Optimal. Leaf size=68 \[ -\frac{a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^2}-\frac{a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^2}+\frac{\sinh (c+d x)}{b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.176147, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6742, 2637, 3303, 3298, 3301} \[ -\frac{a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^2}-\frac{a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^2}+\frac{\sinh (c+d x)}{b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 2637
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x \cosh (c+d x)}{a+b x} \, dx &=\int \left (\frac{\cosh (c+d x)}{b}-\frac{a \cosh (c+d x)}{b (a+b x)}\right ) \, dx\\ &=\frac{\int \cosh (c+d x) \, dx}{b}-\frac{a \int \frac{\cosh (c+d x)}{a+b x} \, dx}{b}\\ &=\frac{\sinh (c+d x)}{b d}-\frac{\left (a \cosh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cosh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b}-\frac{\left (a \sinh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sinh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b}\\ &=-\frac{a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^2}+\frac{\sinh (c+d x)}{b d}-\frac{a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.135784, size = 64, normalized size = 0.94 \[ \frac{-a d \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )-a d \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+b \sinh (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 114, normalized size = 1.7 \begin{align*} -{\frac{{{\rm e}^{-dx-c}}}{2\,bd}}+{\frac{a}{2\,{b}^{2}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{{{\rm e}^{dx+c}}}{2\,bd}}+{\frac{a}{2\,{b}^{2}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.33287, size = 211, normalized size = 3.1 \begin{align*} \frac{1}{2} \, d{\left (\frac{a{\left (\frac{e^{\left (-c + \frac{a d}{b}\right )} E_{1}\left (\frac{{\left (b x + a\right )} d}{b}\right )}{b} + \frac{e^{\left (c - \frac{a d}{b}\right )} E_{1}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{b d} - \frac{\frac{{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac{{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}}{b} + \frac{2 \, a \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{2} d}\right )} +{\left (\frac{x}{b} - \frac{a \log \left (b x + a\right )}{b^{2}}\right )} \cosh \left (d x + c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.9967, size = 251, normalized size = 3.69 \begin{align*} -\frac{{\left (a d{\rm Ei}\left (\frac{b d x + a d}{b}\right ) + a d{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) - 2 \, b \sinh \left (d x + c\right ) -{\left (a d{\rm Ei}\left (\frac{b d x + a d}{b}\right ) - a d{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{2 \, b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \cosh{\left (c + d x \right )}}{a + b x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18626, size = 78, normalized size = 1.15 \begin{align*} -\frac{a{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} + a{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )}}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]