Optimal. Leaf size=112 \[ -\frac{a f \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}-\frac{f \cosh (c+d x)}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac{e+f x}{2 b d (a+b \sinh (c+d x))^2} \]
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Rubi [A] time = 0.0951033, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5464, 2664, 12, 2660, 618, 204} \[ -\frac{a f \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{b d^2 \left (a^2+b^2\right )^{3/2}}-\frac{f \cosh (c+d x)}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac{e+f x}{2 b d (a+b \sinh (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 5464
Rule 2664
Rule 12
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))^3} \, dx &=-\frac{e+f x}{2 b d (a+b \sinh (c+d x))^2}+\frac{f \int \frac{1}{(a+b \sinh (c+d x))^2} \, dx}{2 b d}\\ &=-\frac{e+f x}{2 b d (a+b \sinh (c+d x))^2}-\frac{f \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}+\frac{f \int \frac{a}{a+b \sinh (c+d x)} \, dx}{2 b \left (a^2+b^2\right ) d}\\ &=-\frac{e+f x}{2 b d (a+b \sinh (c+d x))^2}-\frac{f \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}+\frac{(a f) \int \frac{1}{a+b \sinh (c+d x)} \, dx}{2 b \left (a^2+b^2\right ) d}\\ &=-\frac{e+f x}{2 b d (a+b \sinh (c+d x))^2}-\frac{f \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac{(i a 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 \left (a^2+b^2\right ) d^2}\\ &=-\frac{e+f x}{2 b d (a+b \sinh (c+d x))^2}-\frac{f \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}+\frac{(2 i a 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 \left (a^2+b^2\right ) d^2}\\ &=-\frac{a f \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac{e+f x}{2 b d (a+b \sinh (c+d x))^2}-\frac{f \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.373767, size = 112, normalized size = 1. \[ -\frac{\frac{\frac{2 a f \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+\frac{d (e+f x)}{(a+b \sinh (c+d x))^2}}{b}+\frac{f \cosh (c+d x)}{\left (a^2+b^2\right ) (a+b \sinh (c+d x))}}{2 d^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 308, normalized size = 2.8 \begin{align*} -{\frac{2\,{a}^{2}dfx{{\rm e}^{2\,dx+2\,c}}+2\,{b}^{2}dfx{{\rm e}^{2\,dx+2\,c}}+2\,{a}^{2}de{{\rm e}^{2\,dx+2\,c}}-abf{{\rm e}^{3\,dx+3\,c}}+2\,{b}^{2}de{{\rm e}^{2\,dx+2\,c}}-2\,{a}^{2}f{{\rm e}^{2\,dx+2\,c}}+{b}^{2}f{{\rm e}^{2\,dx+2\,c}}+3\,fa{{\rm e}^{dx+c}}b-f{b}^{2}}{{d}^{2}b \left ( b{{\rm e}^{2\,dx+2\,c}}+2\,a{{\rm e}^{dx+c}}-b \right ) ^{2} \left ({a}^{2}+{b}^{2} \right ) }}+{\frac{af}{2\,{d}^{2}b}\ln \left ({{\rm e}^{dx+c}}+{\frac{1}{b} \left ( a \left ({a}^{2}+{b}^{2} \right ) ^{{\frac{3}{2}}}-{a}^{4}-2\,{a}^{2}{b}^{2}-{b}^{4} \right ) \left ({a}^{2}+{b}^{2} \right ) ^{-{\frac{3}{2}}}} \right ) \left ({a}^{2}+{b}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{af}{2\,{d}^{2}b}\ln \left ({{\rm e}^{dx+c}}+{\frac{1}{b} \left ( a \left ({a}^{2}+{b}^{2} \right ) ^{{\frac{3}{2}}}+{a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) \left ({a}^{2}+{b}^{2} \right ) ^{-{\frac{3}{2}}}} \right ) \left ({a}^{2}+{b}^{2} \right ) ^{-{\frac{3}{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 = 2.21918, size = 2811, normalized size = 25.1 \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(-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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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