Optimal. Leaf size=147 \[ \frac{a^2 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{a^2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^4}-\frac{a^2 \cosh (c+d x)}{b^3 (a+b x)}-\frac{2 a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^3}-\frac{2 a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^3}+\frac{\sinh (c+d x)}{b^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.374579, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {6742, 2637, 3297, 3303, 3298, 3301} \[ \frac{a^2 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{a^2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^4}-\frac{a^2 \cosh (c+d x)}{b^3 (a+b x)}-\frac{2 a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^3}-\frac{2 a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^3}+\frac{\sinh (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 2637
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^2 \cosh (c+d x)}{(a+b x)^2} \, dx &=\int \left (\frac{\cosh (c+d x)}{b^2}+\frac{a^2 \cosh (c+d x)}{b^2 (a+b x)^2}-\frac{2 a \cosh (c+d x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{\int \cosh (c+d x) \, dx}{b^2}-\frac{(2 a) \int \frac{\cosh (c+d x)}{a+b x} \, dx}{b^2}+\frac{a^2 \int \frac{\cosh (c+d x)}{(a+b x)^2} \, dx}{b^2}\\ &=-\frac{a^2 \cosh (c+d x)}{b^3 (a+b x)}+\frac{\sinh (c+d x)}{b^2 d}+\frac{\left (a^2 d\right ) \int \frac{\sinh (c+d x)}{a+b x} \, dx}{b^3}-\frac{\left (2 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^2}-\frac{\left (2 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^2}\\ &=-\frac{a^2 \cosh (c+d x)}{b^3 (a+b x)}-\frac{2 a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{\sinh (c+d x)}{b^2 d}-\frac{2 a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{\left (a^2 d \cosh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sinh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b^3}+\frac{\left (a^2 d \sinh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cosh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b^3}\\ &=-\frac{a^2 \cosh (c+d x)}{b^3 (a+b x)}-\frac{2 a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{a^2 d \text{Chi}\left (\frac{a d}{b}+d x\right ) \sinh \left (c-\frac{a d}{b}\right )}{b^4}+\frac{\sinh (c+d x)}{b^2 d}+\frac{a^2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^4}-\frac{2 a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.692046, size = 115, normalized size = 0.78 \[ \frac{b \left (\frac{b \sinh (c+d x)}{d}-\frac{a^2 \cosh (c+d x)}{a+b x}\right )+a \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \sinh \left (c-\frac{a d}{b}\right )-2 b \cosh \left (c-\frac{a d}{b}\right )\right )+a \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \cosh \left (c-\frac{a d}{b}\right )-2 b \sinh \left (c-\frac{a d}{b}\right )\right )}{b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.039, size = 254, normalized size = 1.7 \begin{align*} -{\frac{{{\rm e}^{-dx-c}}}{2\,d{b}^{2}}}-{\frac{d{{\rm e}^{-dx-c}}{a}^{2}}{2\,{b}^{3} \left ( bdx+da \right ) }}+{\frac{d{a}^{2}}{2\,{b}^{4}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{a}{{b}^{3}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{{{\rm e}^{dx+c}}}{2\,d{b}^{2}}}-{\frac{d{{\rm e}^{dx+c}}{a}^{2}}{2\,{b}^{4}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}-{\frac{d{a}^{2}}{2\,{b}^{4}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }+{\frac{a}{{b}^{3}}{{\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 [A] time = 1.34762, size = 319, normalized size = 2.17 \begin{align*} \frac{1}{2} \,{\left (a^{2}{\left (\frac{e^{\left (-c + \frac{a d}{b}\right )} E_{1}\left (\frac{{\left (b x + a\right )} d}{b}\right )}{b^{4}} - \frac{e^{\left (c - \frac{a d}{b}\right )} E_{1}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{b^{4}}\right )} + \frac{2 \, 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^{2} 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^{2}} + \frac{4 \, a \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{3} d}\right )} d -{\left (\frac{a^{2}}{b^{4} x + a b^{3}} - \frac{x}{b^{2}} + \frac{2 \, a \log \left (b x + a\right )}{b^{3}}\right )} \cosh \left (d x + c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.99239, size = 570, normalized size = 3.88 \begin{align*} -\frac{2 \, a^{2} b d \cosh \left (d x + c\right ) -{\left ({\left (a^{3} d^{2} - 2 \, a^{2} b d +{\left (a^{2} b d^{2} - 2 \, a b^{2} d\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\left (a^{3} d^{2} + 2 \, a^{2} b d +{\left (a^{2} b d^{2} + 2 \, a b^{2} d\right )} x\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) - 2 \,{\left (b^{3} x + a b^{2}\right )} \sinh \left (d x + c\right ) +{\left ({\left (a^{3} d^{2} - 2 \, a^{2} b d +{\left (a^{2} b d^{2} - 2 \, a b^{2} d\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\left (a^{3} d^{2} + 2 \, a^{2} b d +{\left (a^{2} b d^{2} + 2 \, a b^{2} d\right )} x\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{2 \,{\left (b^{5} d x + a b^{4} d\right )}} \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^{2} \cosh{\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22872, size = 387, normalized size = 2.63 \begin{align*} \frac{a^{2} b d x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a^{2} b d x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + a^{3} d{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - 2 \, a b^{2} x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a^{3} d{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - 2 \, a b^{2} x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - 2 \, a^{2} b{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - 2 \, a^{2} b{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - a^{2} b e^{\left (d x + c\right )} - a^{2} b e^{\left (-d x - c\right )}}{2 \,{\left (b^{5} x + a b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]