Optimal. Leaf size=101 \[ \frac{\sinh \left (\frac{b}{d}\right ) (b c-a d) \text{Chi}\left (\frac{b c-a d}{d (c+d x)}\right )}{d^2}-\frac{\cosh \left (\frac{b}{d}\right ) (b c-a d) \text{Shi}\left (\frac{b c-a d}{d (c+d x)}\right )}{d^2}+\frac{(c+d x) \cosh \left (\frac{a+b x}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.176673, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5608, 3297, 3303, 3298, 3301} \[ \frac{\sinh \left (\frac{b}{d}\right ) (b c-a d) \text{Chi}\left (\frac{b c-a d}{d (c+d x)}\right )}{d^2}-\frac{\cosh \left (\frac{b}{d}\right ) (b c-a d) \text{Shi}\left (\frac{b c-a d}{d (c+d x)}\right )}{d^2}+\frac{(c+d x) \cosh \left (\frac{a+b x}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 5608
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \cosh \left (\frac{a+b x}{c+d x}\right ) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cosh \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) \cosh \left (\frac{a+b x}{c+d x}\right )}{d}+\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{b}{d}-\frac{(b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(c+d x) \cosh \left (\frac{a+b x}{c+d x}\right )}{d}-\frac{\left ((b c-a d) \cosh \left (\frac{b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{(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{b}{d}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{(b c-a d) x}{d}\right )}{x} \, dx,x,\frac{1}{c+d x}\right )}{d^2}\\ &=\frac{(c+d x) \cosh \left (\frac{a+b x}{c+d x}\right )}{d}+\frac{(b c-a d) \text{Chi}\left (\frac{b c-a d}{d (c+d x)}\right ) \sinh \left (\frac{b}{d}\right )}{d^2}-\frac{(b c-a d) \cosh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c-a d}{d (c+d x)}\right )}{d^2}\\ \end{align*}
Mathematica [B] time = 0.351606, size = 373, normalized size = 3.69 \[ \frac{(b c-a d) \left (\sinh \left (\frac{b}{d}\right )-\cosh \left (\frac{b}{d}\right )\right ) \text{Chi}\left (\frac{b c-a d}{x d^2+c d}\right )+(b c-a d) \left (\sinh \left (\frac{b}{d}\right )+\cosh \left (\frac{b}{d}\right )\right ) \text{Chi}\left (\frac{a d-b c}{d (c+d x)}\right )-a d \sinh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c-a d}{x d^2+c d}\right )+b c \sinh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c-a d}{x d^2+c d}\right )+a d \cosh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c-a d}{x d^2+c d}\right )-b c \cosh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{b c-a d}{x d^2+c d}\right )+2 d^2 x \cosh \left (\frac{a+b x}{c+d x}\right )-a d \sinh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{a d-b c}{d (c+d x)}\right )+b c \sinh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{a d-b c}{d (c+d x)}\right )-a d \cosh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{a d-b c}{d (c+d x)}\right )+b c \cosh \left (\frac{b}{d}\right ) \text{Shi}\left (\frac{a d-b c}{d (c+d x)}\right )+2 c d \cosh \left (\frac{a+b x}{c+d x}\right )}{2 d^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 347, normalized size = 3.4 \begin{align*}{\frac{a}{2}{{\rm e}^{-{\frac{bx+a}{dx+c}}}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}} \right ) ^{-1}}-{\frac{bc}{2\,d}{{\rm e}^{-{\frac{bx+a}{dx+c}}}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}} \right ) ^{-1}}-{\frac{a}{2\,d}{{\rm e}^{-{\frac{b}{d}}}}{\it Ei} \left ( 1,{\frac{ad-bc}{d \left ( dx+c \right ) }} \right ) }+{\frac{bc}{2\,{d}^{2}}{{\rm e}^{-{\frac{b}{d}}}}{\it Ei} \left ( 1,{\frac{ad-bc}{d \left ( dx+c \right ) }} \right ) }+{\frac{dxa}{2\,ad-2\,bc}{{\rm e}^{{\frac{bx+a}{dx+c}}}}}-{\frac{bcx}{2\,ad-2\,bc}{{\rm e}^{{\frac{bx+a}{dx+c}}}}}+{\frac{ac}{2\,ad-2\,bc}{{\rm e}^{{\frac{bx+a}{dx+c}}}}}-{\frac{{c}^{2}b}{2\,d \left ( ad-bc \right ) }{{\rm e}^{{\frac{bx+a}{dx+c}}}}}+{\frac{a}{2\,d}{{\rm e}^{{\frac{b}{d}}}}{\it Ei} \left ( 1,-{\frac{ad-bc}{d \left ( dx+c \right ) }} \right ) }-{\frac{bc}{2\,{d}^{2}}{{\rm e}^{{\frac{b}{d}}}}{\it Ei} \left ( 1,-{\frac{ad-bc}{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*} \int \cosh \left (\frac{b x + a}{d x + c}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9023, size = 344, normalized size = 3.41 \begin{align*} \frac{2 \,{\left (d^{2} x + c d\right )} \cosh \left (\frac{b x + a}{d x + c}\right ) -{\left ({\left (b c - a d\right )}{\rm Ei}\left (\frac{b c - a d}{d^{2} x + c d}\right ) -{\left (b c - a d\right )}{\rm Ei}\left (-\frac{b c - a d}{d^{2} x + c d}\right )\right )} \cosh \left (\frac{b}{d}\right ) +{\left ({\left (b c - a d\right )}{\rm Ei}\left (\frac{b c - a d}{d^{2} x + c d}\right ) +{\left (b c - a d\right )}{\rm Ei}\left (-\frac{b c - a d}{d^{2} x + c d}\right )\right )} \sinh \left (\frac{b}{d}\right )}{2 \, d^{2}} \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 \cosh \left (\frac{b x + a}{d x + c}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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