Optimal. Leaf size=270 \[ -\frac {b \cosh (c) \text {Chi}(d x)}{a^2}+\frac {b \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}-\frac {b \sinh (c) \text {Shi}(d x)}{a^2}-\frac {b \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a}-\frac {\cosh (c+d x)}{2 a x^2}-\frac {d \sinh (c+d x)}{2 a x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.50, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5293, 3297, 3303, 3298, 3301} \[ -\frac {b \cosh (c) \text {Chi}(d x)}{a^2}+\frac {b \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}-\frac {b \sinh (c) \text {Shi}(d x)}{a^2}-\frac {b \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a}-\frac {\cosh (c+d x)}{2 a x^2}-\frac {d \sinh (c+d x)}{2 a x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5293
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx &=\int \left (\frac {\cosh (c+d x)}{a x^3}-\frac {b \cosh (c+d x)}{a^2 x}+\frac {b^2 x \cosh (c+d x)}{a^2 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\cosh (c+d x)}{x^3} \, dx}{a}-\frac {b \int \frac {\cosh (c+d x)}{x} \, dx}{a^2}+\frac {b^2 \int \frac {x \cosh (c+d x)}{a+b x^2} \, dx}{a^2}\\ &=-\frac {\cosh (c+d x)}{2 a x^2}+\frac {b^2 \int \left (-\frac {\cosh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{a^2}+\frac {d \int \frac {\sinh (c+d x)}{x^2} \, dx}{2 a}-\frac {(b \cosh (c)) \int \frac {\cosh (d x)}{x} \, dx}{a^2}-\frac {(b \sinh (c)) \int \frac {\sinh (d x)}{x} \, dx}{a^2}\\ &=-\frac {\cosh (c+d x)}{2 a x^2}-\frac {b \cosh (c) \text {Chi}(d x)}{a^2}-\frac {d \sinh (c+d x)}{2 a x}-\frac {b \sinh (c) \text {Shi}(d x)}{a^2}-\frac {b^{3/2} \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 a^2}+\frac {b^{3/2} \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 a^2}+\frac {d^2 \int \frac {\cosh (c+d x)}{x} \, dx}{2 a}\\ &=-\frac {\cosh (c+d x)}{2 a x^2}-\frac {b \cosh (c) \text {Chi}(d x)}{a^2}-\frac {d \sinh (c+d x)}{2 a x}-\frac {b \sinh (c) \text {Shi}(d x)}{a^2}+\frac {\left (d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx}{2 a}+\frac {\left (b^{3/2} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 a^2}-\frac {\left (b^{3/2} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 a^2}+\frac {\left (d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx}{2 a}+\frac {\left (b^{3/2} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 a^2}+\frac {\left (b^{3/2} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 a^2}\\ &=-\frac {\cosh (c+d x)}{2 a x^2}-\frac {b \cosh (c) \text {Chi}(d x)}{a^2}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a}+\frac {b \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2}-\frac {d \sinh (c+d x)}{2 a x}-\frac {b \sinh (c) \text {Shi}(d x)}{a^2}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a}-\frac {b \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.57, size = 257, normalized size = 0.95 \[ \frac {-\left (x^2 \cosh (c) \left (2 b-a d^2\right ) \text {Chi}(d x)\right )+b x^2 \cosh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right )+b x^2 \cosh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+i b x^2 \sinh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-i b x^2 \sinh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+a d^2 x^2 \sinh (c) \text {Shi}(d x)-a d x \sinh (c+d x)-a \cosh (c+d x)-2 b x^2 \sinh (c) \text {Shi}(d x)}{2 a^2 x^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.49, size = 583, normalized size = 2.16 \[ -\frac {2 \, a d x \sinh \left (d x + c\right ) + 2 \, a \cosh \left (d x + c\right ) - {\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (a d^{2} - 2 \, b\right )} x^{2} {\rm Ei}\left (d x\right ) + {\left (a d^{2} - 2 \, b\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - {\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (a d^{2} - 2 \, b\right )} x^{2} {\rm Ei}\left (d x\right ) - {\left (a d^{2} - 2 \, b\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c) + {\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{4 \, {\left (a^{2} x^{2} \cosh \left (d x + c\right )^{2} - a^{2} x^{2} \sinh \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.12, size = 330, normalized size = 1.22 \[ \frac {d \,{\mathrm e}^{-d x -c}}{4 a x}-\frac {{\mathrm e}^{-d x -c}}{4 a \,x^{2}}-\frac {d^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{4 a}+\frac {{\mathrm e}^{-c} \Ei \left (1, d x \right ) b}{2 a^{2}}-\frac {b \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 a^{2}}-\frac {b \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 a^{2}}-\frac {d \,{\mathrm e}^{d x +c}}{4 a x}-\frac {{\mathrm e}^{d x +c}}{4 a \,x^{2}}-\frac {d^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{4 a}+\frac {{\mathrm e}^{c} \Ei \left (1, -d x \right ) b}{2 a^{2}}-\frac {b \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 a^{2}}-\frac {b \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x^3\,\left (b\,x^2+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\left (c + d x \right )}}{x^{3} \left (a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________