Optimal. Leaf size=874 \[ \frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}-\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}+\frac {\text {Chi}(d x) \sinh (c) d}{a^3}+\frac {7 \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^3}+\frac {7 \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^3}+\frac {\sinh (c+d x) d}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sinh (c+d x) d}{16 (-a)^{5/2} \left (\sqrt {b} x+\sqrt {-a}\right )}+\frac {\cosh (c) \text {Shi}(d x) d}{a^3}-\frac {7 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^3}+\frac {7 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^3}-\frac {\cosh (c+d x)}{a^3 x}+\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {b} x+\sqrt {-a}\right )^2}+\frac {15 \sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}-\frac {15 \sqrt {b} \cosh \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 {15 \sqrt {b} \sinh \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 {15 \sqrt {b} \sinh \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 = 2.69, antiderivative size = 874, normalized size of antiderivative = 1.00, number of steps used = 60, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5293, 3297, 3303, 3298, 3301, 5281} \[ \frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}-\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}+\frac {\text {Chi}(d x) \sinh (c) d}{a^3}+\frac {7 \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^3}+\frac {7 \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^3}+\frac {\sinh (c+d x) d}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sinh (c+d x) d}{16 (-a)^{5/2} \left (\sqrt {b} x+\sqrt {-a}\right )}+\frac {\cosh (c) \text {Shi}(d x) d}{a^3}-\frac {7 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^3}+\frac {7 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^3}-\frac {\cosh (c+d x)}{a^3 x}+\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {b} x+\sqrt {-a}\right )^2}+\frac {15 \sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}-\frac {15 \sqrt {b} \cosh \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 {15 \sqrt {b} \sinh \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 {15 \sqrt {b} \sinh \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 5281
Rule 5293
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx &=\int \left (\frac {\cosh (c+d x)}{a^3 x^2}-\frac {b \cosh (c+d x)}{a \left (a+b x^2\right )^3}-\frac {b \cosh (c+d x)}{a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\cosh (c+d x)}{x^2} \, dx}{a^3}-\frac {b \int \frac {\cosh (c+d x)}{a+b x^2} \, dx}{a^3}-\frac {b \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a^2}-\frac {b \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx}{a}\\ &=-\frac {\cosh (c+d x)}{a^3 x}-\frac {b \int \left (\frac {\sqrt {-a} \cosh (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \cosh (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{a^3}-\frac {b \int \left (-\frac {b \cosh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \cosh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {b \cosh (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{a^2}-\frac {b \int \left (-\frac {b^{3/2} \cosh (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}-b x\right )^3}-\frac {3 b \cosh (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b^{3/2} \cosh (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}+b x\right )^3}-\frac {3 b \cosh (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {3 b \cosh (c+d x)}{8 a^2 \left (-a b-b^2 x^2\right )}\right ) \, dx}{a}+\frac {d \int \frac {\sinh (c+d x)}{x} \, dx}{a^3}\\ &=-\frac {\cosh (c+d x)}{a^3 x}-\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{7/2}}-\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{7/2}}+\frac {\left (3 b^2\right ) \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 a^3}+\frac {\left (3 b^2\right ) \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 a^3}+\frac {b^2 \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{4 a^3}+\frac {b^2 \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{4 a^3}+\frac {\left (3 b^2\right ) \int \frac {\cosh (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^3}+\frac {b^2 \int \frac {\cosh (c+d x)}{-a b-b^2 x^2} \, dx}{2 a^3}-\frac {b^{5/2} \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^3} \, dx}{8 (-a)^{5/2}}-\frac {b^{5/2} \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^3} \, dx}{8 (-a)^{5/2}}+\frac {(d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx}{a^3}+\frac {(d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx}{a^3}\\ &=-\frac {\cosh (c+d x)}{a^3 x}-\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )^2}-\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \text {Chi}(d x) \sinh (c)}{a^3}+\frac {d \cosh (c) \text {Shi}(d x)}{a^3}+\frac {\left (3 b^2\right ) \int \left (-\frac {\sqrt {-a} \cosh (c+d x)}{2 a b \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {-a} \cosh (c+d x)}{2 a b \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{8 a^3}+\frac {b^2 \int \left (-\frac {\sqrt {-a} \cosh (c+d x)}{2 a b \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {-a} \cosh (c+d x)}{2 a b \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{2 a^3}-\frac {(3 b d) \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a^3}+\frac {(3 b d) \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a^3}-\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{4 a^3}+\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{4 a^3}+\frac {\left (b^{3/2} d\right ) \int \frac {\sinh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 (-a)^{5/2}}-\frac {\left (b^{3/2} d\right ) \int \frac {\sinh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 (-a)^{5/2}}-\frac {\left (b \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)^{7/2}}-\frac {\left (b \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)^{7/2}}-\frac {\left (b \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)^{7/2}}+\frac {\left (b \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)^{7/2}}\\ &=-\frac {\cosh (c+d x)}{a^3 x}-\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )^2}-\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {\sqrt {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)^{7/2}}-\frac {\sqrt {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)^{7/2}}+\frac {d \text {Chi}(d x) \sinh (c)}{a^3}+\frac {d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \cosh (c) \text {Shi}(d x)}{a^3}-\frac {\sqrt {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)^{7/2}}-\frac {\sqrt {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)^{7/2}}-\frac {(3 b) \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{7/2}}-\frac {(3 b) \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{7/2}}-\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 (-a)^{7/2}}-\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 (-a)^{7/2}}-\frac {\left (\sqrt {b} d^2\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 (-a)^{5/2}}-\frac {\left (\sqrt {b} d^2\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 (-a)^{5/2}}+\frac {\left (3 