Optimal. Leaf size=133 \[ \frac {\sqrt {\pi } f^a e^{-\frac {e^2}{4 c \log (f)}-d} \text {erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{d-\frac {e^2}{4 c \log (f)}} \text {erfi}\left (\frac {2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.20, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5512, 2287, 2234, 2204} \[ \frac {\sqrt {\pi } f^a e^{-\frac {e^2}{4 c \log (f)}-d} \text {Erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{d-\frac {e^2}{4 c \log (f)}} \text {Erfi}\left (\frac {2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2287
Rule 5512
Rubi steps
\begin {align*} \int f^{a+c x^2} \sinh (d+e x) \, dx &=\int \left (-\frac {1}{2} e^{-d-e x} f^{a+c x^2}+\frac {1}{2} e^{d+e x} f^{a+c x^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int e^{-d-e x} f^{a+c x^2} \, dx\right )+\frac {1}{2} \int e^{d+e x} f^{a+c x^2} \, dx\\ &=-\left (\frac {1}{2} \int e^{-d-e x+a \log (f)+c x^2 \log (f)} \, dx\right )+\frac {1}{2} \int e^{d+e x+a \log (f)+c x^2 \log (f)} \, dx\\ &=-\left (\frac {1}{2} \left (e^{-d-\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(-e+2 c x \log (f))^2}{4 c \log (f)}} \, dx\right )+\frac {1}{2} \left (e^{d-\frac {e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(e+2 c x \log (f))^2}{4 c \log (f)}} \, dx\\ &=\frac {e^{-d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{d-\frac {e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 104, normalized size = 0.78 \[ \frac {\sqrt {\pi } f^a e^{-\frac {e^2}{4 c \log (f)}} \left ((\sinh (d)-\cosh (d)) \text {erfi}\left (\frac {2 c x \log (f)-e}{2 \sqrt {c} \sqrt {\log (f)}}\right )+(\sinh (d)+\cosh (d)) \text {erfi}\left (\frac {2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 217, normalized size = 1.63 \[ -\frac {\sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (\frac {4 \, a c \log \relax (f)^{2} + 4 \, c d \log \relax (f) - e^{2}}{4 \, c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \relax (f)^{2} + 4 \, c d \log \relax (f) - e^{2}}{4 \, c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) + e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) - \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (\frac {4 \, a c \log \relax (f)^{2} - 4 \, c d \log \relax (f) - e^{2}}{4 \, c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \relax (f)^{2} - 4 \, c d \log \relax (f) - e^{2}}{4 \, c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) - e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right )}{4 \, c \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 132, normalized size = 0.99 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {e}{c \log \relax (f)}\right )}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} + 4 \, c d \log \relax (f) - e^{2}}{4 \, c \log \relax (f)}\right )}}{4 \, \sqrt {-c \log \relax (f)}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x - \frac {e}{c \log \relax (f)}\right )}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} - 4 \, c d \log \relax (f) - e^{2}}{4 \, c \log \relax (f)}\right )}}{4 \, \sqrt {-c \log \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 117, normalized size = 0.88 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 d \ln \relax (f ) c -e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {e}{2 \sqrt {-c \ln \relax (f )}}\right )}{4 \sqrt {-c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d \ln \relax (f ) c +e^{2}}{4 \ln \relax (f ) c}} \erf \left (\sqrt {-c \ln \relax (f )}\, x +\frac {e}{2 \sqrt {-c \ln \relax (f )}}\right )}{4 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 105, normalized size = 0.79 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (d - \frac {e^{2}}{4 \, c \log \relax (f)}\right )}}{4 \, \sqrt {-c \log \relax (f)}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x + \frac {e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (-d - \frac {e^{2}}{4 \, c \log \relax (f)}\right )}}{4 \, \sqrt {-c \log \relax (f)}} \]
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 \[ \int f^{c\,x^2+a}\,\mathrm {sinh}\left (d+e\,x\right ) \,d x \]
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 \[ \int f^{a + c x^{2}} \sinh {\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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