Optimal. Leaf size=95 \[ \frac{2 a \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{b^3 d}+\frac{x \left (2 a^2+b^2\right )}{2 b^3}-\frac{\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d} \]
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Rubi [A] time = 0.180593, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2865, 2735, 2660, 618, 204} \[ \frac{2 a \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{b^3 d}+\frac{x \left (2 a^2+b^2\right )}{2 b^3}-\frac{\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d} \]
Antiderivative was successfully verified.
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Rule 2865
Rule 2735
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac{\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}+\frac{i \int \frac{i a b-i \left (2 a^2+b^2\right ) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{2 b^2}\\ &=\frac{\left (2 a^2+b^2\right ) x}{2 b^3}-\frac{\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}-\frac{\left (a \left (a^2+b^2\right )\right ) \int \frac{1}{a+b \sinh (c+d x)} \, dx}{b^3}\\ &=\frac{\left (2 a^2+b^2\right ) x}{2 b^3}-\frac{\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}+\frac{\left (2 i a \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{b^3 d}\\ &=\frac{\left (2 a^2+b^2\right ) x}{2 b^3}-\frac{\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}-\frac{\left (4 i a \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{b^3 d}\\ &=\frac{\left (2 a^2+b^2\right ) x}{2 b^3}+\frac{2 a \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{b^3 d}-\frac{\cosh (c+d x) (2 a-b \sinh (c+d x))}{2 b^2 d}\\ \end{align*}
Mathematica [A] time = 0.427519, size = 109, normalized size = 1.15 \[ \frac{8 a \sqrt{-a^2-b^2} \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{-a^2-b^2}}\right )+4 a^2 c+4 a^2 d x-4 a b \cosh (c+d x)+b^2 \sinh (2 (c+d x))+2 b^2 c+2 b^2 d x}{4 b^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.035, size = 260, normalized size = 2.7 \begin{align*} -{\frac{1}{2\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{1}{2\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{a}{d{b}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{{a}^{2}}{d{b}^{3}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{1}{2\,bd}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{1}{2\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{1}{2\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{a}{d{b}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{{a}^{2}}{d{b}^{3}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{1}{2\,bd}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-2\,{\frac{a\sqrt{{a}^{2}+{b}^{2}}}{d{b}^{3}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( 1/2\,dx+c/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.27919, size = 1150, normalized size = 12.11 \begin{align*} \frac{b^{2} \cosh \left (d x + c\right )^{4} + b^{2} \sinh \left (d x + c\right )^{4} + 4 \,{\left (2 \, a^{2} + b^{2}\right )} d x \cosh \left (d x + c\right )^{2} - 4 \, a b \cosh \left (d x + c\right )^{3} + 4 \,{\left (b^{2} \cosh \left (d x + c\right ) - a b\right )} \sinh \left (d x + c\right )^{3} - 4 \, a b \cosh \left (d x + c\right ) + 2 \,{\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \,{\left (2 \, a^{2} + b^{2}\right )} d x - 6 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 8 \,{\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2}\right )} \sqrt{a^{2} + b^{2}} \log \left (\frac{b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \,{\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt{a^{2} + b^{2}}{\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \,{\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right ) - b^{2} + 4 \,{\left (b^{2} \cosh \left (d x + c\right )^{3} + 2 \,{\left (2 \, a^{2} + b^{2}\right )} d x \cosh \left (d x + c\right ) - 3 \, a b \cosh \left (d x + c\right )^{2} - a b\right )} \sinh \left (d x + c\right )}{8 \,{\left (b^{3} d \cosh \left (d x + c\right )^{2} + 2 \, b^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{3} d \sinh \left (d x + c\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15216, size = 223, normalized size = 2.35 \begin{align*} \frac{{\left (2 \, a^{2} + b^{2}\right )}{\left (d x + c\right )}}{2 \, b^{3} d} - \frac{{\left (4 \, a b e^{\left (d x + c\right )} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, b^{3} d} - \frac{{\left (a^{3} + a b^{2}\right )} \log \left (\frac{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}} b^{3} d} + \frac{b d e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a d e^{\left (d x + c\right )}}{8 \, b^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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