Optimal. Leaf size=303 \[ -\frac {\cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}-\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}+\frac {\sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}-\frac {\sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}+\frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a} \]
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Rubi [A] time = 0.52, antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5293, 3303, 3298, 3301} \[ -\frac {\cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}-\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}+\frac {\sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a}-\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}-\frac {\sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a}+\frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5293
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{x \left (a+b x^3\right )} \, dx &=\int \left (\frac {\cosh (c+d x)}{a x}-\frac {b x^2 \cosh (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx\\ &=\frac {\int \frac {\cosh (c+d x)}{x} \, dx}{a}-\frac {b \int \frac {x^2 \cosh (c+d x)}{a+b x^3} \, dx}{a}\\ &=-\frac {b \int \left (\frac {\cosh (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cosh (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cosh (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{a}+\frac {\cosh (c) \int \frac {\cosh (d x)}{x} \, dx}{a}+\frac {\sinh (c) \int \frac {\sinh (d x)}{x} \, dx}{a}\\ &=\frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a}-\frac {\sqrt [3]{b} \int \frac {\cosh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a}-\frac {\sqrt [3]{b} \int \frac {\cosh (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a}-\frac {\sqrt [3]{b} \int \frac {\cosh (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a}\\ &=\frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a}-\frac {\left (\sqrt [3]{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}-\frac {\left (\sqrt [3]{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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a}-\frac {\left (\sqrt [3]{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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a}-\frac {\left (\sqrt [3]{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}-\frac {\left (i \sqrt [3]{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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a}-\frac {\left (i \sqrt [3]{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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a}\\ &=\frac {\cosh (c) \text {Chi}(d x)}{a}-\frac {\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}-\frac {\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}-\frac {\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}+\frac {\sinh (c) \text {Shi}(d x)}{a}+\frac {\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}-\frac {\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}-\frac {\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}\\ \end {align*}
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Mathematica [C] time = 0.28, size = 186, normalized size = 0.61 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,-\sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))-\cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))\& \right ]+\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))+\cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))\& \right ]-6 \cosh (c) \text {Chi}(d x)-6 \sinh (c) \text {Shi}(d x)}{6 a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 530, normalized size = 1.75 \[ -\frac {{\rm Ei}\left (d x - \frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + c\right ) + {\rm Ei}\left (-d x - \frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - c\right ) + {\rm Ei}\left (d x + \frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - c\right ) + {\rm Ei}\left (-d x + \frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \cosh \left (\frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + c\right ) + {\rm Ei}\left (-d x + \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) \cosh \left (c + \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) - 3 \, {\left ({\rm Ei}\left (d x\right ) + {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) + {\rm Ei}\left (d x + \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) \cosh \left (-c + \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) + {\rm Ei}\left (d x - \frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + c\right ) + {\rm Ei}\left (-d x - \frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - c\right ) - {\rm Ei}\left (d x + \frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - c\right ) - {\rm Ei}\left (-d x + \frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right ) \sinh \left (\frac {1}{2} \, \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + c\right ) - {\rm Ei}\left (-d x + \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) \sinh \left (c + \left (-\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) - 3 \, {\left ({\rm Ei}\left (d x\right ) - {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c) - {\rm Ei}\left (d x + \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right ) \sinh \left (-c + \left (\frac {a d^{3}}{b}\right )^{\frac {1}{3}}\right )}{6 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.11, size = 138, normalized size = 0.46 \[ \frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }{\mathrm e}^{-\textit {\_R1}} \Ei \left (1, d x -\textit {\_R1} +c \right )}{6 a}-\frac {{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2 a}+\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }{\mathrm e}^{\textit {\_R1}} \Ei \left (1, -d x +\textit {\_R1} -c \right )}{6 a}-\frac {{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (b x^{3} + a\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x\,\left (b\,x^3+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\left (c + d x \right )}}{x \left (a + b x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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