Optimal. Leaf size=172 \[ \frac {1}{6} a^2 d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+a b d^2 \cosh (c) \text {Chi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)-\frac {a b \cosh (c+d x)}{x^2}-\frac {a b d \sinh (c+d x)}{x}+b^2 d \sinh (c) \text {Chi}(d x)+b^2 d \cosh (c) \text {Shi}(d x)-\frac {b^2 \cosh (c+d x)}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.43, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6742, 3297, 3303, 3298, 3301} \[ \frac {1}{6} a^2 d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {a^2 d \sinh (c+d x)}{6 x^2}-\frac {a^2 \cosh (c+d x)}{3 x^3}+a b d^2 \cosh (c) \text {Chi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)-\frac {a b \cosh (c+d x)}{x^2}-\frac {a b d \sinh (c+d x)}{x}+b^2 d \sinh (c) \text {Chi}(d x)+b^2 d \cosh (c) \text {Shi}(d x)-\frac {b^2 \cosh (c+d x)}{x} \]
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)^2 \cosh (c+d x)}{x^4} \, dx &=\int \left (\frac {a^2 \cosh (c+d x)}{x^4}+\frac {2 a b \cosh (c+d x)}{x^3}+\frac {b^2 \cosh (c+d x)}{x^2}\right ) \, dx\\ &=a^2 \int \frac {\cosh (c+d x)}{x^4} \, dx+(2 a b) \int \frac {\cosh (c+d x)}{x^3} \, dx+b^2 \int \frac {\cosh (c+d x)}{x^2} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a b \cosh (c+d x)}{x^2}-\frac {b^2 \cosh (c+d x)}{x}+\frac {1}{3} \left (a^2 d\right ) \int \frac {\sinh (c+d x)}{x^3} \, dx+(a b d) \int \frac {\sinh (c+d x)}{x^2} \, dx+\left (b^2 d\right ) \int \frac {\sinh (c+d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a b \cosh (c+d x)}{x^2}-\frac {b^2 \cosh (c+d x)}{x}-\frac {a^2 d \sinh (c+d x)}{6 x^2}-\frac {a b d \sinh (c+d x)}{x}+\frac {1}{6} \left (a^2 d^2\right ) \int \frac {\cosh (c+d x)}{x^2} \, dx+\left (a b d^2\right ) \int \frac {\cosh (c+d x)}{x} \, dx+\left (b^2 d \cosh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx+\left (b^2 d \sinh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a b \cosh (c+d x)}{x^2}-\frac {b^2 \cosh (c+d x)}{x}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}+b^2 d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{6 x^2}-\frac {a b d \sinh (c+d x)}{x}+b^2 d \cosh (c) \text {Shi}(d x)+\frac {1}{6} \left (a^2 d^3\right ) \int \frac {\sinh (c+d x)}{x} \, dx+\left (a b d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\left (a b d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a b \cosh (c+d x)}{x^2}-\frac {b^2 \cosh (c+d x)}{x}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}+a b d^2 \cosh (c) \text {Chi}(d x)+b^2 d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{6 x^2}-\frac {a b d \sinh (c+d x)}{x}+b^2 d \cosh (c) \text {Shi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{6} \left (a^2 d^3 \cosh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx+\frac {1}{6} \left (a^2 d^3 \sinh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a b \cosh (c+d x)}{x^2}-\frac {b^2 \cosh (c+d x)}{x}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}+a b d^2 \cosh (c) \text {Chi}(d x)+b^2 d \text {Chi}(d x) \sinh (c)+\frac {1}{6} a^2 d^3 \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{6 x^2}-\frac {a b d \sinh (c+d x)}{x}+b^2 d \cosh (c) \text {Shi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.49, size = 154, normalized size = 0.90 \[ -\frac {-d x^3 \text {Chi}(d x) \left (\sinh (c) \left (a^2 d^2+6 b^2\right )+6 a b d \cosh (c)\right )-d x^3 \text {Shi}(d x) \left (a^2 d^2 \cosh (c)+6 a b d \sinh (c)+6 b^2 \cosh (c)\right )+a^2 d^2 x^2 \cosh (c+d x)+a^2 d x \sinh (c+d x)+2 a^2 \cosh (c+d x)+6 a b d x^2 \sinh (c+d