Optimal. Leaf size=271 \[ -\frac {3 \sqrt {\pi } f^a e^{-\frac {e^2}{4 c \log (f)}-d} \text {erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{-\frac {9 e^2}{4 c \log (f)}-3 d} \text {erfi}\left (\frac {3 e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 \sqrt {\pi } f^a e^{d-\frac {e^2}{4 c \log (f)}} \text {erfi}\left (\frac {2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{3 d-\frac {9 e^2}{4 c \log (f)}} \text {erfi}\left (\frac {2 c x \log (f)+3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.35, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5512, 2287, 2234, 2204} \[ -\frac {3 \sqrt {\pi } f^a e^{-\frac {e^2}{4 c \log (f)}-d} \text {Erfi}\left (\frac {e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{-\frac {9 e^2}{4 c \log (f)}-3 d} \text {Erfi}\left (\frac {3 e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 \sqrt {\pi } f^a e^{d-\frac {e^2}{4 c \log (f)}} \text {Erfi}\left (\frac {2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{3 d-\frac {9 e^2}{4 c \log (f)}} \text {Erfi}\left (\frac {2 c x \log (f)+3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
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 ^3(d+e x) \, dx &=\int \left (-\frac {1}{8} e^{-3 d-3 e x} f^{a+c x^2}+\frac {3}{8} e^{-d-e x} f^{a+c x^2}-\frac {3}{8} e^{d+e x} f^{a+c x^2}+\frac {1}{8} e^{3 d+3 e x} f^{a+c x^2}\right ) \, dx\\ &=-\left (\frac {1}{8} \int e^{-3 d-3 e x} f^{a+c x^2} \, dx\right )+\frac {1}{8} \int e^{3 d+3 e x} f^{a+c x^2} \, dx+\frac {3}{8} \int e^{-d-e x} f^{a+c x^2} \, dx-\frac {3}{8} \int e^{d+e x} f^{a+c x^2} \, dx\\ &=-\left (\frac {1}{8} \int e^{-3 d-3 e x+a \log (f)+c x^2 \log (f)} \, dx\right )+\frac {1}{8} \int e^{3 d+3 e x+a \log (f)+c x^2 \log (f)} \, dx+\frac {3}{8} \int e^{-d-e x+a \log (f)+c x^2 \log (f)} \, dx-\frac {3}{8} \int e^{d+e x+a \log (f)+c x^2 \log (f)} \, dx\\ &=-\left (\frac {1}{8} \left (e^{-3 d-\frac {9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(-3 e+2 c x \log (f))^2}{4 c \log (f)}} \, dx\right )+\frac {1}{8} \left (e^{3 d-\frac {9 e^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(3 e+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{8} \left (3 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 {1}{8} \left (3 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 {3 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 )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-3 d-\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {3 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 )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{3 d-\frac {9 e^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 214, normalized size = 0.79 \[ \frac {\sqrt {\pi } f^a e^{-\frac {9 e^2}{4 c \log (f)}} \left ((\sinh (d)+\cosh (d)) \left (3 (\cosh (2 d)-\sinh (2 d)) e^{\frac {2 e^2}{c \log (f)}} \text {erfi}\left (\frac {2 c x \log (f)-e}{2 \sqrt {c} \sqrt {\log (f)}}\right )+(\sinh (2 d)+\cosh (2 d)) \text {erfi}\left (\frac {2 c x \log (f)+3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )-3 e^{\frac {2 e^2}{c \log (f)}} \text {erfi}\left (\frac {2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )\right )+(\sinh (3 d)-\cosh (3 d)) \text {erfi}\left (\frac {2 c x \log (f)-3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 427, normalized size = 1.58 \[ -\frac {\sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (\frac {4 \, a c \log \relax (f)^{2} + 12 \, c d \log \relax (f) - 9 \, e^{2}}{4 \, c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \relax (f)^{2} + 12 \, c d \log \relax (f) - 9 \, e^{2}}{4 \, c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) + 3 \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) - 3 \, \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (\frac {4 \, a c \log \relax (f)^{2} + 4 \, c d \log \relax (f) - e^{2}}{4 \, c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \relax (f)^{2} + 4 \, c d \log \relax (f) - e^{2}}{4 \, c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) + e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) + 3 \, \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (\frac {4 \, a c \log \relax (f)^{2} - 4 \, c d \log \relax (f) - e^{2}}{4 \, c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \relax (f)^{2} - 4 \, c d \log \relax (f) - e^{2}}{4 \, c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) - e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) - \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (\frac {4 \, a c \log \relax (f)^{2} - 12 \, c d \log \relax (f) - 9 \, e^{2}}{4 \, c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (\frac {4 \, a c \log \relax (f)^{2} - 12 \, c d \log \relax (f) - 9 \, e^{2}}{4 \, c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x \log \relax (f) - 3 \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right )}{16 \, c \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 264, normalized size = 0.97 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {3 \, e}{c \log \relax (f)}\right )}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} + 12 \, c d \log \relax (f) - 9 \, e^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {e}{c \log \relax (f)}\right )}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} + 4 \, c d \log \relax (f) - e^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x - \frac {e}{c \log \relax (f)}\right )}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} - 4 \, c d \log \relax (f) - e^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x - \frac {3 \, e}{c \log \relax (f)}\right )}\right ) e^{\left (\frac {4 \, a c \log \relax (f)^{2} - 12 \, c d \log \relax (f) - 9 \, e^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 234, normalized size = 0.86 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {3 d \ln \relax (f ) c -\frac {9 e^{2}}{4}}{c \ln \relax (f )}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {3 e}{2 \sqrt {-c \ln \relax (f )}}\right )}{16 \sqrt {-c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {3 \left (4 d \ln \relax (f ) c +3 e^{2}\right )}{4 \ln \relax (f ) c}} \erf \left (\sqrt {-c \ln \relax (f )}\, x +\frac {3 e}{2 \sqrt {-c \ln \relax (f )}}\right )}{16 \sqrt {-c \ln \relax (f )}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {4 d \ln \relax (f ) c +e^{2}}{4 \ln \relax (f ) c}} \erf \left (\sqrt {-c \ln \relax (f )}\, x +\frac {e}{2 \sqrt {-c \ln \relax (f )}}\right )}{16 \sqrt {-c \ln \relax (f )}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {4 d \ln \relax (f ) c -e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {e}{2 \sqrt {-c \ln \relax (f )}}\right )}{16 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 211, normalized size = 0.78 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {3 \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (3 \, d - \frac {9 \, e^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} - \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (d - \frac {e^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x + \frac {e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (-d - \frac {e^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x + \frac {3 \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (-3 \, d - \frac {9 \, e^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int f^{c\,x^2+a}\,{\mathrm {sinh}\left (d+e\,x\right )}^3 \,d x \]
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 \[ \int f^{a + c x^{2}} \sinh ^{3}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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