Optimal. Leaf size=136 \[ -\frac {3}{8} \sqrt {\pi } e^{-a} \sqrt {b} \text {erf}\left (\sqrt {b} x\right )-\frac {1}{8} \sqrt {3 \pi } e^{-3 a} \sqrt {b} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )+\frac {3}{8} \sqrt {\pi } e^a \sqrt {b} \text {erfi}\left (\sqrt {b} x\right )+\frac {1}{8} \sqrt {3 \pi } e^{3 a} \sqrt {b} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )-\frac {\cosh ^3\left (a+b x^2\right )}{x} \]
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Rubi [A] time = 0.11, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5331, 5618, 5298, 2204, 2205} \[ -\frac {3}{8} \sqrt {\pi } e^{-a} \sqrt {b} \text {Erf}\left (\sqrt {b} x\right )-\frac {1}{8} \sqrt {3 \pi } e^{-3 a} \sqrt {b} \text {Erf}\left (\sqrt {3} \sqrt {b} x\right )+\frac {3}{8} \sqrt {\pi } e^a \sqrt {b} \text {Erfi}\left (\sqrt {b} x\right )+\frac {1}{8} \sqrt {3 \pi } e^{3 a} \sqrt {b} \text {Erfi}\left (\sqrt {3} \sqrt {b} x\right )-\frac {\cosh ^3\left (a+b x^2\right )}{x} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 5298
Rule 5331
Rule 5618
Rubi steps
\begin {align*} \int \frac {\cosh ^3\left (a+b x^2\right )}{x^2} \, dx &=-\frac {\cosh ^3\left (a+b x^2\right )}{x}+(6 b) \int \cosh ^2\left (a+b x^2\right ) \sinh \left (a+b x^2\right ) \, dx\\ &=-\frac {\cosh ^3\left (a+b x^2\right )}{x}+(6 b) \int \left (\frac {1}{4} \sinh \left (a+b x^2\right )+\frac {1}{4} \sinh \left (3 a+3 b x^2\right )\right ) \, dx\\ &=-\frac {\cosh ^3\left (a+b x^2\right )}{x}+\frac {1}{2} (3 b) \int \sinh \left (a+b x^2\right ) \, dx+\frac {1}{2} (3 b) \int \sinh \left (3 a+3 b x^2\right ) \, dx\\ &=-\frac {\cosh ^3\left (a+b x^2\right )}{x}-\frac {1}{4} (3 b) \int e^{-3 a-3 b x^2} \, dx-\frac {1}{4} (3 b) \int e^{-a-b x^2} \, dx+\frac {1}{4} (3 b) \int e^{a+b x^2} \, dx+\frac {1}{4} (3 b) \int e^{3 a+3 b x^2} \, dx\\ &=-\frac {\cosh ^3\left (a+b x^2\right )}{x}-\frac {3}{8} \sqrt {b} e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )-\frac {1}{8} \sqrt {b} e^{-3 a} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {b} x\right )+\frac {3}{8} \sqrt {b} e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )+\frac {1}{8} \sqrt {b} e^{3 a} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )\\ \end {align*}
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Mathematica [A] time = 0.38, size = 204, normalized size = 1.50 \[ \frac {3 \sqrt {\pi } \sqrt {b} x (\sinh (a)-\cosh (a)) \text {erf}\left (\sqrt {b} x\right )+\sqrt {3 \pi } \sqrt {b} x (\sinh (3 a)-\cosh (3 a)) \text {erf}\left (\sqrt {3} \sqrt {b} x\right )+3 \sqrt {\pi } \sqrt {b} x \sinh (a) \text {erfi}\left (\sqrt {b} x\right )+\sqrt {3 \pi } \sqrt {b} x \sinh (3 a) \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )+3 \sqrt {\pi } \sqrt {b} x \cosh (a) \text {erfi}\left (\sqrt {b} x\right )+\sqrt {3 \pi } \sqrt {b} x \cosh (3 a) \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )-6 \cosh \left (a+b x^2\right )-2 \cosh \left (3 \left (a+b x^2\right )\right )}{8 x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 891, normalized size = 6.55 \[ \text {result too large to display} \]
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 \[ \int \frac {\cosh \left (b x^{2} + a\right )^{3}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 149, normalized size = 1.10 \[ -\frac {{\mathrm e}^{-3 a} {\mathrm e}^{-3 b \,x^{2}}}{8 x}-\frac {{\mathrm e}^{-3 a} \sqrt {b}\, \sqrt {\pi }\, \sqrt {3}\, \erf \left (x \sqrt {3}\, \sqrt {b}\right )}{8}-\frac {3 \,{\mathrm e}^{-a} {\mathrm e}^{-b \,x^{2}}}{8 x}-\frac {3 \,{\mathrm e}^{-a} \sqrt {b}\, \sqrt {\pi }\, \erf \left (x \sqrt {b}\right )}{8}-\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{b \,x^{2}}}{8 x}+\frac {3 \,{\mathrm e}^{a} b \sqrt {\pi }\, \erf \left (\sqrt {-b}\, x \right )}{8 \sqrt {-b}}-\frac {{\mathrm e}^{3 a} {\mathrm e}^{3 b \,x^{2}}}{8 x}+\frac {3 \,{\mathrm e}^{3 a} b \sqrt {\pi }\, \erf \left (\sqrt {-3 b}\, x \right )}{8 \sqrt {-3 b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 102, normalized size = 0.75 \[ -\frac {\sqrt {3} \sqrt {b x^{2}} e^{\left (-3 \, a\right )} \Gamma \left (-\frac {1}{2}, 3 \, b x^{2}\right )}{16 \, x} - \frac {\sqrt {3} \sqrt {-b x^{2}} e^{\left (3 \, a\right )} \Gamma \left (-\frac {1}{2}, -3 \, b x^{2}\right )}{16 \, x} - \frac {3 \, \sqrt {b x^{2}} e^{\left (-a\right )} \Gamma \left (-\frac {1}{2}, b x^{2}\right )}{16 \, x} - \frac {3 \, \sqrt {-b x^{2}} e^{a} \Gamma \left (-\frac {1}{2}, -b x^{2}\right )}{16 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (b\,x^2+a\right )}^3}{x^2} \,d x \]
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 \[ \int \frac {\cosh ^{3}{\left (a + b x^{2} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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