x)+6 a b x \cosh (c+d x)+6 b^2 x^2 \cosh (c+d x)}{6 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 194, normalized size = 1.13 \[ -\frac {2 \, {\left (6 \, a b x + {\left (a^{2} d^{2} + 6 \, b^{2}\right )} x^{2} + 2 \, a^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{3} + 6 \, a b d^{2} + 6 \, b^{2} d\right )} x^{3} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{3} - 6 \, a b d^{2} + 6 \, b^{2} d\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) + 2 \, {\left (6 \, a b d x^{2} + a^{2} d x\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{3} + 6 \, a b d^{2} + 6 \, b^{2} d\right )} x^{3} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{3} - 6 \, a b d^{2} + 6 \, b^{2} d\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{12 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 285, normalized size = 1.66 \[ -\frac {a^{2} d^{3} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d^{3} x^{3} {\rm Ei}\left (d x\right ) e^{c} - 6 \, a b d^{2} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 6 \, a b d^{2} x^{3} {\rm Ei}\left (d x\right ) e^{c} + 6 \, b^{2} d x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 6 \, b^{2} d x^{3} {\rm Ei}\left (d x\right ) e^{c} + a^{2} d^{2} x^{2} e^{\left (d x + c\right )} + a^{2} d^{2} x^{2} e^{\left (-d x - c\right )} + 6 \, a b d x^{2} e^{\left (d x + c\right )} - 6 \, a b d x^{2} e^{\left (-d x - c\right )} + a^{2} d x e^{\left (d x + c\right )} + 6 \, b^{2} x^{2} e^{\left (d x + c\right )} - a^{2} d x e^{\left (-d x - c\right )} + 6 \, b^{2} x^{2} e^{\left (-d x - c\right )} + 6 \, a b x e^{\left (d x + c\right )} + 6 \, a b x e^{\left (-d x - c\right )} + 2 \, a^{2} e^{\left (d x + c\right )} + 2 \, a^{2} e^{\left (-d x - c\right )}}{12 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.13, size = 287, normalized size = 1.67 \[ -\frac {d^{2} a^{2} {\mathrm e}^{-d x -c}}{12 x}+\frac {d \,a^{2} {\mathrm e}^{-d x -c}}{12 x^{2}}-\frac {a^{2} {\mathrm e}^{-d x -c}}{6 x^{3}}-\frac {b^{2} {\mathrm e}^{-d x -c}}{2 x}+\frac {d \,b^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}+\frac {d a b \,{\mathrm e}^{-d x -c}}{2 x}-\frac {a b \,{\mathrm e}^{-d x -c}}{2 x^{2}}-\frac {d^{2} a b \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}+\frac {d^{3} a^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{12}-\frac {a^{2} {\mathrm e}^{d x +c}}{6 x^{3}}-\frac {a b \,{\mathrm e}^{d x +c}}{2 x^{2}}-\frac {d a b \,{\mathrm e}^{d x +c}}{2 x}-\frac {d^{2} a b \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}-\frac {d^{2} a^{2} {\mathrm e}^{d x +c}}{12 x}-\frac {d^{3} a^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{12}-\frac {d \,a^{2} {\mathrm e}^{d x +c}}{12 x^{2}}-\frac {b^{2} {\mathrm e}^{d x +c}}{2 x}-\frac {d \,b^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 117, normalized size = 0.68 \[ \frac {1}{6} \, {\left (a^{2} d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - a^{2} d^{2} e^{c} \Gamma \left (-2, -d x\right ) + 3 \, a b d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + 3 \, a b d e^{c} \Gamma \left (-1, -d x\right ) - 3 \, b^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 3 \, b^{2} {\rm Ei}\left (d x\right ) e^{c}\right )} d - \frac {{\left (3 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )} \cosh \left (d x + c\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (a+b\,x\right )}^2}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x\right )^{2} \cosh {\left (c + d x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________