2} b^{2} e^{\left (4 \, x\right )} - 3 \, b^{4} e^{\left (4 \, x\right )} + 8 \, a^{3} b e^{\left (3 \, x\right )} + 4 \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{4} e^{\left (2 \, x\right )} - 3 \, b^{4} e^{\left (2 \, x\right )} + 6 \, a b^{3} e^{x} + a^{4} + 3 \, a^{2} b^{2} - 2 \, b^{4}\right )}}{3 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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