Optimal. Leaf size=410 \[ \frac{\sqrt [3]{-1} b^{2/3} \cosh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac{(-1)^{2/3} b^{2/3} \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-x d-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac{b^{2/3} \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac{\sqrt [3]{-1} b^{2/3} \sinh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac{b^{2/3} \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac{(-1)^{2/3} b^{2/3} \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a}-\frac{\cosh (c+d x)}{2 a x^2}-\frac{d \sinh (c+d x)}{2 a x} \]
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Rubi [A] time = 0.639085, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 18, 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} \[ \frac{\sqrt [3]{-1} b^{2/3} \cosh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Chi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac{(-1)^{2/3} b^{2/3} \cosh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (-x d-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac{b^{2/3} \cosh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac{\sqrt [3]{-1} b^{2/3} \sinh \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Shi}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac{b^{2/3} \sinh \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac{(-1)^{2/3} b^{2/3} \sinh \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Shi}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a}-\frac{\cosh (c+d x)}{2 a x^2}-\frac{d \sinh (c+d x)}{2 a x} \]
Antiderivative was successfully verified.
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Rule 5293
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rule 5281
Rubi steps
\begin{align*} \int \frac{\cosh (c+d x)}{x^3 \left (a+b x^3\right )} \, dx &=\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\\ &=\frac{\int \frac{\cosh (c+d x)}{x^3} \, dx}{a}-\frac{b \int \frac{\cosh (c+d x)}{a+b x^3} \, dx}{a}\\ &=-\frac{\cosh (c+d x)}{2 a x^2}-\frac{b \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}{a}+\frac{d \int \frac{\sinh (c+d x)}{x^2} \, dx}{2 a}\\ &=-\frac{\cosh (c+d x)}{2 a x^2}-\frac{d \sinh (c+d x)}{2 a x}+\frac{b \int \frac{\cosh (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 a^{5/3}}+\frac{b \int \frac{\cosh (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{5/3}}+\frac{b \int \frac{\cosh (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{5/3}}+\frac{d^2 \int \frac{\cosh (c+d x)}{x} \, dx}{2 a}\\ &=-\frac{\cosh (c+d x)}{2 a x^2}-\frac{d \sinh (c+d x)}{2 a x}+\frac{\left (d^2 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx}{2 a}+\frac{\left (b \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}{3 a^{5/3}}+\frac{\left (b \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}{3 a^{5/3}}+\frac{\left (b \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}{3 a^{5/3}}+\frac{\left (d^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx}{2 a}+\frac{\left (b \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}{3 a^{5/3}}+\frac{\left (i b \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}{3 a^{5/3}}+\frac{\left (i b \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}{3 a^{5/3}}\\ &=-\frac{\cosh (c+d x)}{2 a x^2}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a}+\frac{\sqrt [3]{-1} b^{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 )}{3 a^{5/3}}-\frac{(-1)^{2/3} b^{2/3} \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 )}{3 a^{5/3}}-\frac{b^{2/3} \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 )}{3 a^{5/3}}-\frac{d \sinh (c+d x)}{2 a x}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a}-\frac{\sqrt [3]{-1} b^{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 )}{3 a^{5/3}}-\frac{b^{2/3} \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 )}{3 a^{5/3}}-\frac{(-1)^{2/3} b^{2/3} \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 )}{3 a^{5/3}}\\ \end{align*}
Mathematica [C] time = 0.37738, size = 237, normalized size = 0.58 \[ -\frac{x^2 \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-\sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))-\cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]+x^2 \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{\sinh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\cosh (\text{$\#$1} d+c) \text{Chi}(d (x-\text{$\#$1}))+\sinh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))+\cosh (\text{$\#$1} d+c) \text{Shi}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]-3 d^2 x^2 \cosh (c) \text{Chi}(d x)-3 d^2 x^2 \sinh (c) \text{Shi}(d x)+3 d x \sinh (c+d x)+3 \cosh (c+d x)}{6 a x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.054, size = 240, normalized size = 0.6 \begin{align*}{\frac{d{{\rm e}^{-dx-c}}}{4\,ax}}-{\frac{{{\rm e}^{-dx-c}}}{4\,a{x}^{2}}}-{\frac{{d}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{4\,a}}+{\frac{{d}^{2}}{6\,a}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{3}-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }{\frac{{{\rm e}^{-{\it \_R1}}}{\it Ei} \left ( 1,dx-{\it \_R1}+c \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}}-{\frac{d{{\rm e}^{dx+c}}}{4\,ax}}-{\frac{{{\rm e}^{dx+c}}}{4\,a{x}^{2}}}-{\frac{{d}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{4\,a}}+{\frac{{d}^{2}}{6\,a}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{3}-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-b{c}^{3} \right ) }{\frac{{{\rm e}^{{\it \_R1}}}{\it Ei} \left ( 1,-dx+{\it \_R1}-c \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}} \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 = 2.13039, size = 3071, normalized size = 7.49 \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^{3} + a\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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