Optimal. Leaf size=791 \[ \text{result too large to display} \]
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Rubi [A] time = 1.86841, antiderivative size = 791, normalized size of antiderivative = 1., 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 5293
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rule 5289
Rule 5280
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*}
Mathematica [C] time = 3.71688, size = 998, normalized size = 1.26 \[ -\frac{-i a \sinh (c) \left (\text{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right )-\text{CosIntegral}\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{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right )+\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\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{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right )-\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\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{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right )+\text{CosIntegral}\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{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right )-\text{CosIntegral}\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{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right )+\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\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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Maple [B] time = 0.152, size = 1294, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.20699, size = 4891, normalized size = 6.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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