Optimal. Leaf size=791 \[ -\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {3 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^4}+\frac {3 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^4}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}-\frac {3 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^4}+\frac {3 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^4}-\frac {d^2 \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^3}+\frac {d^2 \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^3}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}-\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}+\frac {9 \sqrt {b} d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\frac {9 \sqrt {b} d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}+\frac {9 \sqrt {b} d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}+\frac {9 \sqrt {b} d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}} \]
[Out]
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Rubi [A] time = 1.87, antiderivative size = 791, normalized size of antiderivative = 1.00, number of steps used = 46, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5293, 3297, 3303, 3298, 3301, 5289, 5280} \[ -\frac {d^2 \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^3}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {3 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^4}+\frac {3 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^4}+\frac {d^2 \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^3}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}-\frac {3 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^4}+\frac {3 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^4}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}-\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {d \sinh (c+d x)}{2 a^3 x}+\frac {9 \sqrt {b} d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\frac {9 \sqrt {b} d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}+\frac {9 \sqrt {b} d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}+\frac {9 \sqrt {b} d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5280
Rule 5289
Rule 5293
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )^3} \, dx &=\int \left (\frac {\cosh (c+d x)}{a^3 x^3}-\frac {3 b \cosh (c+d x)}{a^4 x}+\frac {b^2 x \cosh (c+d x)}{a^2 \left (a+b x^2\right )^3}+\frac {2 b^2 x \cosh (c+d x)}{a^3 \left (a+b x^2\right )^2}+\frac {3 b^2 x \cosh (c+d x)}{a^4 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\cosh (c+d x)}{x^3} \, dx}{a^3}-\frac {(3 b) \int \frac {\cosh (c+d x)}{x} \, dx}{a^4}+\frac {\left (3 b^2\right ) \int \frac {x \cosh (c+d x)}{a+b x^2} \, dx}{a^4}+\frac {\left (2 b^2\right ) \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a^3}+\frac {b^2 \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx}{a^2}\\ &=-\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}+\frac {\left (3 b^2\right ) \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^4}+\frac {d \int \frac {\sinh (c+d x)}{x^2} \, dx}{2 a^3}+\frac {(b d) \int \frac {\sinh (c+d x)}{a+b x^2} \, dx}{a^3}+\frac {(b d) \int \frac {\sinh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 a^2}-\frac {(3 b \cosh (c)) \int \frac {\cosh (d x)}{x} \, dx}{a^4}-\frac {(3 b \sinh (c)) \int \frac {\sinh (d x)}{x} \, dx}{a^4}\\ &=-\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}-\frac {\left (3 b^{3/2}\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 a^4}+\frac {\left (3 b^{3/2}\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 a^4}+\frac {(b d) \int \left (\frac {\sqrt {-a} \sinh (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \sinh (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{a^3}+\frac {(b d) \int \left (-\frac {b \sinh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \sinh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {b \sinh (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{4 a^2}+\frac {d^2 \int \frac {\cosh (c+d x)}{x} \, dx}{2 a^3}\\ &=-\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}+\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{7/2}}+\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{7/2}}-\frac {\left (b^2 d\right ) \int \frac {\sinh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 a^3}-\frac {\left (b^2 d\right ) \int \frac {\sinh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 a^3}-\frac {\left (b^2 d\right ) \int \frac {\sinh (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^3}+\frac {\left (d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx}{2 a^3}+\frac {\left (3 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^4}-\frac {\left (3 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^4}+\frac {\left (d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx}{2 a^3}+\frac {\left (3 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^4}+\frac {\left (3 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^4}\\ &=-\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}+\frac {3 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^4}+\frac {3 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^4}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}-\frac {3 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^4}+\frac {3 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^4}-\frac {\left (b^2 d\right ) \int \left (-\frac {\sqrt {-a} \sinh (c+d x)}{2 a b \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {-a} \sinh (c+d x)}{2 a b \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{8 a^3}+\frac {\left (b d^2\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a^3}-\frac {\left (b d^2\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a^3}+\frac {\left (b d \cosh \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)^{7/2}}-\frac {\left (b d \cosh \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)^{7/2}}+\frac {\left (b d \sinh \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)^{7/2}}+\frac {\left (b d \sinh \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)^{7/2}}\\ &=-\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}+\frac {3 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^4}+\frac {3 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^4}+\frac {\sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{7/2}}-\frac {\sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{7/2}}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}+\frac {\sqrt {b} d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{7/2}}-\frac {3 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^4}+\frac {\sqrt {b} d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{7/2}}+\frac {3 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^4}+\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{7/2}}+\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{7/2}}-\frac {\left (b d^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}+b x} \, dx}{16 a^3}+\frac {\left (b d^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}-b x} \, dx}{16 a^3}-\frac {\left (b d^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}+b x} \, dx}{16 a^3}-\frac {\left (b d^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}-b x} \, dx}{16 a^3}\\ &=-\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}+\frac {3 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^4}-\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}+\frac {3 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^4}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^3}+\frac {\sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{7/2}}-\frac {\sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 (-a)^{7/2}}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}+\frac {\sqrt {b} d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{7/2}}-\frac {3 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^4}+\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}+\frac {\sqrt {b} d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{7/2}}+\frac {3 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^4}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^3}+\frac {\left (b d \cosh \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}{16 (-a)^{7/2}}-\frac {\left (b d \cosh \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}{16 (-a)^{7/2}}+\frac {\left (b d \sinh \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}{16 (-a)^{7/2}}+\frac {\left (b d \sinh \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}{16 (-a)^{7/2}}\\ &=-\frac {\cosh (c+d x)}{2 a^3 x^2}-\frac {b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cosh (c) \text {Chi}(d x)}{a^4}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a^3}+\frac {3 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^4}-\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}+\frac {3 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^4}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^3}+\frac {9 \sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\frac {9 \sqrt {b} d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\frac {d \sinh (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \sinh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 b \sinh (c) \text {Shi}(d x)}{a^4}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a^3}+\frac {9 \sqrt {b} d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}-\frac {3 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^4}+\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}+\frac {9 \sqrt {b} d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{7/2}}+\frac {3 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^4}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^3}\\ \end {align*}
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Mathematica [C] time = 4.02, size = 998, normalized size = 1.26 \[ -\frac {-i a \sinh (c) \left (\text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )-\text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )\right ) d^2+a \cosh (c) \left (\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right )+\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+\sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )\right )\right ) d^2-9 i \sqrt {a} \sqrt {b} \sinh (c) \left (\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right )-\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+\sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-\text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )\right )\right ) d-9 \sqrt {a} \sqrt {b} \cosh (c) \left (\text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+\text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )-\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )\right )\right ) d+\frac {2 a \cosh (d x) \left (2 \left (6 b^2 x^4+9 a b x^2+2 a^2\right ) \cosh (c)+d x \left (3 b^2 x^4+7 a b x^2+4 a^2\right ) \sinh (c)\right )}{x^2 \left (b x^2+a\right )^2}+\frac {2 a \left (d x \left (3 b^2 x^4+7 a b x^2+4 a^2\right ) \cosh (c)+2 \left (6 b^2 x^4+9 a b x^2+2 a^2\right ) \sinh (c)\right ) \sinh (d x)}{x^2 \left (b x^2+a\right )^2}+8 \left (6 b-a d^2\right ) (\cosh (c) \text {Chi}(d x)+\sinh (c) \text {Shi}(d x))+24 i b \sinh (c) \left (\text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )-\text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )\right )-24 b \cosh (c) \left (\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right )+\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+\sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )\right )\right )}{16 a^4} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.53, size = 2363, normalized size = 2.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.23, size = 1294, normalized size = 1.64 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x^3\,{\left (b\,x^2+a\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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