Optimal. Leaf size=1147 \[ \text{result too large to display} \]
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Rubi [A] time = 3.19773, antiderivative size = 1147, normalized size of antiderivative = 1., number of steps used = 89, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {5291, 5293, 3297, 3303, 3298, 3301, 5292, 5290, 5281} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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[Out]
Rule 5291
Rule 5293
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rule 5292
Rule 5290
Rule 5281
Rubi steps
\begin{align*} \int \frac{x \cosh (c+d x)}{\left (a+b x^3\right )^3} \, dx &=-\frac{\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}-\frac{\int \frac{\cosh (c+d x)}{x^2 \left (a+b x^3\right )^2} \, dx}{6 b}+\frac{d \int \frac{\sinh (c+d x)}{x \left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac{\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\cosh (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac{d \sinh (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}+\frac{2 \int \frac{\cosh (c+d x)}{x^5 \left (a+b x^3\right )} \, dx}{9 b^2}-\frac{d \int \frac{\sinh (c+d x)}{x^4 \left (a+b x^3\right )} \, dx}{18 b^2}-\frac{d \int \frac{\sinh (c+d x)}{x^4 \left (a+b x^3\right )} \, dx}{6 b^2}+\frac{d^2 \int \frac{\cosh (c+d x)}{x^3 \left (a+b x^3\right )} \, dx}{18 b^2}\\ &=-\frac{\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\cosh (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac{d \sinh (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}+\frac{2 \int \left (\frac{\cosh (c+d x)}{a x^5}-\frac{b \cosh (c+d x)}{a^2 x^2}+\frac{b^2 x \cosh (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{9 b^2}-\frac{d \int \left (\frac{\sinh (c+d x)}{a x^4}-\frac{b \sinh (c+d x)}{a^2 x}+\frac{b^2 x^2 \sinh (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 b^2}-\frac{d \int \left (\frac{\sinh (c+d x)}{a x^4}-\frac{b \sinh (c+d x)}{a^2 x}+\frac{b^2 x^2 \sinh (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{6 b^2}+\frac{d^2 \int \left (\frac{\cosh (c+d x)}{a x^3}-\frac{b \cosh (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{18 b^2}\\ &=-\frac{\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\cosh (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac{d \sinh (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}+\frac{2 \int \frac{x \cosh (c+d x)}{a+b x^3} \, dx}{9 a^2}+\frac{2 \int \frac{\cosh (c+d x)}{x^5} \, dx}{9 a b^2}-\frac{2 \int \frac{\cosh (c+d x)}{x^2} \, dx}{9 a^2 b}-\frac{d \int \frac{x^2 \sinh (c+d x)}{a+b x^3} \, dx}{18 a^2}-\frac{d \int \frac{x^2 \sinh (c+d x)}{a+b x^3} \, dx}{6 a^2}-\frac{d \int \frac{\sinh (c+d x)}{x^4} \, dx}{18 a b^2}-\frac{d \int \frac{\sinh (c+d x)}{x^4} \, dx}{6 a b^2}+\frac{d \int \frac{\sinh (c+d x)}{x} \, dx}{18 a^2 b}+\frac{d \int \frac{\sinh (c+d x)}{x} \, dx}{6 a^2 b}+\frac{d^2 \int \frac{\cosh (c+d x)}{x^3} \, dx}{18 a b^2}-\frac{d^2 \int \frac{\cosh (c+d x)}{a+b x^3} \, dx}{18 a b}\\ &=-\frac{\cosh (c+d x)}{18 a b^2 x^4}-\frac{d^2 \cosh (c+d x)}{36 a b^2 x^2}+\frac{2 \cosh (c+d x)}{9 a^2 b x}-\frac{\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\cosh (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac{2 d \sinh (c+d x)}{27 a b^2 x^3}-\frac{d \sinh (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}+\frac{2 \int \left (-\frac{\cosh (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{(-1)^{2/3} \cosh (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac{\sqrt [3]{-1} \cosh (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{9 a^2}-\frac{d \int \left (\frac{\sinh (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\sinh (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\sinh (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{18 a^2}-\frac{d \int \left (\frac{\sinh (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\sinh (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\sinh (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{6 a^2}+\frac{d \int \frac{\sinh (c+d x)}{x^4} \, dx}{18 a b^2}-\frac{(2 d) \int \frac{\sinh (c+d x)}{x} \, dx}{9 a^2 b}-\frac{d^2 \int \frac{\cosh (c+d x)}{x^3} \, dx}{54 a b^2}-\frac{d^2 \int \frac{\cosh (c+d x)}{x^3} \, dx}{18 a b^2}-\frac{d^2 \int \left (-\frac{\cosh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\cosh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac{\cosh (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{18 a b}+\frac{d^3 \int \frac{\sinh (c+d x)}{x^2} \, dx}{36 a b^2}+\frac{(d \cosh (c)) \int \frac{\sinh (d x)}{x} \, dx}{18 a^2 b}+\frac{(d \cosh (c)) \int \frac{\sinh (d x)}{x} \, dx}{6 a^2 b}+\frac{(d \sinh (c)) \int \frac{\cosh (d x)}{x} \, dx}{18 a^2 b}+\frac{(d \sinh (c)) \int \frac{\cosh (d x)}{x} \, dx}{6 a^2 b}\\ &=-\frac{\cosh (c+d x)}{18 a b^2 x^4}+\frac{d^2 \cosh (c+d x)}{108 a b^2 x^2}+\frac{2 \cosh (c+d x)}{9 a^2 b x}-\frac{\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\cosh (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac{2 d \text{Chi}(d x) \sinh (c)}{9 a^2 b}+\frac{d \sinh (c+d x)}{18 a b^2 x^3}-\frac{d^3 \sinh (c+d x)}{36 a b^2 x}-\frac{d \sinh (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}+\frac{2 d \cosh (c) \text{Shi}(d x)}{9 a^2 b}-\frac{2 \int \frac{\cosh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac{\left (2 \sqrt [3]{-1}\right ) \int \frac{\cosh (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac{\left (2 (-1)^{2/3}\right ) \int \frac{\cosh (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac{d \int \frac{\sinh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{d \int \frac{\sinh (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{d \int \frac{\sinh (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{d \int \frac{\sinh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac{d \int \frac{\sinh (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac{d \int \frac{\sinh (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}+\frac{d^2 \int \frac{\cosh (c+d x)}{x^3} \, dx}{54 a b^2}+\frac{d^2 \int \frac{\cosh (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{54 a^{5/3} b}+\frac{d^2 \int \frac{\cosh (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}+\frac{d^2 \int \frac{\cosh (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}-\frac{d^3 \int \frac{\sinh (c+d x)}{x^2} \, dx}{108 a b^2}-\frac{d^3 \int \frac{\sinh (c+d x)}{x^2} \, dx}{36 a b^2}+\frac{d^4 \int \frac{\cosh (c+d x)}{x} \, dx}{36 a b^2}-\frac{(2 d \cosh (c)) \int \frac{\sinh (d x)}{x} \, dx}{9 a^2 b}-\frac{(2 d \sinh (c)) \int \frac{\cosh (d x)}{x} \, dx}{9 a^2 b}\\ &=-\frac{\cosh (c+d x)}{18 a b^2 x^4}+\frac{2 \cosh (c+d x)}{9 a^2 b x}-\frac{\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\cosh (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac{d \sinh (c+d x)}{18 a b^2 x^3}+\frac{d^3 \sinh (c+d x)}{108 a b^2 x}-\frac{d \sinh (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}+\frac{d^3 \int \frac{\sinh (c+d x)}{x^2} \, dx}{108 a b^2}-\frac{d^4 \int \frac{\cosh (c+d x)}{x} \, dx}{108 a b^2}-\frac{d^4 \int \frac{\cosh (c+d x)}{x} \, dx}{36 a b^2}+\frac{\left (d^4 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx}{36 a b^2}-\frac{\left (2 \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cosh \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac{\left (d \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sinh \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{\left (d \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sinh \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}+\frac{\left (d^2 \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cosh \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{54 a^{5/3} b}+\frac{\left (2 \sqrt [3]{-1} \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac{\left (i d \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{\left (i d \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}+\frac{\left (d^2 \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}-\frac{\left (2 (-1)^{2/3} \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac{\left (i d \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{\left (i d \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}+\frac{\left (d^2 \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}+\frac{\left (d^4 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx}{36 a b^2}-\frac{\left (2 \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sinh \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac{\left (d \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cosh \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{\left (d \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cosh \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}+\frac{\left (d^2 \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sinh \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{54 a^{5/3} b}+\frac{\left (2 (-1)^{5/6} \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac{\left (d \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{\left (d \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}+\frac{\left (i d^2 \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{(-1)^{5/6} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}+\frac{\left (2 \sqrt [6]{-1} \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac{\left (d \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{\left (d \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}+\frac{\left (i d^2 \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [6]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-i d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}\\ &=-\frac{\cosh (c+d x)}{18 a b^2 x^4}+\frac{2 \cosh (c+d x)}{9 a^2 b x}-\frac{\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\cosh (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac{d^4 \cosh (c) \text{Chi}(d x)}{36 a b^2}-\frac{2 (-1)^{2/3} \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}+\frac{\sqrt [3]{-1} d^2 \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{5/3} b^{4/3}}+\frac{2 \sqrt [3]{-1} \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}-\frac{(-1)^{2/3} d^2 \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{5/3} b^{4/3}}-\frac{2 \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}-\frac{d^2 \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}-\frac{2 d \text{Chi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^2 b}-\frac{2 d \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^2 b}-\frac{2 d \text{Chi}\left (-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^2 b}+\frac{d \sinh (c+d x)}{18 a b^2 x^3}-\frac{d \sinh (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}+\frac{d^4 \sinh (c) \text{Shi}(d x)}{36 a b^2}+\frac{2 d \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}+\frac{2 (-1)^{2/3} \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}-\frac{\sqrt [3]{-1} d^2 \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{5/3} b^{4/3}}-\frac{2 d \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}-\frac{2 \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}-\frac{d^2 \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}-\frac{2 d \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac{2 \sqrt [3]{-1} \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}-\frac{(-1)^{2/3} d^2 \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}+\frac{d^4 \int \frac{\cosh (c+d x)}{x} \, dx}{108 a b^2}-\frac{\left (d^4 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx}{108 a b^2}-\frac{\left (d^4 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx}{36 a