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}+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}+b x} \, dx}{4 a^3}+\frac {\left (3 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}-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}-b x} \, dx}{4 a^3}+\frac {\left (3 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}+b x} \, dx}{16 a^3}+\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}+b x} \, dx}{4 a^3}-\frac {\left (3 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}-b x} \, dx}{16 a^3}-\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}-b x} \, dx}{4 a^3}\\ &=-\frac {\cosh (c+d x)}{a^3 x}-\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )^2}-\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {\sqrt {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)^{7/2}}-\frac {\sqrt {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)^{7/2}}+\frac {d \text {Chi}(d x) \sinh (c)}{a^3}+\frac {7 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^3}+\frac {7 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^3}+\frac {d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \cosh (c) \text {Shi}(d x)}{a^3}-\frac {7 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^3}-\frac {\sqrt {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)^{7/2}}+\frac {7 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^3}-\frac {\sqrt {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)^{7/2}}-\frac {\left (3 b \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}{16 (-a)^{7/2}}-\frac {\left (b \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}{4 (-a)^{7/2}}-\frac {\left (\sqrt {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)^{5/2}}-\frac {\left (3 b \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}{16 (-a)^{7/2}}-\frac {\left (b \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}{4 (-a)^{7/2}}-\frac {\left (\sqrt {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)^{5/2}}-\frac {\left (3 b \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}{16 (-a)^{7/2}}-\frac {\left (b \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}{4 (-a)^{7/2}}-\frac {\left (\sqrt {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)^{5/2}}+\frac {\left (3 b \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}{16 (-a)^{7/2}}+\frac {\left (b \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}{4 (-a)^{7/2}}+\frac {\left (\sqrt {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)^{5/2}}\\ &=-\frac {\cosh (c+d x)}{a^3 x}-\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )^2}-\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {15 \sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}+\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)^{5/2} \sqrt {b}}-\frac {15 \sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{7/2}}-\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)^{5/2} \sqrt {b}}+\frac {d \text {Chi}(d x) \sinh (c)}{a^3}+\frac {7 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^3}+\frac {7 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^3}+\frac {d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \cosh (c) \text {Shi}(d x)}{a^3}-\frac {7 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^3}-\frac {15 \sqrt {b} \sinh \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 {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)^{5/2} \sqrt {b}}+\frac {7 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^3}-\frac {15 \sqrt {b} \sinh \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 {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)^{5/2} \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 2.94, size = 1009, normalized size = 1.15 \[ \frac {-\frac {30 \sqrt {a} b^2 \cosh (c) \cosh (d x) x^3}{\left (b x^2+a\right )^2}-\frac {30 \sqrt {a} b^2 \sinh (c) \sinh (d x) x^3}{\left (b x^2+a\right )^2}-\frac {2 a^{3/2} b d \cosh (d x) \sinh (c) x^2}{\left (b x^2+a\right )^2}-\frac {2 a^{3/2} b d \cosh (c) \sinh (d x) x^2}{\left (b x^2+a\right )^2}-\frac {50 a^{3/2} b \cosh (c) \cosh (d x) x}{\left (b x^2+a\right )^2}-\frac {50 a^{3/2} b \sinh (c) \sinh (d x) x}{\left (b x^2+a\right )^2}-\frac {2 a^{5/2} d \cosh (d x) \sinh (c)}{\left (b x^2+a\right )^2}+16 \sqrt {a} d \text {Chi}(d x) \sinh (c)+\frac {\text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (7 \sqrt {a} \sqrt {b} d \sinh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )-i \left (15 b-a d^2\right ) \cosh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )}{\sqrt {b}}+\frac {\text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (i \left (15 b-a d^2\right ) \cosh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )+7 \sqrt {a} \sqrt {b} d \sinh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )}{\sqrt {b}}-\frac {2 a^{5/2} d \cosh (c) \sinh (d x)}{\left (b x^2+a\right )^2}+16 \sqrt {a} d \cosh (c) \text {Shi}(d x)+7 i \sqrt {a} d \cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \cosh (c) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\frac {i a d^2 \cosh (c) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )}{\sqrt {b}}-15 i \sqrt {b} \cosh (c) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-\frac {a d^2 \cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sinh (c) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )}{\sqrt {b}}+15 \sqrt {b} \cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sinh (c) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+7 \sqrt {a} d \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sinh (c) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-7 i \sqrt {a} d \cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \cosh (c) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-\frac {i a d^2 \cosh (c) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )}{\sqrt {b}}+15 i \sqrt {b} \cosh (c) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-\frac {a d^2 \cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sinh (c) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )}{\sqrt {b}}+15 \sqrt {b} \cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sinh (c) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+7 \sqrt {a} d \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sinh (c) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-\frac {16 a^{5/2} \cosh (c) \cosh (d x)}{\left (b x^2+a\right )^2 x}-\frac {16 a^{5/2} \sinh (c) \sinh (d x)}{\left (b x^2+a\right )^2 x}}{16 a^{7/2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.61, size = 2346, normalized size = 2.68 \[ \text {result too large to display} \]
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 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 1178, normalized size = 1.35 \[ \text {result too large to display} \]
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 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x^{2}}\,{d x} \]
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 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x^2\,{\left (b\,x^2+a\right )}^3} \,d x \]
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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