Optimal. Leaf size=283 \[ \frac {\cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac {\cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}+\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}-\frac {\sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}+\frac {\sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b} \]
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Rubi [A] time = 0.46, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5293, 3303, 3298, 3301} \[ \frac {\cosh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Chi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac {\cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (-x d-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}+\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Chi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}-\frac {\sinh \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Shi}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b}+\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b}+\frac {\sinh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Shi}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5293
Rubi steps
\begin {align*} \int \frac {x^2 \cosh (c+d x)}{a+b x^3} \, dx &=\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\\ &=\frac {\int \frac {\cosh (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac {\int \frac {\cosh (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac {\int \frac {\cosh (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}\\ &=\frac {\cosh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\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 b^{2/3}}+\frac {\cosh \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\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 b^{2/3}}+\frac {\cosh \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\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 b^{2/3}}+\frac {\sinh \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\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 b^{2/3}}+\frac {\left (i \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 b^{2/3}}+\frac {\left (i \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 b^{2/3}}\\ &=\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 b}+\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 b}+\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 b}-\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 b}+\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 b}+\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 b}\\ \end {align*}
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Mathematica [C] time = 0.24, size = 170, normalized size = 0.60 \[ \frac {\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,-\sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))-\cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))\& \right ]+\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\sinh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\cosh (\text {$\#$1} d+c) \text {Chi}(d (x-\text {$\#$1}))+\sinh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))+\cosh (\text {$\#$1} d+c) \text {Shi}(d (x-\text {$\#$1}))\& \right ]}{6 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 500, normalized size = 1.77 \[ \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 ) + {\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 ) - {\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 \, b} \]
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 \[ \int \frac {x^{2} \cosh \left (d x + c\right )}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 423, normalized size = 1.49 \[ -\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\textit {\_R1}^{2} {\mathrm e}^{-\textit {\_R1}} \Ei \left (1, d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{6 b}-\frac {c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {{\mathrm e}^{-\textit {\_R1}} \Ei \left (1, d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{6 b}+\frac {c \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\textit {\_R1} \,{\mathrm e}^{-\textit {\_R1}} \Ei \left (1, d x -\textit {\_R1} +c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b}-\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\textit {\_R1}^{2} {\mathrm e}^{\textit {\_R1}} \Ei \left (1, -d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{6 b}-\frac {c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {{\mathrm e}^{\textit {\_R1}} \Ei \left (1, -d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{6 b}+\frac {c \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\textit {\_R1} \,{\mathrm e}^{\textit {\_R1}} \Ei \left (1, -d x +\textit {\_R1} -c \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b} \]
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 \[ \frac {{\left (d x^{2} e^{\left (2 \, c\right )} + x e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} - {\left (d x^{2} - x\right )} e^{\left (-d x\right )}}{2 \, {\left (b d^{2} x^{3} e^{c} + a d^{2} e^{c}\right )}} + \frac {1}{2} \, \int \frac {{\left (2 \, b x^{3} e^{c} - 3 \, a d x e^{c} - a e^{c}\right )} e^{\left (d x\right )}}{b^{2} d^{2} x^{6} + 2 \, a b d^{2} x^{3} + a^{2} d^{2}}\,{d x} + \frac {1}{2} \, \int \frac {{\left (2 \, b x^{3} + 3 \, a d x - a\right )} e^{\left (-d x\right )}}{b^{2} d^{2} x^{6} e^{c} + 2 \, a b d^{2} x^{3} e^{c} + a^{2} d^{2} e^{c}}\,{d x} \]
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 \[ \int \frac {x^2\,\mathrm {cosh}\left (c+d\,x\right )}{b\,x^3+a} \,d x \]
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 \[ \int \frac {x^{2} \cosh {\left (c + d x \right )}}{a + b x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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