Optimal. Leaf size=107 \[ \frac{\sinh \left (\frac{2 b}{d}\right ) (b c-a d) \text{Chi}\left (\frac{2 (b c-a d)}{d (c+d x)}\right )}{d^2}-\frac{\cosh \left (\frac{2 b}{d}\right ) (b c-a d) \text{Shi}\left (\frac{2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac{(c+d x) \sinh ^2\left (\frac{a+b x}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.18226, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5607, 3313, 12, 3303, 3298, 3301} \[ \frac{\sinh \left (\frac{2 b}{d}\right ) (b c-a d) \text{Chi}\left (\frac{2 (b c-a d)}{d (c+d x)}\right )}{d^2}-\frac{\cosh \left (\frac{2 b}{d}\right ) (b c-a d) \text{Shi}\left (\frac{2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac{(c+d x) \sinh ^2\left (\frac{a+b x}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 5607
Rule 3313
Rule 12
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \sinh ^2\left (\frac{a+b x}{c+d x}\right ) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\sinh ^2\left (\frac{b}{d}-\frac{(b c-a d) x}{d}\right )}{x^2} \, dx,x,\frac{1}{c+d x}\right )}{d}\\ &=\frac{(c+d x) \sinh ^2\left (\frac{a+b x}{c+d x}\right )}{d}-\frac{(2 i (b c-a d)) \operatorname{Subst}\left (\int \frac{i \sinh \left (\frac{2 b}{d}-\frac{2 (b c-a d) x}{d}\right )}{2 x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(c+d x) \sinh ^2\left (\frac{a+b x}{c+d x}\right )}{d}+\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{2 b}{d}-\frac{2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(c+d x) \sinh ^2\left (\frac{a+b x}{c+d x}\right )}{d}-\frac{\left ((b c-a d) \cosh \left (\frac{2 b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}+\frac{\left ((b c-a d) \sinh \left (\frac{2 b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(b c-a d) \text{Chi}\left (\frac{2 (b c-a d)}{d (c+d x)}\right ) \sinh \left (\frac{2 b}{d}\right )}{d^2}+\frac{(c+d x) \sinh ^2\left (\frac{a+b x}{c+d x}\right )}{d}-\frac{(b c-a d) \cosh \left (\frac{2 b}{d}\right ) \text{Shi}\left (\frac{2 (b c-a d)}{d (c+d x)}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.848931, size = 112, normalized size = 1.05 \[ \frac{2 \sinh \left (\frac{2 b}{d}\right ) (b c-a d) \text{Chi}\left (\frac{2 (a d-b c)}{d (c+d x)}\right )+2 \cosh \left (\frac{2 b}{d}\right ) (b c-a d) \text{Shi}\left (\frac{2 (a d-b c)}{d (c+d x)}\right )+d \left ((c+d x) \cosh \left (\frac{2 (a+b x)}{c+d x}\right )-d x\right )}{2 d^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.086, size = 358, normalized size = 3.4 \begin{align*} -{\frac{x}{2}}+{\frac{a}{4}{{\rm e}^{-2\,{\frac{bx+a}{dx+c}}}} \left ({\frac{da}{dx+c}}-{\frac{cb}{dx+c}} \right ) ^{-1}}-{\frac{cb}{4\,d}{{\rm e}^{-2\,{\frac{bx+a}{dx+c}}}} \left ({\frac{da}{dx+c}}-{\frac{cb}{dx+c}} \right ) ^{-1}}-{\frac{a}{2\,d}{{\rm e}^{-2\,{\frac{b}{d}}}}{\it Ei} \left ( 1,2\,{\frac{da-cb}{d \left ( dx+c \right ) }} \right ) }+{\frac{cb}{2\,{d}^{2}}{{\rm e}^{-2\,{\frac{b}{d}}}}{\it Ei} \left ( 1,2\,{\frac{da-cb}{d \left ( dx+c \right ) }} \right ) }+{\frac{dxa}{4\,da-4\,cb}{{\rm e}^{2\,{\frac{bx+a}{dx+c}}}}}-{\frac{bcx}{4\,da-4\,cb}{{\rm e}^{2\,{\frac{bx+a}{dx+c}}}}}+{\frac{ac}{4\,da-4\,cb}{{\rm e}^{2\,{\frac{bx+a}{dx+c}}}}}-{\frac{b{c}^{2}}{4\,d \left ( da-cb \right ) }{{\rm e}^{2\,{\frac{bx+a}{dx+c}}}}}+{\frac{a}{2\,d}{{\rm e}^{2\,{\frac{b}{d}}}}{\it Ei} \left ( 1,-2\,{\frac{da-cb}{d \left ( dx+c \right ) }} \right ) }-{\frac{cb}{2\,{d}^{2}}{{\rm e}^{2\,{\frac{b}{d}}}}{\it Ei} \left ( 1,-2\,{\frac{da-cb}{d \left ( dx+c \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, x + \frac{1}{4} \, \int e^{\left (\frac{2 \, b c}{d^{2} x + c d} - \frac{2 \, a}{d x + c} - \frac{2 \, b}{d}\right )}\,{d x} + \frac{1}{4} \, \int e^{\left (-\frac{2 \, b c}{d^{2} x + c d} + \frac{2 \, a}{d x + c} + \frac{2 \, b}{d}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06622, size = 775, normalized size = 7.24 \begin{align*} -\frac{d^{2} x -{\left (d^{2} x + c d\right )} \cosh \left (\frac{b x + a}{d x + c}\right )^{2} -{\left (d^{2} x -{\left (b c - a d\right )}{\rm Ei}\left (-\frac{2 \,{\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac{2 \, b}{d}\right ) + c d\right )} \sinh \left (\frac{b x + a}{d x + c}\right )^{2} -{\left ({\left (b c - a d\right )}{\rm Ei}\left (-\frac{2 \,{\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac{b x + a}{d x + c}\right )^{2} -{\left (b c - a d\right )}{\rm Ei}\left (\frac{2 \,{\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \cosh \left (\frac{2 \, b}{d}\right ) -{\left ({\left (b c - a d\right )}{\rm Ei}\left (-\frac{2 \,{\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac{b x + a}{d x + c}\right )^{2} -{\left (b c - a d\right )}{\rm Ei}\left (-\frac{2 \,{\left (b c - a d\right )}}{d^{2} x + c d}\right ) \sinh \left (\frac{b x + a}{d x + c}\right )^{2} +{\left (b c - a d\right )}{\rm Ei}\left (\frac{2 \,{\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \sinh \left (\frac{2 \, b}{d}\right )}{2 \,{\left (d^{2} \cosh \left (\frac{b x + a}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac{b x + a}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (\frac{b x + a}{d x + c}\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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