b^2}-\frac{\left (d^4 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx}{108 a b^2}-\frac{\left (d^4 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx}{36 a b^2}\\ &=-\frac{\cosh (c+d x)}{18 a b^2 x^4}+\frac{2 \cosh (c+d x)}{9 a^2 b x}-\frac{\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\cosh (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac{d^4 \cosh (c) \text{Chi}(d x)}{108 a b^2}-\frac{2 (-1)^{2/3} \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}+\frac{\sqrt [3]{-1} d^2 \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{5/3} b^{4/3}}+\frac{2 \sqrt [3]{-1} \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}-\frac{(-1)^{2/3} d^2 \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{5/3} b^{4/3}}-\frac{2 \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}-\frac{d^2 \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}-\frac{2 d \text{Chi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^2 b}-\frac{2 d \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^2 b}-\frac{2 d \text{Chi}\left (-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^2 b}+\frac{d \sinh (c+d x)}{18 a b^2 x^3}-\frac{d \sinh (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac{d^4 \sinh (c) \text{Shi}(d x)}{108 a b^2}+\frac{2 d \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}+\frac{2 (-1)^{2/3} \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}-\frac{\sqrt [3]{-1} d^2 \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{5/3} b^{4/3}}-\frac{2 d \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}-\frac{2 \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}-\frac{d^2 \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}-\frac{2 d \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac{2 \sqrt [3]{-1} \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}-\frac{(-1)^{2/3} d^2 \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}+\frac{\left (d^4 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx}{108 a b^2}+\frac{\left (d^4 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx}{108 a b^2}\\ &=-\frac{\cosh (c+d x)}{18 a b^2 x^4}+\frac{2 \cosh (c+d x)}{9 a^2 b x}-\frac{\cosh (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\cosh (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac{2 (-1)^{2/3} \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}+\frac{\sqrt [3]{-1} d^2 \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{5/3} b^{4/3}}+\frac{2 \sqrt [3]{-1} \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}-\frac{(-1)^{2/3} d^2 \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{5/3} b^{4/3}}-\frac{2 \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}-\frac{d^2 \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}-\frac{2 d \text{Chi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^2 b}-\frac{2 d \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^2 b}-\frac{2 d \text{Chi}\left (-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^2 b}+\frac{d \sinh (c+d x)}{18 a b^2 x^3}-\frac{d \sinh (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}+\frac{2 d \cosh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}+\frac{2 (-1)^{2/3} \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}-\frac{\sqrt [3]{-1} d^2 \sinh \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{5/3} b^{4/3}}-\frac{2 d \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}-\frac{2 \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}-\frac{d^2 \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}-\frac{2 d \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac{2 \sqrt [3]{-1} \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}-\frac{(-1)^{2/3} d^2 \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.457546, size = 669, normalized size = 0.58 \[ \frac{\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-4 \text{$\#$1}^2 b d \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+4 \text{$\#$1}^2 b d \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+4 \text{$\#$1}^2 b d \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-4 \text{$\#$1}^2 b d \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+a d^2 \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))-a d^2 \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))-a d^2 \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+a d^2 \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+4 \text{$\#$1} b \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+4 \text{$\#$1} b \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]-\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{4 \text{$\#$1}^2 b d \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+4 \text{$\#$1}^2 b d \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+4 \text{$\#$1}^2 b d \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+4 \text{$\#$1}^2 b d \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+a d^2 \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+a d^2 \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+a d^2 \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+a d^2 \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]+\frac{6 b \cosh (d x) \left (a d \sinh (c) \left (a+b x^3\right )+b x^2 \cosh (c) \left (7 a+4 b x^3\right )\right )}{\left (a+b x^3\right )^2}+\frac{6 b \sinh (d x) \left (a d \cosh (c) \left (a+b x^3\right )+b x^2 \sinh (c) \left (7 a+4 b x^3\right )\right )}{\left (a+b x^3\right )^2}}{108 a^2 b^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.063, size = 810, normalized size = 0.7 \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 = 3.0848, size = 10103, normalized size = 8.81 \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{x \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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