Optimal. Leaf size=56 \[ a \cosh (c) \text {Chi}(d x)+a \sinh (c) \text {Shi}(d x)+\frac {2 b \sinh (c+d x)}{d^3}-\frac {2 b x \cosh (c+d x)}{d^2}+\frac {b x^2 \sinh (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5287, 3303, 3298, 3301, 3296, 2637} \[ a \cosh (c) \text {Chi}(d x)+a \sinh (c) \text {Shi}(d x)+\frac {2 b \sinh (c+d x)}{d^3}-\frac {2 b x \cosh (c+d x)}{d^2}+\frac {b x^2 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2637
Rule 3296
Rule 3298
Rule 3301
Rule 3303
Rule 5287
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x} \, dx &=\int \left (\frac {a \cosh (c+d x)}{x}+b x^2 \cosh (c+d x)\right ) \, dx\\ &=a \int \frac {\cosh (c+d x)}{x} \, dx+b \int x^2 \cosh (c+d x) \, dx\\ &=\frac {b x^2 \sinh (c+d x)}{d}-\frac {(2 b) \int x \sinh (c+d x) \, dx}{d}+(a \cosh (c)) \int \frac {\cosh (d x)}{x} \, dx+(a \sinh (c)) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {2 b x \cosh (c+d x)}{d^2}+a \cosh (c) \text {Chi}(d x)+\frac {b x^2 \sinh (c+d x)}{d}+a \sinh (c) \text {Shi}(d x)+\frac {(2 b) \int \cosh (c+d x) \, dx}{d^2}\\ &=-\frac {2 b x \cosh (c+d x)}{d^2}+a \cosh (c) \text {Chi}(d x)+\frac {2 b \sinh (c+d x)}{d^3}+\frac {b x^2 \sinh (c+d x)}{d}+a \sinh (c) \text {Shi}(d x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 49, normalized size = 0.88 \[ a \cosh (c) \text {Chi}(d x)+a \sinh (c) \text {Shi}(d x)+\frac {b \left (\left (d^2 x^2+2\right ) \sinh (c+d x)-2 d x \cosh (c+d x)\right )}{d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.55, size = 86, normalized size = 1.54 \[ -\frac {4 \, b d x \cosh \left (d x + c\right ) - {\left (a d^{3} {\rm Ei}\left (d x\right ) + a d^{3} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - 2 \, {\left (b d^{2} x^{2} + 2 \, b\right )} \sinh \left (d x + c\right ) - {\left (a d^{3} {\rm Ei}\left (d x\right ) - a d^{3} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{2 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 109, normalized size = 1.95 \[ \frac {b d^{2} x^{2} e^{\left (d x + c\right )} - b d^{2} x^{2} e^{\left (-d x - c\right )} + a d^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d^{3} {\rm Ei}\left (d x\right ) e^{c} - 2 \, b d x e^{\left (d x + c\right )} - 2 \, b d x e^{\left (-d x - c\right )} + 2 \, b e^{\left (d x + c\right )} - 2 \, b e^{\left (-d x - c\right )}}{2 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.12, size = 113, normalized size = 2.02 \[ -\frac {b \,{\mathrm e}^{d x +c} x}{d^{2}}+\frac {b \,{\mathrm e}^{d x +c} x^{2}}{2 d}-\frac {b \,{\mathrm e}^{-d x -c} x^{2}}{2 d}-\frac {b \,{\mathrm e}^{-d x -c} x}{d^{2}}-\frac {a \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}-\frac {b \,{\mathrm e}^{-d x -c}}{d^{3}}-\frac {a \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}+\frac {b \,{\mathrm e}^{d x +c}}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.40, size = 140, normalized size = 2.50 \[ -\frac {1}{6} \, {\left (b {\left (\frac {{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} e^{\left (d x\right )}}{d^{4}} + \frac {{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} e^{\left (-d x - c\right )}}{d^{4}}\right )} + \frac {2 \, a \cosh \left (d x + c\right ) \log \left (x^{3}\right )}{d} - \frac {3 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a}{d}\right )} d + \frac {1}{3} \, {\left (b x^{3} + a \log \left (x^{3}\right )\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ a\,\mathrm {coshint}\left (d\,x\right )\,\mathrm {cosh}\relax (c)+a\,\mathrm {sinhint}\left (d\,x\right )\,\mathrm {sinh}\relax (c)+\frac {b\,\left (2\,\mathrm {sinh}\left (c+d\,x\right )+d^2\,x^2\,\mathrm {sinh}\left (c+d\,x\right )-2\,d\,x\,\mathrm {cosh}\left (c+d\,x\right )\right )}{d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.91, size = 66, normalized size = 1.18 \[ a \sinh {\relax (c )} \operatorname {Shi}{\left (d x \right )} + a \cosh {\relax (c )} \operatorname {Chi}\left (d x\right ) + b \left (\begin {cases} \frac {x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 \sinh {\left (c + d x \right )}}{d^{3}} & \text {for}\: d \neq 0 \\\frac {x^{3} \cosh {\relax (c )}}{3} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________