Optimal. Leaf size=80 \[ \frac{b c \sinh \left (\frac{2 b}{d}\right ) \text{Chi}\left (\frac{2 b c}{d (c+d x)}\right )}{d^2}-\frac{b c \cosh \left (\frac{2 b}{d}\right ) \text{Shi}\left (\frac{2 b c}{d (c+d x)}\right )}{d^2}+\frac{(c+d x) \sinh ^2\left (\frac{b x}{c+d x}\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.145793, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {5607, 3313, 12, 3303, 3298, 3301} \[ \frac{b c \sinh \left (\frac{2 b}{d}\right ) \text{Chi}\left (\frac{2 b c}{d (c+d x)}\right )}{d^2}-\frac{b c \cosh \left (\frac{2 b}{d}\right ) \text{Shi}\left (\frac{2 b c}{d (c+d x)}\right )}{d^2}+\frac{(c+d x) \sinh ^2\left (\frac{b x}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5607
Rule 3313
Rule 12
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \sinh ^2\left (\frac{b x}{c+d x}\right ) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\sinh ^2\left (\frac{b}{d}-\frac{b c x}{d}\right )}{x^2} \, dx,x,\frac{1}{c+d x}\right )}{d}\\ &=\frac{(c+d x) \sinh ^2\left (\frac{b x}{c+d x}\right )}{d}-\frac{(2 i b c) \operatorname{Subst}\left (\int \frac{i \sinh \left (\frac{2 b}{d}-\frac{2 b c x}{d}\right )}{2 x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(c+d x) \sinh ^2\left (\frac{b x}{c+d x}\right )}{d}+\frac{(b c) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{2 b}{d}-\frac{2 b c x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(c+d x) \sinh ^2\left (\frac{b x}{c+d x}\right )}{d}-\frac{\left (b c \cosh \left (\frac{2 b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{2 b c x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}+\frac{\left (b c \sinh \left (\frac{2 b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{2 b c x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{b c \text{Chi}\left (\frac{2 b c}{d (c+d x)}\right ) \sinh \left (\frac{2 b}{d}\right )}{d^2}+\frac{(c+d x) \sinh ^2\left (\frac{b x}{c+d x}\right )}{d}-\frac{b c \cosh \left (\frac{2 b}{d}\right ) \text{Shi}\left (\frac{2 b c}{d (c+d x)}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.358976, size = 85, normalized size = 1.06 \[ \frac{2 b c \sinh \left (\frac{2 b}{d}\right ) \text{Chi}\left (\frac{2 b c}{d (c+d x)}\right )-2 b c \cosh \left (\frac{2 b}{d}\right ) \text{Shi}\left (\frac{2 b c}{d (c+d x)}\right )+d \left ((c+d x) \cosh \left (\frac{2 b x}{c+d x}\right )-d x\right )}{2 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.087, size = 120, normalized size = 1.5 \begin{align*} -{\frac{x}{2}}+{\frac{dx+c}{4\,d}{{\rm e}^{-2\,{\frac{bx}{dx+c}}}}}+{\frac{cb}{2\,{d}^{2}}{{\rm e}^{-2\,{\frac{b}{d}}}}{\it Ei} \left ( 1,-2\,{\frac{cb}{d \left ( dx+c \right ) }} \right ) }+{\frac{x}{4}{{\rm e}^{2\,{\frac{bx}{dx+c}}}}}+{\frac{c}{4\,d}{{\rm e}^{2\,{\frac{bx}{dx+c}}}}}-{\frac{cb}{2\,{d}^{2}}{{\rm e}^{2\,{\frac{b}{d}}}}{\it Ei} \left ( 1,2\,{\frac{cb}{d \left ( dx+c \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, b c \int \frac{x e^{\left (\frac{2 \, b c}{d^{2} x + c d}\right )}}{d^{2} x^{2} e^{\left (\frac{2 \, b}{d}\right )} + 2 \, c d x e^{\left (\frac{2 \, b}{d}\right )} + c^{2} e^{\left (\frac{2 \, b}{d}\right )}}\,{d x} - \frac{1}{2} \, b c \int \frac{x e^{\left (-\frac{2 \, b c}{d^{2} x + c d} + \frac{2 \, b}{d}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} + \frac{1}{4} \,{\left (x e^{\left (\frac{2 \, b c}{d^{2} x + c d}\right )} + x e^{\left (-\frac{2 \, b c}{d^{2} x + c d} + \frac{4 \, b}{d}\right )}\right )} e^{\left (-\frac{2 \, b}{d}\right )} - \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.11719, size = 589, normalized size = 7.36 \begin{align*} -\frac{d^{2} x -{\left (d^{2} x + c d\right )} \cosh \left (\frac{b x}{d x + c}\right )^{2} +{\left (b c{\rm Ei}\left (-\frac{2 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac{2 \, b}{d}\right ) - d^{2} x - c d\right )} \sinh \left (\frac{b x}{d x + c}\right )^{2} -{\left (b c{\rm Ei}\left (-\frac{2 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac{b x}{d x + c}\right )^{2} - b c{\rm Ei}\left (\frac{2 \, b c}{d^{2} x + c d}\right )\right )} \cosh \left (\frac{2 \, b}{d}\right ) -{\left (b c{\rm Ei}\left (-\frac{2 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac{b x}{d x + c}\right )^{2} - b c{\rm Ei}\left (-\frac{2 \, b c}{d^{2} x + c d}\right ) \sinh \left (\frac{b x}{d x + c}\right )^{2} + b c{\rm Ei}\left (\frac{2 \, b c}{d^{2} x + c d}\right )\right )} \sinh \left (\frac{2 \, b}{d}\right )}{2 \,{\left (d^{2} \cosh \left (\frac{b x}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac{b x}{d x + c}\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (\frac{b x}{d x + c}\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]