Optimal. Leaf size=88 \[ -\frac {1}{2} \sqrt {\frac {\pi }{2}} e^{-2 a} \sqrt {b} \text {erf}\left (\sqrt {2} \sqrt {b} x\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} e^{2 a} \sqrt {b} \text {erfi}\left (\sqrt {2} \sqrt {b} x\right )-\frac {\sinh ^2\left (a+b x^2\right )}{x} \]
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Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5330, 5617, 5314, 5298, 2204, 2205} \[ -\frac {1}{2} \sqrt {\frac {\pi }{2}} e^{-2 a} \sqrt {b} \text {Erf}\left (\sqrt {2} \sqrt {b} x\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} e^{2 a} \sqrt {b} \text {Erfi}\left (\sqrt {2} \sqrt {b} x\right )-\frac {\sinh ^2\left (a+b x^2\right )}{x} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 5298
Rule 5314
Rule 5330
Rule 5617
Rubi steps
\begin {align*} \int \frac {\sinh ^2\left (a+b x^2\right )}{x^2} \, dx &=-\frac {\sinh ^2\left (a+b x^2\right )}{x}+(4 b) \int \cosh \left (a+b x^2\right ) \sinh \left (a+b x^2\right ) \, dx\\ &=-\frac {\sinh ^2\left (a+b x^2\right )}{x}+(2 b) \int \sinh \left (2 \left (a+b x^2\right )\right ) \, dx\\ &=-\frac {\sinh ^2\left (a+b x^2\right )}{x}+(2 b) \int \sinh \left (2 a+2 b x^2\right ) \, dx\\ &=-\frac {\sinh ^2\left (a+b x^2\right )}{x}-b \int e^{-2 a-2 b x^2} \, dx+b \int e^{2 a+2 b x^2} \, dx\\ &=-\frac {1}{2} \sqrt {b} e^{-2 a} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {b} x\right )+\frac {1}{2} \sqrt {b} e^{2 a} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {b} x\right )-\frac {\sinh ^2\left (a+b x^2\right )}{x}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 94, normalized size = 1.07 \[ \frac {\sqrt {2 \pi } \sqrt {b} x (\sinh (2 a)-\cosh (2 a)) \text {erf}\left (\sqrt {2} \sqrt {b} x\right )+\sqrt {2 \pi } \sqrt {b} x (\sinh (2 a)+\cosh (2 a)) \text {erfi}\left (\sqrt {2} \sqrt {b} x\right )-4 \sinh ^2\left (a+b x^2\right )}{4 x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 396, normalized size = 4.50 \[ -\frac {\cosh \left (b x^{2} + a\right )^{4} + 4 \, \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )^{3} + \sinh \left (b x^{2} + a\right )^{4} + \sqrt {2} \sqrt {\pi } {\left (x \cosh \left (b x^{2} + a\right )^{2} \cosh \left (2 \, a\right ) + x \cosh \left (b x^{2} + a\right )^{2} \sinh \left (2 \, a\right ) + {\left (x \cosh \left (2 \, a\right ) + x \sinh \left (2 \, a\right )\right )} \sinh \left (b x^{2} + a\right )^{2} + 2 \, {\left (x \cosh \left (b x^{2} + a\right ) \cosh \left (2 \, a\right ) + x \cosh \left (b x^{2} + a\right ) \sinh \left (2 \, a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {-b} \operatorname {erf}\left (\sqrt {2} \sqrt {-b} x\right ) + \sqrt {2} \sqrt {\pi } {\left (x \cosh \left (b x^{2} + a\right )^{2} \cosh \left (2 \, a\right ) - x \cosh \left (b x^{2} + a\right )^{2} \sinh \left (2 \, a\right ) + {\left (x \cosh \left (2 \, a\right ) - x \sinh \left (2 \, a\right )\right )} \sinh \left (b x^{2} + a\right )^{2} + 2 \, {\left (x \cosh \left (b x^{2} + a\right ) \cosh \left (2 \, a\right ) - x \cosh \left (b x^{2} + a\right ) \sinh \left (2 \, a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {b} \operatorname {erf}\left (\sqrt {2} \sqrt {b} x\right ) + 2 \, {\left (3 \, \cosh \left (b x^{2} + a\right )^{2} - 1\right )} \sinh \left (b x^{2} + a\right )^{2} - 2 \, \cosh \left (b x^{2} + a\right )^{2} + 4 \, {\left (\cosh \left (b x^{2} + a\right )^{3} - \cosh \left (b x^{2} + a\right )\right )} \sinh \left (b x^{2} + a\right ) + 1}{4 \, {\left (x \cosh \left (b x^{2} + a\right )^{2} + 2 \, x \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right ) + x \sinh \left (b x^{2} + a\right )^{2}\right )}} \]
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 \[ \int \frac {\sinh \left (b x^{2} + a\right )^{2}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 86, normalized size = 0.98 \[ \frac {1}{2 x}-\frac {{\mathrm e}^{-2 a} {\mathrm e}^{-2 b \,x^{2}}}{4 x}-\frac {{\mathrm e}^{-2 a} \sqrt {b}\, \sqrt {\pi }\, \sqrt {2}\, \erf \left (x \sqrt {2}\, \sqrt {b}\right )}{4}-\frac {{\mathrm e}^{2 a} {\mathrm e}^{2 b \,x^{2}}}{4 x}+\frac {{\mathrm e}^{2 a} b \sqrt {\pi }\, \erf \left (\sqrt {-2 b}\, x \right )}{2 \sqrt {-2 b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 61, normalized size = 0.69 \[ -\frac {\sqrt {2} \sqrt {b x^{2}} e^{\left (-2 \, a\right )} \Gamma \left (-\frac {1}{2}, 2 \, b x^{2}\right )}{8 \, x} - \frac {\sqrt {2} \sqrt {-b x^{2}} e^{\left (2 \, a\right )} \Gamma \left (-\frac {1}{2}, -2 \, b x^{2}\right )}{8 \, x} + \frac {1}{2 \, x} \]
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 \[ \int \frac {{\mathrm {sinh}\left (b\,x^2+a\right )}^2}{x^2} \,d x \]
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 \[ \int \frac {\sinh ^{2}{\left (a + b x^{2} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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