Optimal. Leaf size=81 \[ \frac{b \sinh \left (2 a-\frac{2 b c}{d}\right ) \text{Chi}\left (\frac{2 b c}{d}+2 b x\right )}{d^2}+\frac{b \cosh \left (2 a-\frac{2 b c}{d}\right ) \text{Shi}\left (\frac{2 b c}{d}+2 b x\right )}{d^2}-\frac{\sinh ^2(a+b x)}{d (c+d x)} \]
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Rubi [A] time = 0.154983, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {3313, 12, 3303, 3298, 3301} \[ \frac{b \sinh \left (2 a-\frac{2 b c}{d}\right ) \text{Chi}\left (\frac{2 b c}{d}+2 b x\right )}{d^2}+\frac{b \cosh \left (2 a-\frac{2 b c}{d}\right ) \text{Shi}\left (\frac{2 b c}{d}+2 b x\right )}{d^2}-\frac{\sinh ^2(a+b x)}{d (c+d x)} \]
Antiderivative was successfully verified.
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Rule 3313
Rule 12
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\sinh ^2(a+b x)}{(c+d x)^2} \, dx &=-\frac{\sinh ^2(a+b x)}{d (c+d x)}-\frac{(2 i b) \int \frac{i \sinh (2 a+2 b x)}{2 (c+d x)} \, dx}{d}\\ &=-\frac{\sinh ^2(a+b x)}{d (c+d x)}+\frac{b \int \frac{\sinh (2 a+2 b x)}{c+d x} \, dx}{d}\\ &=-\frac{\sinh ^2(a+b x)}{d (c+d x)}+\frac{\left (b \cosh \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d}+\frac{\left (b \sinh \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d}\\ &=\frac{b \text{Chi}\left (\frac{2 b c}{d}+2 b x\right ) \sinh \left (2 a-\frac{2 b c}{d}\right )}{d^2}-\frac{\sinh ^2(a+b x)}{d (c+d x)}+\frac{b \cosh \left (2 a-\frac{2 b c}{d}\right ) \text{Shi}\left (\frac{2 b c}{d}+2 b x\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.44441, size = 75, normalized size = 0.93 \[ \frac{b \sinh \left (2 a-\frac{2 b c}{d}\right ) \text{Chi}\left (\frac{2 b (c+d x)}{d}\right )+b \cosh \left (2 a-\frac{2 b c}{d}\right ) \text{Shi}\left (\frac{2 b (c+d x)}{d}\right )-\frac{d \sinh ^2(a+b x)}{c+d x}}{d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.072, size = 152, normalized size = 1.9 \begin{align*}{\frac{1}{2\,d \left ( dx+c \right ) }}-{\frac{b{{\rm e}^{-2\,bx-2\,a}}}{ \left ( 4\,bdx+4\,cb \right ) d}}+{\frac{b}{2\,{d}^{2}}{{\rm e}^{-2\,{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,2\,bx+2\,a-2\,{\frac{da-cb}{d}} \right ) }-{\frac{b{{\rm e}^{2\,bx+2\,a}}}{4\,{d}^{2}} \left ({\frac{cb}{d}}+bx \right ) ^{-1}}-{\frac{b}{2\,{d}^{2}}{{\rm e}^{2\,{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,-2\,bx-2\,a-2\,{\frac{-da+cb}{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38888, size = 119, normalized size = 1.47 \begin{align*} -\frac{e^{\left (-2 \, a + \frac{2 \, b c}{d}\right )} E_{2}\left (\frac{2 \,{\left (d x + c\right )} b}{d}\right )}{4 \,{\left (d x + c\right )} d} - \frac{e^{\left (2 \, a - \frac{2 \, b c}{d}\right )} E_{2}\left (-\frac{2 \,{\left (d x + c\right )} b}{d}\right )}{4 \,{\left (d x + c\right )} d} + \frac{1}{2 \,{\left (d^{2} x + c d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.72063, size = 365, normalized size = 4.51 \begin{align*} -\frac{d \cosh \left (b x + a\right )^{2} + d \sinh \left (b x + a\right )^{2} -{\left ({\left (b d x + b c\right )}{\rm Ei}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) -{\left (b d x + b c\right )}{\rm Ei}\left (-\frac{2 \,{\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) -{\left ({\left (b d x + b c\right )}{\rm Ei}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) +{\left (b d x + b c\right )}{\rm Ei}\left (-\frac{2 \,{\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) - d}{2 \,{\left (d^{3} x + c d^{2}\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 \frac{\sinh ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \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{\sinh \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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