Optimal. Leaf size=110 \[ \frac {1}{4} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {b^2 \log ^2(f)}{4 f}} \text {erfi}\left (\frac {b \log (f)+2 f x}{2 \sqrt {f}}\right )-\frac {1}{4} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {b^2 \log ^2(f)}{4 f}-d} \text {erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5512, 2287, 2234, 2205, 2204} \[ \frac {1}{4} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {b^2 \log ^2(f)}{4 f}} \text {Erfi}\left (\frac {b \log (f)+2 f x}{2 \sqrt {f}}\right )-\frac {1}{4} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {b^2 \log ^2(f)}{4 f}-d} \text {Erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right ) \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 2234
Rule 2287
Rule 5512
Rubi steps
\begin {align*} \int f^{a+b x} \sinh \left (d+f x^2\right ) \, dx &=\int \left (-\frac {1}{2} e^{-d-f x^2} f^{a+b x}+\frac {1}{2} e^{d+f x^2} f^{a+b x}\right ) \, dx\\ &=-\left (\frac {1}{2} \int e^{-d-f x^2} f^{a+b x} \, dx\right )+\frac {1}{2} \int e^{d+f x^2} f^{a+b x} \, dx\\ &=-\left (\frac {1}{2} \int e^{-d-f x^2+a \log (f)+b x \log (f)} \, dx\right )+\frac {1}{2} \int e^{d+f x^2+a \log (f)+b x \log (f)} \, dx\\ &=\frac {1}{2} \left (e^{d-\frac {b^2 \log ^2(f)}{4 f}} f^a\right ) \int e^{\frac {(2 f x+b \log (f))^2}{4 f}} \, dx-\frac {1}{2} \left (e^{-d+\frac {b^2 \log ^2(f)}{4 f}} f^a\right ) \int e^{-\frac {(-2 f x+b \log (f))^2}{4 f}} \, dx\\ &=-\frac {1}{4} e^{-d+\frac {b^2 \log ^2(f)}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{4} e^{d-\frac {b^2 \log ^2(f)}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {2 f x+b \log (f)}{2 \sqrt {f}}\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 103, normalized size = 0.94 \[ \frac {1}{4} \sqrt {\pi } f^{a-\frac {1}{2}} e^{-\frac {b^2 \log ^2(f)}{4 f}} \left ((\sinh (d)+\cosh (d)) \text {erfi}\left (\frac {b \log (f)+2 f x}{2 \sqrt {f}}\right )-e^{\frac {b^2 \log ^2(f)}{2 f}} (\cosh (d)-\sinh (d)) \text {erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 213, normalized size = 1.94 \[ -\frac {\sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} - 4 \, a f \log \relax (f) - 4 \, d f}{4 \, f}\right ) \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \relax (f)\right )} \sqrt {-f}}{2 \, f}\right ) - \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \relax (f)^{2} + 4 \, a f \log \relax (f) - 4 \, d f}{4 \, f}\right ) \operatorname {erf}\left (-\frac {2 \, f x - b \log \relax (f)}{2 \, \sqrt {f}}\right ) - \sqrt {\pi } \sqrt {f} \operatorname {erf}\left (-\frac {2 \, f x - b \log \relax (f)}{2 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} + 4 \, a f \log \relax (f) - 4 \, d f}{4 \, f}\right ) - \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \relax (f)\right )} \sqrt {-f}}{2 \, f}\right ) \sinh \left (\frac {b^{2} \log \relax (f)^{2} - 4 \, a f \log \relax (f) - 4 \, d f}{4 \, f}\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 106, normalized size = 0.96 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {f} {\left (2 \, x - \frac {b \log \relax (f)}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \relax (f)^{2} + 4 \, a f \log \relax (f) - 4 \, d f}{4 \, f}\right )}}{4 \, \sqrt {f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-f} {\left (2 \, x + \frac {b \log \relax (f)}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a f \log \relax (f) - 4 \, d f}{4 \, f}\right )}}{4 \, \sqrt {-f}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 100, normalized size = 0.91 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-4 d f}{4 f}} \erf \left (-\sqrt {-f}\, x +\frac {\ln \relax (f ) b}{2 \sqrt {-f}}\right )}{4 \sqrt {-f}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {\ln \relax (f )^{2} b^{2}-4 d f}{4 f}} \erf \left (-\sqrt {f}\, x +\frac {\ln \relax (f ) b}{2 \sqrt {f}}\right )}{4 \sqrt {f}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 90, normalized size = 0.82 \[ -\frac {1}{4} \, \sqrt {\pi } f^{a - \frac {1}{2}} \operatorname {erf}\left (\sqrt {f} x - \frac {b \log \relax (f)}{2 \, \sqrt {f}}\right ) e^{\left (\frac {b^{2} \log \relax (f)^{2}}{4 \, f} - d\right )} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-f} x - \frac {b \log \relax (f)}{2 \, \sqrt {-f}}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2}}{4 \, f} + d\right )}}{4 \, \sqrt {-f}} \]
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 \[ \int f^{a+b\,x}\,\mathrm {sinh}\left (f\,x^2+d\right ) \,d x \]
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 \[ \int f^{a + b x} \sinh {\left (d + f x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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