Optimal. Leaf size=148 \[ \frac {1}{8} \sqrt {\frac {\pi }{2}} f^{a-\frac {1}{2}} e^{\frac {b^2 \log ^2(f)}{8 f}-2 d} \text {erf}\left (\frac {4 f x-b \log (f)}{2 \sqrt {2} \sqrt {f}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} f^{a-\frac {1}{2}} e^{2 d-\frac {b^2 \log ^2(f)}{8 f}} \text {erfi}\left (\frac {b \log (f)+4 f x}{2 \sqrt {2} \sqrt {f}}\right )-\frac {f^{a+b x}}{2 b \log (f)} \]
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Rubi [A] time = 0.19, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5512, 2194, 2287, 2234, 2205, 2204} \[ \frac {1}{8} \sqrt {\frac {\pi }{2}} f^{a-\frac {1}{2}} e^{\frac {b^2 \log ^2(f)}{8 f}-2 d} \text {Erf}\left (\frac {4 f x-b \log (f)}{2 \sqrt {2} \sqrt {f}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} f^{a-\frac {1}{2}} e^{2 d-\frac {b^2 \log ^2(f)}{8 f}} \text {Erfi}\left (\frac {b \log (f)+4 f x}{2 \sqrt {2} \sqrt {f}}\right )-\frac {f^{a+b x}}{2 b \log (f)} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2204
Rule 2205
Rule 2234
Rule 2287
Rule 5512
Rubi steps
\begin {align*} \int f^{a+b x} \sinh ^2\left (d+f x^2\right ) \, dx &=\int \left (-\frac {1}{2} f^{a+b x}+\frac {1}{4} e^{-2 d-2 f x^2} f^{a+b x}+\frac {1}{4} e^{2 d+2 f x^2} f^{a+b x}\right ) \, dx\\ &=\frac {1}{4} \int e^{-2 d-2 f x^2} f^{a+b x} \, dx+\frac {1}{4} \int e^{2 d+2 f x^2} f^{a+b x} \, dx-\frac {1}{2} \int f^{a+b x} \, dx\\ &=-\frac {f^{a+b x}}{2 b \log (f)}+\frac {1}{4} \int e^{-2 d-2 f x^2+a \log (f)+b x \log (f)} \, dx+\frac {1}{4} \int e^{2 d+2 f x^2+a \log (f)+b x \log (f)} \, dx\\ &=-\frac {f^{a+b x}}{2 b \log (f)}+\frac {1}{4} \left (e^{2 d-\frac {b^2 \log ^2(f)}{8 f}} f^a\right ) \int e^{\frac {(4 f x+b \log (f))^2}{8 f}} \, dx+\frac {1}{4} \left (e^{-2 d+\frac {b^2 \log ^2(f)}{8 f}} f^a\right ) \int e^{-\frac {(-4 f x+b \log (f))^2}{8 f}} \, dx\\ &=\frac {1}{8} e^{-2 d+\frac {b^2 \log ^2(f)}{8 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {4 f x-b \log (f)}{2 \sqrt {2} \sqrt {f}}\right )+\frac {1}{8} e^{2 d-\frac {b^2 \log ^2(f)}{8 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {4 f x+b \log (f)}{2 \sqrt {2} \sqrt {f}}\right )-\frac {f^{a+b x}}{2 b \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 149, normalized size = 1.01 \[ \frac {1}{16} f^a \left (\frac {\sqrt {2 \pi } e^{\frac {b^2 \log ^2(f)}{8 f}} (\cosh (2 d)-\sinh (2 d)) \text {erf}\left (\frac {4 f x-b \log (f)}{2 \sqrt {2} \sqrt {f}}\right )}{\sqrt {f}}+\frac {\sqrt {2 \pi } e^{-\frac {b^2 \log ^2(f)}{8 f}} (\sinh (2 d)+\cosh (2 d)) \text {erfi}\left (\frac {b \log (f)+4 f x}{2 \sqrt {2} \sqrt {f}}\right )}{\sqrt {f}}-\frac {8 f^{b x}}{b \log (f)}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 278, normalized size = 1.88 \[ -\frac {\sqrt {2} \sqrt {\pi } b \sqrt {-f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} - 8 \, a f \log \relax (f) - 16 \, d f}{8 \, f}\right ) \operatorname {erf}\left (\frac {\sqrt {2} {\left (4 \, f x + b \log \relax (f)\right )} \sqrt {-f}}{4 \, f}\right ) \log \relax (f) + \sqrt {2} \sqrt {\pi } b \sqrt {f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} + 8 \, a f \log \relax (f) - 16 \, d f}{8 \, f}\right ) \operatorname {erf}\left (-\frac {\sqrt {2} {\left (4 \, f x - b \log \relax (f)\right )}}{4 \, \sqrt {f}}\right ) \log \relax (f) + \sqrt {2} \sqrt {\pi } b \sqrt {f} \operatorname {erf}\left (-\frac {\sqrt {2} {\left (4 \, f x - b \log \relax (f)\right )}}{4 \, \sqrt {f}}\right ) \log \relax (f) \sinh \left (\frac {b^{2} \log \relax (f)^{2} + 8 \, a f \log \relax (f) - 16 \, d f}{8 \, f}\right ) - \sqrt {2} \sqrt {\pi } b \sqrt {-f} \operatorname {erf}\left (\frac {\sqrt {2} {\left (4 \, f x + b \log \relax (f)\right )} \sqrt {-f}}{4 \, f}\right ) \log \relax (f) \sinh \left (\frac {b^{2} \log \relax (f)^{2} - 8 \, a f \log \relax (f) - 16 \, d f}{8 \, f}\right ) + 8 \, f \cosh \left ({\left (b x + a\right )} \log \relax (f)\right ) + 8 \, f \sinh \left ({\left (b x + a\right )} \log \relax (f)\right )}{16 \, b f \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.18, size = 356, normalized size = 2.41 \[ -\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} \sqrt {f} {\left (4 \, x - \frac {b \log \relax (f)}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \relax (f)^{2} + 8 \, a f \log \relax (f) - 16 \, d f}{8 \, f}\right )}}{16 \, \sqrt {f}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} \sqrt {-f} {\left (4 \, x + \frac {b \log \relax (f)}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 8 \, a f \log \relax (f) - 16 \, d f}{8 \, f}\right )}}{16 \, \sqrt {-f}} - {\left (\frac {2 \, b \cos \left (-\frac {1}{2} \, \pi b x \mathrm {sgn}\relax (f) + \frac {1}{2} \, \pi b x - \frac {1}{2} \, \pi a \mathrm {sgn}\relax (f) + \frac {1}{2} \, \pi a\right ) \log \left ({\left | f \right |}\right )}{4 \, b^{2} \log \left ({\left | f \right |}\right )^{2} + {\left (\pi b \mathrm {sgn}\relax (f) - \pi b\right )}^{2}} - \frac {{\left (\pi b \mathrm {sgn}\relax (f) - \pi b\right )} \sin \left (-\frac {1}{2} \, \pi b x \mathrm {sgn}\relax (f) + \frac {1}{2} \, \pi b x - \frac {1}{2} \, \pi a \mathrm {sgn}\relax (f) + \frac {1}{2} \, \pi a\right )}{4 \, b^{2} \log \left ({\left | f \right |}\right )^{2} + {\left (\pi b \mathrm {sgn}\relax (f) - \pi b\right )}^{2}}\right )} e^{\left (b x \log \left ({\left | f \right |}\right ) + a \log \left ({\left | f \right |}\right )\right )} - \frac {1}{2} i \, {\left (\frac {2 i \, e^{\left (\frac {1}{2} i \, \pi b x \mathrm {sgn}\relax (f) - \frac {1}{2} i \, \pi b x + \frac {1}{2} i \, \pi a \mathrm {sgn}\relax (f) - \frac {1}{2} i \, \pi a\right )}}{2 i \, \pi b \mathrm {sgn}\relax (f) - 2 i \, \pi b + 4 \, b \log \left ({\left | f \right |}\right )} - \frac {2 i \, e^{\left (-\frac {1}{2} i \, \pi b x \mathrm {sgn}\relax (f) + \frac {1}{2} i \, \pi b x - \frac {1}{2} i \, \pi a \mathrm {sgn}\relax (f) + \frac {1}{2} i \, \pi a\right )}}{-2 i \, \pi b \mathrm {sgn}\relax (f) + 2 i \, \pi b + 4 \, b \log \left ({\left | f \right |}\right )}\right )} e^{\left (b x \log \left ({\left | f \right |}\right ) + a \log \left ({\left | f \right |}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 126, normalized size = 0.85 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {\ln \relax (f )^{2} b^{2}-16 d f}{8 f}} \sqrt {2}\, \erf \left (-\sqrt {2}\, \sqrt {f}\, x +\frac {\ln \relax (f ) b \sqrt {2}}{4 \sqrt {f}}\right )}{16 \sqrt {f}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-16 d f}{8 f}} \erf \left (-\sqrt {-2 f}\, x +\frac {\ln \relax (f ) b}{2 \sqrt {-2 f}}\right )}{8 \sqrt {-2 f}}-\frac {f^{a} f^{b x}}{2 \ln \relax (f ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 127, normalized size = 0.86 \[ \frac {\sqrt {2} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {2} \sqrt {f} x - \frac {\sqrt {2} b \log \relax (f)}{4 \, \sqrt {f}}\right ) e^{\left (\frac {b^{2} \log \relax (f)^{2}}{8 \, f} - 2 \, d\right )}}{16 \, \sqrt {f}} + \frac {\sqrt {2} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {2} \sqrt {-f} x - \frac {\sqrt {2} b \log \relax (f)}{4 \, \sqrt {-f}}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2}}{8 \, f} + 2 \, d\right )}}{16 \, \sqrt {-f}} - \frac {f^{b x + a}}{2 \, b \log \relax (f)} \]
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 \[ \int f^{a+b\,x}\,{\mathrm {sinh}\left (f\,x^2+d\right )}^2 \,d x \]
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 \[ \int f^{a + b x} \sinh ^{2}{\left (d + f x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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