Optimal. Leaf size=1147 \[ \frac {\sqrt [3]{-1} \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 ) d^2}{54 a^{5/3} b^{4/3}}-\frac {(-1)^{2/3} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}-\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}-\frac {\sqrt [3]{-1} \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 ) d^2}{54 a^{5/3} b^{4/3}}-\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}-\frac {(-1)^{2/3} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}-\frac {2 \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{27 a^2 b}-\frac {2 \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 ) d}{27 a^2 b}-\frac {2 \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{27 a^2 b}-\frac {\sinh (c+d x) d}{18 b^2 x^3 \left (b x^3+a\right )}+\frac {\sinh (c+d x) d}{18 a b^2 x^3}+\frac {2 \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 ) d}{27 a^2 b}-\frac {2 \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{27 a^2 b}-\frac {2 \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{27 a^2 b}+\frac {2 \cosh (c+d x)}{9 a^2 b x}+\frac {\cosh (c+d x)}{18 b^2 x^4 \left (b x^3+a\right )}-\frac {\cosh (c+d x)}{6 b x \left (b x^3+a\right )^2}-\frac {\cosh (c+d x)}{18 a b^2 x^4}-\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 {2 \sqrt [3]{-1} \cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}-\frac {2 \cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\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 {2 \sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {2 \sqrt [3]{-1} \sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}} \]
[Out]
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Rubi [A] time = 3.20, antiderivative size = 1147, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5281
Rule 5290
Rule 5291
Rule 5292
Rule 5293
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*}
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Mathematica [C] time = 0.50, 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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fricas [B] time = 0.68, size = 4691, normalized size = 4.09 \[ \text {result too large to display} \]
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 \[ \int \frac {x \cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 810, normalized size = 0.71 \[ \frac {d^{6} {\mathrm e}^{-d x -c} b \,x^{5}}{9 a^{2} \left (b^{2} d^{6} x^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}-\frac {d^{7} {\mathrm e}^{-d x -c} x^{3}}{36 a \left (b^{2} d^{6} x^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}+\frac {7 d^{6} {\mathrm e}^{-d x -c} x^{2}}{36 a \left (b^{2} d^{6} x^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}-\frac {d^{7} {\mathrm e}^{-d x -c}}{36 b \left (b^{2} d^{6} x^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}-\frac {d \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1}^{2} b c -2 \textit {\_R1} b \,c^{2}-a \,d^{3}+b \,c^{3}+4 \textit {\_R1}^{2} b -2 \textit {\_R1} b c -2 b \,c^{2}+4 \textit {\_R1} b +6 c b \right ) {\mathrm e}^{-\textit {\_R1}} \Ei \left (1, d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{108 a^{2} b^{2}}+\frac {d c \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}+6 \textit {\_R1} -6 c +10\right ) {\mathrm e}^{-\textit {\_R1}} \Ei \left (1, d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{108 a^{2} b}+\frac {d^{6} {\mathrm e}^{d x +c} b \,x^{5}}{9 a^{2} \left (b^{2} d^{6} x^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}+\frac {d^{7} {\mathrm e}^{d x +c} x^{3}}{36 a \left (b^{2} d^{6} x^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}+\frac {7 d^{6} {\mathrm e}^{d x +c} x^{2}}{36 a \left (b^{2} d^{6} x^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}+\frac {d^{7} {\mathrm e}^{d x +c}}{36 b \left (b^{2} d^{6} x^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}-\frac {d \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1}^{2} b c -2 \textit {\_R1} b \,c^{2}-a \,d^{3}+b \,c^{3}-4 \textit {\_R1}^{2} b +2 \textit {\_R1} b c +2 b \,c^{2}+4 \textit {\_R1} b +6 c b \right ) {\mathrm e}^{\textit {\_R1}} \Ei \left (1, -d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{108 a^{2} b^{2}}+\frac {d c \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}-6 \textit {\_R1} +6 c +10\right ) {\mathrm e}^{\textit {\_R1}} \Ei \left (1, -d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{108 a^{2} b} \]
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 \[ \frac {x e^{\left (d x + 2 \, c\right )} - x e^{\left (-d x\right )}}{2 \, {\left (b^{3} d x^{9} e^{c} + 3 \, a b^{2} d x^{6} e^{c} + 3 \, a^{2} b d x^{3} e^{c} + a^{3} d e^{c}\right )}} + \frac {1}{2} \, \int \frac {{\left (8 \, b x^{3} e^{c} - a e^{c}\right )} e^{\left (d x\right )}}{b^{4} d x^{12} + 4 \, a b^{3} d x^{9} + 6 \, a^{2} b^{2} d x^{6} + 4 \, a^{3} b d x^{3} + a^{4} d}\,{d x} - \frac {1}{2} \, \int \frac {{\left (8 \, b x^{3} - a\right )} e^{\left (-d x\right )}}{b^{4} d x^{12} e^{c} + 4 \, a b^{3} d x^{9} e^{c} + 6 \, a^{2} b^{2} d x^{6} e^{c} + 4 \, a^{3} b d x^{3} e^{c} + a^{4} d e^{c}}\,{d x} \]
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 \[ \int \frac {x\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^3} \,d x \]
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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