Optimal. Leaf size=58 \[ \frac{1}{2} a \left (a^2+3 b^2\right ) \tan ^{-1}(\sinh (x))-\frac{1}{2} a b^2 \sinh (x)-\frac{1}{2} \text{sech}^2(x) (b-a \sinh (x)) (a+b \sinh (x))^2+b^3 \log (\cosh (x)) \]
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Rubi [A] time = 0.106695, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {4391, 2668, 739, 774, 635, 204, 260} \[ \frac{1}{2} a \left (a^2+3 b^2\right ) \tan ^{-1}(\sinh (x))-\frac{1}{2} a b^2 \sinh (x)-\frac{1}{2} \text{sech}^2(x) (b-a \sinh (x)) (a+b \sinh (x))^2+b^3 \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 4391
Rule 2668
Rule 739
Rule 774
Rule 635
Rule 204
Rule 260
Rubi steps
\begin{align*} \int (a \text{sech}(x)+b \tanh (x))^3 \, dx &=\int \text{sech}^3(x) (a+b \sinh (x))^3 \, dx\\ &=b^3 \operatorname{Subst}\left (\int \frac{(a+x)^3}{\left (-b^2-x^2\right )^2} \, dx,x,b \sinh (x)\right )\\ &=-\frac{1}{2} \text{sech}^2(x) (b-a \sinh (x)) (a+b \sinh (x))^2+\frac{1}{2} b \operatorname{Subst}\left (\int \frac{(a+x) \left (-a^2-2 b^2+a x\right )}{-b^2-x^2} \, dx,x,b \sinh (x)\right )\\ &=-\frac{1}{2} a b^2 \sinh (x)-\frac{1}{2} \text{sech}^2(x) (b-a \sinh (x)) (a+b \sinh (x))^2-\frac{1}{2} b \operatorname{Subst}\left (\int \frac{a b^2-a \left (-a^2-2 b^2\right )+2 b^2 x}{-b^2-x^2} \, dx,x,b \sinh (x)\right )\\ &=-\frac{1}{2} a b^2 \sinh (x)-\frac{1}{2} \text{sech}^2(x) (b-a \sinh (x)) (a+b \sinh (x))^2-b^3 \operatorname{Subst}\left (\int \frac{x}{-b^2-x^2} \, dx,x,b \sinh (x)\right )-\frac{1}{2} \left (a b \left (a^2+3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-b^2-x^2} \, dx,x,b \sinh (x)\right )\\ &=\frac{1}{2} a \left (a^2+3 b^2\right ) \tan ^{-1}(\sinh (x))+b^3 \log (\cosh (x))-\frac{1}{2} a b^2 \sinh (x)-\frac{1}{2} \text{sech}^2(x) (b-a \sinh (x)) (a+b \sinh (x))^2\\ \end{align*}
Mathematica [B] time = 1.86863, size = 194, normalized size = 3.34 \[ \frac{1}{4} \left (\frac{2 a^4 b \text{sech}^2(x)}{a^2+b^2}+\frac{b \left (\left (a^3+3 a b^2-2 \left (-b^2\right )^{3/2}\right ) \log \left (\sqrt{-b^2}-b \sinh (x)\right )-\left (a^3+3 a b^2+2 \left (-b^2\right )^{3/2}\right ) \log \left (\sqrt{-b^2}+b \sinh (x)\right )\right )}{\sqrt{-b^2}}-\frac{2 b \tanh ^2(x) \left (-2 a^2 b^2-4 a^4+a b^3 \sinh (x)+b^4\right )}{a^2+b^2}+\frac{a \tanh (x) \text{sech}(x) \left (-4 a^2 b^2+2 a^4+b^4 \cosh (2 x)-7 b^4\right )}{a^2+b^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 79, normalized size = 1.4 \begin{align*}{\frac{{a}^{3}{\rm sech} \left (x\right )\tanh \left ( x \right ) }{2}}+{a}^{3}\arctan \left ({{\rm e}^{x}} \right ) +{\frac{3\,{a}^{2}b \left ( \sinh \left ( x \right ) \right ) ^{2}}{2\, \left ( \cosh \left ( x \right ) \right ) ^{2}}}-3\,{\frac{a{b}^{2}\sinh \left ( x \right ) }{ \left ( \cosh \left ( x \right ) \right ) ^{2}}}+{\frac{3\,a{b}^{2}{\rm sech} \left (x\right )\tanh \left ( x \right ) }{2}}+3\,a{b}^{2}\arctan \left ({{\rm e}^{x}} \right ) +{b}^{3}\ln \left ( \cosh \left ( x \right ) \right ) -{\frac{{b}^{3} \left ( \tanh \left ( x \right ) \right ) ^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.57354, size = 162, normalized size = 2.79 \begin{align*} \frac{3}{2} \, a^{2} b \tanh \left (x\right )^{2} + b^{3}{\left (x + \frac{2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + \log \left (e^{\left (-2 \, x\right )} + 1\right )\right )} - 3 \, a b^{2}{\left (\frac{e^{\left (-x\right )} - e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + \arctan \left (e^{\left (-x\right )}\right )\right )} + a^{3}{\left (\frac{e^{\left (-x\right )} - e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} - \arctan \left (e^{\left (-x\right )}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.47664, size = 1359, normalized size = 23.43 \begin{align*} \text{result too large to display} \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 )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14454, size = 158, normalized size = 2.72 \begin{align*} \frac{1}{2} \, b^{3} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right ) + \frac{1}{4} \,{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )}{\left (a^{3} + 3 \, a b^{2}\right )} - \frac{b^{3}{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 2 \, a^{3}{\left (e^{\left (-x\right )} - e^{x}\right )} - 6 \, a b^{2}{\left (e^{\left (-x\right )} - e^{x}\right )} + 12 \, a^{2} b}{2 \,{\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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