Optimal. Leaf size=101 \[ \frac{4}{3} a b \left (2 a^2-b^2\right ) \sinh (x)+\frac{1}{3} a^2 \left (3 a^2-2 b^2\right ) \sinh (x) \cosh (x)-\frac{1}{3} \text{csch}(x) (a \cosh (x)+b)^2 \left (\left (3 a^2-2 b^2\right ) \cosh (x)+a b\right )+a^4 x-\frac{1}{3} \text{csch}^3(x) (a \cosh (x)+b)^3 (a+b \cosh (x)) \]
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Rubi [A] time = 0.238612, antiderivative size = 101, 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 = {4392, 2691, 2861, 2734} \[ \frac{4}{3} a b \left (2 a^2-b^2\right ) \sinh (x)+\frac{1}{3} a^2 \left (3 a^2-2 b^2\right ) \sinh (x) \cosh (x)-\frac{1}{3} \text{csch}(x) (a \cosh (x)+b)^2 \left (\left (3 a^2-2 b^2\right ) \cosh (x)+a b\right )+a^4 x-\frac{1}{3} \text{csch}^3(x) (a \cosh (x)+b)^3 (a+b \cosh (x)) \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2691
Rule 2861
Rule 2734
Rubi steps
\begin{align*} \int (a \coth (x)+b \text{csch}(x))^4 \, dx &=\int (i b+i a \cosh (x))^4 \text{csch}^4(x) \, dx\\ &=-\frac{1}{3} (b+a \cosh (x))^3 (a+b \cosh (x)) \text{csch}^3(x)+\frac{1}{3} \int (i b+i a \cosh (x))^2 \left (-3 a^2+2 b^2-a b \cosh (x)\right ) \text{csch}^2(x) \, dx\\ &=-\frac{1}{3} (b+a \cosh (x))^2 \left (a b+\left (3 a^2-2 b^2\right ) \cosh (x)\right ) \text{csch}(x)-\frac{1}{3} (b+a \cosh (x))^3 (a+b \cosh (x)) \text{csch}^3(x)+\frac{1}{3} \int (i b+i a \cosh (x)) \left (-2 i a^2 b-2 i a \left (3 a^2-2 b^2\right ) \cosh (x)\right ) \, dx\\ &=a^4 x-\frac{1}{3} (b+a \cosh (x))^2 \left (a b+\left (3 a^2-2 b^2\right ) \cosh (x)\right ) \text{csch}(x)-\frac{1}{3} (b+a \cosh (x))^3 (a+b \cosh (x)) \text{csch}^3(x)+\frac{4}{3} a b \left (2 a^2-b^2\right ) \sinh (x)+\frac{1}{3} a^2 \left (3 a^2-2 b^2\right ) \cosh (x) \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.265863, size = 95, normalized size = 0.94 \[ -\frac{1}{12} \text{csch}^3(x) \left (6 a^2 b^2 \cosh (3 x)+6 b^2 \left (3 a^2+b^2\right ) \cosh (x)+24 a^3 b \cosh (2 x)-8 a^3 b+9 a^4 x \sinh (x)-3 a^4 x \sinh (3 x)+4 a^4 \cosh (3 x)+16 a b^3-2 b^4 \cosh (3 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 123, normalized size = 1.2 \begin{align*}{a}^{4} \left ( x-{\rm coth} \left (x\right )-{\frac{ \left ({\rm coth} \left (x\right ) \right ) ^{3}}{3}} \right ) +4\,{a}^{3}b \left ( -1/3\,{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{ \left ( \sinh \left ( x \right ) \right ) ^{3}}}-2/3\,{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\sinh \left ( x \right ) }}+2/3\,\sinh \left ( x \right ) \right ) +6\,{a}^{2}{b}^{2} \left ( -1/2\,{\frac{\cosh \left ( x \right ) }{ \left ( \sinh \left ( x \right ) \right ) ^{3}}}-1/2\, \left ( 2/3-1/3\, \left ({\rm csch} \left (x\right ) \right ) ^{2} \right ){\rm coth} \left (x\right ) \right ) +4\,a{b}^{3} \left ( -1/3\,{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{ \left ( \sinh \left ( x \right ) \right ) ^{3}}}+1/3\,{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\sinh \left ( x \right ) }}-1/3\,\sinh \left ( x \right ) \right ) +{b}^{4} \left ({\frac{2}{3}}-{\frac{ \left ({\rm csch} \left (x\right ) \right ) ^{2}}{3}} \right ){\rm coth} \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04159, size = 289, normalized size = 2.86 \begin{align*} -2 \, a^{2} b^{2} \coth \left (x\right )^{3} + \frac{1}{3} \, a^{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^{3} b{\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} \, b^{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 b^{3}}{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 = 2.05329, size = 497, normalized size = 4.92 \begin{align*} -\frac{24 \, a^{3} b \cosh \left (x\right )^{2} - 8 \, a^{3} b + 16 \, a b^{3} + 2 \,{\left (2 \, a^{4} + 3 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )^{3} -{\left (3 \, a^{4} x + 4 \, a^{4} + 6 \, a^{2} b^{2} - 2 \, b^{4}\right )} \sinh \left (x\right )^{3} + 6 \,{\left (4 \, a^{3} b +{\left (2 \, a^{4} + 3 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 6 \,{\left (3 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) + 3 \,{\left (3 \, a^{4} x + 4 \, a^{4} + 6 \, a^{2} b^{2} - 2 \, b^{4} -{\left (3 \, a^{4} x + 4 \, a^{4} + 6 \, a^{2} b^{2} - 2 \, b^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )}{3 \,{\left (\sinh \left (x\right )^{3} + 3 \,{\left (\cosh \left (x\right )^{2} - 1\right )} \sinh \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 \coth{\left (x \right )} + b \operatorname{csch}{\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 = 1.22368, size = 151, normalized size = 1.5 \begin{align*} a^{4} x - \frac{4 \,{\left (6 \, a^{3} b e^{\left (5 \, x\right )} + 3 \, a^{4} e^{\left (4 \, x\right )} + 9 \, a^{2} b^{2} e^{\left (4 \, x\right )} - 4 \, a^{3} b e^{\left (3 \, x\right )} + 8 \, 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^{3} b e^{x} + 2 \, a^{4} + 3 \, a^{2} b^{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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