Optimal. Leaf size=74 \[ -\frac{2 f \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{b d^2 \sqrt{a^2+b^2}}-\frac{e+f x}{b d (a+b \sinh (c+d x))} \]
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Rubi [A] time = 0.0729192, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5464, 2660, 618, 204} \[ -\frac{2 f \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{b d^2 \sqrt{a^2+b^2}}-\frac{e+f x}{b d (a+b \sinh (c+d x))} \]
Antiderivative was successfully verified.
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Rule 5464
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{(e+f x) \cosh (c+d x)}{(a+b \sinh (c+d x))^2} \, dx &=-\frac{e+f x}{b d (a+b \sinh (c+d x))}+\frac{f \int \frac{1}{a+b \sinh (c+d x)} \, dx}{b d}\\ &=-\frac{e+f x}{b d (a+b \sinh (c+d x))}-\frac{(2 i f) \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 d^2}\\ &=-\frac{e+f x}{b d (a+b \sinh (c+d x))}+\frac{(4 i f) \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 d^2}\\ &=-\frac{2 f \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{b \sqrt{a^2+b^2} d^2}-\frac{e+f x}{b d (a+b \sinh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.503442, size = 78, normalized size = 1.05 \[ \frac{\frac{2 f \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}-\frac{d (e+f x)}{a+b \sinh (c+d x)}}{b d^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.214, size = 164, normalized size = 2.2 \begin{align*} -2\,{\frac{ \left ( fx+e \right ){{\rm e}^{dx+c}}}{bd \left ( b{{\rm e}^{2\,dx+2\,c}}+2\,a{{\rm e}^{dx+c}}-b \right ) }}+{\frac{f}{{d}^{2}b}\ln \left ({{\rm e}^{dx+c}}+{\frac{1}{b} \left ( a\sqrt{{a}^{2}+{b}^{2}}-{a}^{2}-{b}^{2} \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}} \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}}-{\frac{f}{{d}^{2}b}\ln \left ({{\rm e}^{dx+c}}+{\frac{1}{b} \left ( a\sqrt{{a}^{2}+{b}^{2}}+{a}^{2}+{b}^{2} \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}} \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}} \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 = 2.1182, size = 1015, normalized size = 13.72 \begin{align*} \frac{{\left (b f \cosh \left (d x + c\right )^{2} + b f \sinh \left (d x + c\right )^{2} + 2 \, a f \cosh \left (d x + c\right ) - b f + 2 \,{\left (b f \cosh \left (d x + c\right ) + a f\right )} \sinh \left (d x + c\right )\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 ) - 2 \,{\left ({\left (a^{2} + b^{2}\right )} d f x +{\left (a^{2} + b^{2}\right )} d e\right )} \cosh \left (d x + c\right ) - 2 \,{\left ({\left (a^{2} + b^{2}\right )} d f x +{\left (a^{2} + b^{2}\right )} d e\right )} \sinh \left (d x + c\right )}{{\left (a^{2} b^{2} + b^{4}\right )} d^{2} \cosh \left (d x + c\right )^{2} +{\left (a^{2} b^{2} + b^{4}\right )} d^{2} \sinh \left (d x + c\right )^{2} + 2 \,{\left (a^{3} b + a b^{3}\right )} d^{2} \cosh \left (d x + c\right ) -{\left (a^{2} b^{2} + b^{4}\right )} d^{2} + 2 \,{\left ({\left (a^{2} b^{2} + b^{4}\right )} d^{2} \cosh \left (d x + c\right ) +{\left (a^{3} b + a b^{3}\right )} d^{2}\right )} \sinh \left (d x + c\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )} \cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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