Optimal. Leaf size=88 \[ \frac {1}{2} a d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a d^2 \sinh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{2 x^2}-\frac {a d \sinh (c+d x)}{2 x}+b d \sinh (c) \text {Chi}(d x)+b d \cosh (c) \text {Shi}(d x)-\frac {b \cosh (c+d x)}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.28, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 3297, 3303, 3298, 3301} \[ \frac {1}{2} a d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a d^2 \sinh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{2 x^2}-\frac {a d \sinh (c+d x)}{2 x}+b d \sinh (c) \text {Chi}(d x)+b d \cosh (c) \text {Shi}(d x)-\frac {b \cosh (c+d x)}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {(a+b x) \cosh (c+d x)}{x^3} \, dx &=\int \left (\frac {a \cosh (c+d x)}{x^3}+\frac {b \cosh (c+d x)}{x^2}\right ) \, dx\\ &=a \int \frac {\cosh (c+d x)}{x^3} \, dx+b \int \frac {\cosh (c+d x)}{x^2} \, dx\\ &=-\frac {a \cosh (c+d x)}{2 x^2}-\frac {b \cosh (c+d x)}{x}+\frac {1}{2} (a d) \int \frac {\sinh (c+d x)}{x^2} \, dx+(b d) \int \frac {\sinh (c+d x)}{x} \, dx\\ &=-\frac {a \cosh (c+d x)}{2 x^2}-\frac {b \cosh (c+d x)}{x}-\frac {a d \sinh (c+d x)}{2 x}+\frac {1}{2} \left (a d^2\right ) \int \frac {\cosh (c+d x)}{x} \, dx+(b d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx+(b d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx\\ &=-\frac {a \cosh (c+d x)}{2 x^2}-\frac {b \cosh (c+d x)}{x}+b d \text {Chi}(d x) \sinh (c)-\frac {a d \sinh (c+d x)}{2 x}+b d \cosh (c) \text {Shi}(d x)+\frac {1}{2} \left (a d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\frac {1}{2} \left (a d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {a \cosh (c+d x)}{2 x^2}-\frac {b \cosh (c+d x)}{x}+\frac {1}{2} a d^2 \cosh (c) \text {Chi}(d x)+b d \text {Chi}(d x) \sinh (c)-\frac {a d \sinh (c+d x)}{2 x}+b d \cosh (c) \text {Shi}(d x)+\frac {1}{2} a d^2 \sinh (c) \text {Shi}(d x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 78, normalized size = 0.89 \[ \frac {d x^2 \text {Chi}(d x) (a d \cosh (c)+2 b \sinh (c))+d x^2 \text {Shi}(d x) (a d \sinh (c)+2 b \cosh (c))-a d x \sinh (c+d x)-a \cosh (c+d x)-2 b x \cosh (c+d x)}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 116, normalized size = 1.32 \[ -\frac {2 \, a d x \sinh \left (d x + c\right ) + 2 \, {\left (2 \, b x + a\right )} \cosh \left (d x + c\right ) - {\left ({\left (a d^{2} + 2 \, b d\right )} x^{2} {\rm Ei}\left (d x\right ) + {\left (a d^{2} - 2 \, b d\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - {\left ({\left (a d^{2} + 2 \, b d\right )} x^{2} {\rm Ei}\left (d x\right ) - {\left (a d^{2} - 2 \, b d\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 134, normalized size = 1.52 \[ \frac {a d^{2} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d^{2} x^{2} {\rm Ei}\left (d x\right ) e^{c} - 2 \, b d x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 2 \, b d x^{2} {\rm Ei}\left (d x\right ) e^{c} - a d x e^{\left (d x + c\right )} + a d x e^{\left (-d x - c\right )} - 2 \, b x e^{\left (d x + c\right )} - 2 \, b x e^{\left (-d x - c\right )} - a e^{\left (d x + c\right )} - a e^{\left (-d x - c\right )}}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 139, normalized size = 1.58 \[ \frac {d a \,{\mathrm e}^{-d x -c}}{4 x}-\frac {a \,{\mathrm e}^{-d x -c}}{4 x^{2}}-\frac {d^{2} a \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{4}-\frac {b \,{\mathrm e}^{-d x -c}}{2 x}+\frac {d b \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {a \,{\mathrm e}^{d x +c}}{4 x^{2}}-\frac {d a \,{\mathrm e}^{d x +c}}{4 x}-\frac {d^{2} a \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{4}-\frac {b \,{\mathrm e}^{d x +c}}{2 x}-\frac {d b \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 66, normalized size = 0.75 \[ \frac {1}{4} \, {\left (a d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + a d e^{c} \Gamma \left (-1, -d x\right ) - 2 \, b {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 2 \, b {\rm Ei}\left (d x\right ) e^{c}\right )} d - \frac {{\left (2 \, b x + a\right )} \cosh \left (d x + c\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (a+b\,x\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________