Optimal. Leaf size=140 \[ \frac {\sqrt {\pi } f^a e^{d-\frac {e^2}{4 (c \log (f)+f)}} \text {erfi}\left (\frac {2 x (c \log (f)+f)+e}{2 \sqrt {c \log (f)+f}}\right )}{4 \sqrt {c \log (f)+f}}-\frac {\sqrt {\pi } f^a e^{\frac {e^2}{4 f-4 c \log (f)}-d} \text {erf}\left (\frac {2 x (f-c \log (f))+e}{2 \sqrt {f-c \log (f)}}\right )}{4 \sqrt {f-c \log (f)}} \]
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Rubi [A] time = 0.32, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5512, 2287, 2234, 2205, 2204} \[ \frac {\sqrt {\pi } f^a e^{d-\frac {e^2}{4 (c \log (f)+f)}} \text {Erfi}\left (\frac {2 x (c \log (f)+f)+e}{2 \sqrt {c \log (f)+f}}\right )}{4 \sqrt {c \log (f)+f}}-\frac {\sqrt {\pi } f^a e^{\frac {e^2}{4 f-4 c \log (f)}-d} \text {Erf}\left (\frac {2 x (f-c \log (f))+e}{2 \sqrt {f-c \log (f)}}\right )}{4 \sqrt {f-c \log (f)}} \]
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+c x^2} \sinh \left (d+e x+f x^2\right ) \, dx &=\int \left (-\frac {1}{2} e^{-d-e x-f x^2} f^{a+c x^2}+\frac {1}{2} e^{d+e x+f x^2} f^{a+c x^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int e^{-d-e x-f x^2} f^{a+c x^2} \, dx\right )+\frac {1}{2} \int e^{d+e x+f x^2} f^{a+c x^2} \, dx\\ &=-\left (\frac {1}{2} \int e^{-d-e x+a \log (f)-x^2 (f-c \log (f))} \, dx\right )+\frac {1}{2} \int e^{d+e x+a \log (f)+x^2 (f+c \log (f))} \, dx\\ &=-\left (\frac {1}{2} \left (e^{-d+\frac {e^2}{4 f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(-e+2 x (-f+c \log (f)))^2}{4 (-f+c \log (f))}\right ) \, dx\right )+\frac {1}{2} \left (e^{d-\frac {e^2}{4 (f+c \log (f))}} f^a\right ) \int \exp \left (\frac {(e+2 x (f+c \log (f)))^2}{4 (f+c \log (f))}\right ) \, dx\\ &=-\frac {e^{-d+\frac {e^2}{4 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {e+2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{4 \sqrt {f-c \log (f)}}+\frac {e^{d-\frac {e^2}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{4 \sqrt {f+c \log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.74, size = 166, normalized size = 1.19 \[ \frac {\sqrt {\pi } f^a e^{-\frac {e^2}{4 (c \log (f)+f)}} \left (\sqrt {f-c \log (f)} (\sinh (d)+\cosh (d)) \text {erfi}\left (\frac {2 c x \log (f)+e+2 f x}{2 \sqrt {c \log (f)+f}}\right )-\sqrt {c \log (f)+f} (\cosh (d)-\sinh (d)) e^{\frac {e^2 f}{2 f^2-2 c^2 \log ^2(f)}} \text {erf}\left (\frac {-2 c x \log (f)+e+2 f x}{2 \sqrt {f-c \log (f)}}\right )\right )}{4 \sqrt {f-c \log (f)} \sqrt {c \log (f)+f}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 322, normalized size = 2.30 \[ \frac {{\left (\sqrt {\pi } {\left (c \log \relax (f) + f\right )} \cosh \left (\frac {4 \, a c \log \relax (f)^{2} - e^{2} + 4 \, d f - 4 \, {\left (c d + a f\right )} \log \relax (f)}{4 \, {\left (c \log \relax (f) - f\right )}}\right ) + \sqrt {\pi } {\left (c \log \relax (f) + f\right )} \sinh \left (\frac {4 \, a c \log \relax (f)^{2} - e^{2} + 4 \, d f - 4 \, {\left (c d + a f\right )} \log \relax (f)}{4 \, {\left (c \log \relax (f) - f\right )}}\right )\right )} \sqrt {-c \log \relax (f) + f} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) - 2 \, f x - e\right )} \sqrt {-c \log \relax (f) + f}}{2 \, {\left (c \log \relax (f) - f\right )}}\right ) - {\left (\sqrt {\pi } {\left (c \log \relax (f) - f\right )} \cosh \left (\frac {4 \, a c \log \relax (f)^{2} - e^{2} + 4 \, d f + 4 \, {\left (c d + a f\right )} \log \relax (f)}{4 \, {\left (c \log \relax (f) + f\right )}}\right ) + \sqrt {\pi } {\left (c \log \relax (f) - f\right )} \sinh \left (\frac {4 \, a c \log \relax (f)^{2} - e^{2} + 4 \, d f + 4 \, {\left (c d + a f\right )} \log \relax (f)}{4 \, {\left (c \log \relax (f) + f\right )}}\right )\right )} \sqrt {-c \log \relax (f) - f} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) + 2 \, f x + e\right )} \sqrt {-c \log \relax (f) - f}}{2 \, {\left (c \log \relax (f) + f\right )}}\right )}{4 \, {\left (c^{2} \log \relax (f)^{2} - f^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 172, normalized size = 1.23 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f) - f} {\left (2 \, x + \frac {e}{c \log \relax (f) + f}\right )}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} + 4 \, c d \log \relax (f) + 4 \, a f \log \relax (f) + 4 \, d f - e^{2}}{4 \, {\left (c \log \relax (f) + f\right )}}\right )}}{4 \, \sqrt {-c \log \relax (f) - f}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f) + f} {\left (2 \, x - \frac {e}{c \log \relax (f) - f}\right )}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} - 4 \, c d \log \relax (f) - 4 \, a f \log \relax (f) + 4 \, d f - e^{2}}{4 \, {\left (c \log \relax (f) - f\right )}}\right )}}{4 \, \sqrt {-c \log \relax (f) + f}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 147, normalized size = 1.05 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 d \ln \relax (f ) c +4 d f -e^{2}}{4 c \ln \relax (f )+4 f}} \erf \left (-\sqrt {-c \ln \relax (f )-f}\, x +\frac {e}{2 \sqrt {-c \ln \relax (f )-f}}\right )}{4 \sqrt {-c \ln \relax (f )-f}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d \ln \relax (f ) c -4 d f +e^{2}}{4 \left (-f +c \ln \relax (f )\right )}} \erf \left (x \sqrt {f -c \ln \relax (f )}+\frac {e}{2 \sqrt {f -c \ln \relax (f )}}\right )}{4 \sqrt {f -c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 127, normalized size = 0.91 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - f} x - \frac {e}{2 \, \sqrt {-c \log \relax (f) - f}}\right ) e^{\left (d - \frac {e^{2}}{4 \, {\left (c \log \relax (f) + f\right )}}\right )}}{4 \, \sqrt {-c \log \relax (f) - f}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + f} x + \frac {e}{2 \, \sqrt {-c \log \relax (f) + f}}\right ) e^{\left (-d - \frac {e^{2}}{4 \, {\left (c \log \relax (f) - f\right )}}\right )}}{4 \, \sqrt {-c \log \relax (f) + f}} \]
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 \[ \int f^{c\,x^2+a}\,\mathrm {sinh}\left (f\,x^2+e\,x+d\right ) \,d x \]
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 \[ \int f^{a + c x^{2}} \sinh {\left (d + e x + f x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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