Integrand size = 14, antiderivative size = 136 \[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^2} \, dx=-\frac {\cosh ^3\left (a+b x^2\right )}{x}-\frac {3}{8} \sqrt {b} e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )-\frac {1}{8} \sqrt {b} e^{-3 a} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {b} x\right )+\frac {3}{8} \sqrt {b} e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )+\frac {1}{8} \sqrt {b} e^{3 a} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {b} x\right ) \]
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Time = 0.09 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5439, 5737, 5406, 2235, 2236} \[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^2} \, dx=-\frac {3}{8} \sqrt {\pi } e^{-a} \sqrt {b} \text {erf}\left (\sqrt {b} x\right )-\frac {1}{8} \sqrt {3 \pi } e^{-3 a} \sqrt {b} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )+\frac {3}{8} \sqrt {\pi } e^a \sqrt {b} \text {erfi}\left (\sqrt {b} x\right )+\frac {1}{8} \sqrt {3 \pi } e^{3 a} \sqrt {b} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )-\frac {\cosh ^3\left (a+b x^2\right )}{x} \]
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Rule 2235
Rule 2236
Rule 5406
Rule 5439
Rule 5737
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.25 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.50 \[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^2} \, dx=\frac {-6 \cosh \left (a+b x^2\right )-2 \cosh \left (3 \left (a+b x^2\right )\right )+3 \sqrt {b} \sqrt {\pi } x \cosh (a) \text {erfi}\left (\sqrt {b} x\right )+\sqrt {b} \sqrt {3 \pi } x \cosh (3 a) \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )+3 \sqrt {b} \sqrt {\pi } x \text {erfi}\left (\sqrt {b} x\right ) \sinh (a)+3 \sqrt {b} \sqrt {\pi } x \text {erf}\left (\sqrt {b} x\right ) (-\cosh (a)+\sinh (a))+\sqrt {b} \sqrt {3 \pi } x \text {erfi}\left (\sqrt {3} \sqrt {b} x\right ) \sinh (3 a)+\sqrt {b} \sqrt {3 \pi } x \text {erf}\left (\sqrt {3} \sqrt {b} x\right ) (-\cosh (3 a)+\sinh (3 a))}{8 x} \]
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Time = 0.18 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.10
method | result | size |
risch | \(-\frac {{\mathrm e}^{-3 a} {\mathrm e}^{-3 b \,x^{2}}}{8 x}-\frac {{\mathrm e}^{-3 a} \sqrt {b}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {erf}\left (x \sqrt {3}\, \sqrt {b}\right )}{8}-\frac {3 \,{\mathrm e}^{-a} {\mathrm e}^{-b \,x^{2}}}{8 x}-\frac {3 \,\operatorname {erf}\left (x \sqrt {b}\right ) \sqrt {b}\, \sqrt {\pi }\, {\mathrm e}^{-a}}{8}-\frac {{\mathrm e}^{3 a} {\mathrm e}^{3 b \,x^{2}}}{8 x}+\frac {3 \,{\mathrm e}^{3 a} b \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-3 b}\, x \right )}{8 \sqrt {-3 b}}-\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{b \,x^{2}}}{8 x}+\frac {3 \,{\mathrm e}^{a} b \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b}\, x \right )}{8 \sqrt {-b}}\) | \(149\) |
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Leaf count of result is larger than twice the leaf count of optimal. 891 vs. \(2 (98) = 196\).
Time = 0.26 (sec) , antiderivative size = 891, normalized size of antiderivative = 6.55 \[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^2} \, dx=\int \frac {\cosh ^{3}{\left (a + b x^{2} \right )}}{x^{2}}\, dx \]
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none
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.75 \[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^2} \, dx=-\frac {\sqrt {3} \sqrt {b x^{2}} e^{\left (-3 \, a\right )} \Gamma \left (-\frac {1}{2}, 3 \, b x^{2}\right )}{16 \, x} - \frac {\sqrt {3} \sqrt {-b x^{2}} e^{\left (3 \, a\right )} \Gamma \left (-\frac {1}{2}, -3 \, b x^{2}\right )}{16 \, x} - \frac {3 \, \sqrt {b x^{2}} e^{\left (-a\right )} \Gamma \left (-\frac {1}{2}, b x^{2}\right )}{16 \, x} - \frac {3 \, \sqrt {-b x^{2}} e^{a} \Gamma \left (-\frac {1}{2}, -b x^{2}\right )}{16 \, x} \]
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\[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^2} \, dx=\int { \frac {\cosh \left (b x^{2} + a\right )^{3}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\cosh ^3\left (a+b x^2\right )}{x^2} \, dx=\int \frac {{\mathrm {cosh}\left (b\,x^2+a\right )}^3}{x^2} \,d x \]
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