Optimal. Leaf size=219 \[ -\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{-\frac {(2 e-b \log (f))^2}{4 c \log (f)}-2 d} \text {erfi}\left (\frac {-b \log (f)-2 c x \log (f)+2 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {(b \log (f)+2 e)^2}{4 c \log (f)}} \text {erfi}\left (\frac {b \log (f)+2 c x \log (f)+2 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.36, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5512, 2234, 2204, 2287} \[ -\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{-\frac {(2 e-b \log (f))^2}{4 c \log (f)}-2 d} \text {Erfi}\left (\frac {-b \log (f)-2 c x \log (f)+2 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {(b \log (f)+2 e)^2}{4 c \log (f)}} \text {Erfi}\left (\frac {b \log (f)+2 c x \log (f)+2 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2287
Rule 5512
Rubi steps
\begin {align*} \int f^{a+b x+c x^2} \sinh ^2(d+e x) \, dx &=\int \left (-\frac {1}{2} f^{a+b x+c x^2}+\frac {1}{4} e^{-2 d-2 e x} f^{a+b x+c x^2}+\frac {1}{4} e^{2 d+2 e x} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac {1}{4} \int e^{-2 d-2 e x} f^{a+b x+c x^2} \, dx+\frac {1}{4} \int e^{2 d+2 e x} f^{a+b x+c x^2} \, dx-\frac {1}{2} \int f^{a+b x+c x^2} \, dx\\ &=\frac {1}{4} \int \exp \left (-2 d+a \log (f)+c x^2 \log (f)-x (2 e-b \log (f))\right ) \, dx+\frac {1}{4} \int \exp \left (2 d+a \log (f)+c x^2 \log (f)+x (2 e+b \log (f))\right ) \, dx-\frac {1}{2} f^{a-\frac {b^2}{4 c}} \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx\\ &=-\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {1}{4} \left (e^{-2 d-\frac {(2 e-b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(-2 e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac {1}{4} \left (e^{2 d-\frac {(2 e+b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(2 e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx\\ &=-\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 d-\frac {(2 e-b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {2 e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{2 d-\frac {(2 e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {2 e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 183, normalized size = 0.84 \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} e^{-\frac {e (b \log (f)+e)}{c \log (f)}} \left (e^{\frac {2 b e}{c}} (\cosh (2 d)-\sinh (2 d)) \text {erfi}\left (\frac {\log (f) (b+2 c x)-2 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )+(\sinh (2 d)+\cosh (2 d)) \text {erfi}\left (\frac {\log (f) (b+2 c x)+2 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )-2 e^{\frac {e (b \log (f)+e)}{c \log (f)}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 343, normalized size = 1.57 \[ \frac {2 \, \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)}{4 \, c}\right ) + \sqrt {\pi } \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)}}{2 \, c}\right ) - \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + 4 \, e^{2} - 4 \, {\left (2 \, c d - b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + 4 \, e^{2} - 4 \, {\left (2 \, c d - b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \relax (f) + 2 \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right ) - \sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + 4 \, e^{2} + 4 \, {\left (2 \, c d - b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right ) + \sqrt {\pi } \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + 4 \, e^{2} + 4 \, {\left (2 \, c d - b e\right )} \log \relax (f)}{4 \, c \log \relax (f)}\right )\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \relax (f) - 2 \, e\right )} \sqrt {-c \log \relax (f)}}{2 \, c \log \relax (f)}\right )}{8 \, c \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 225, normalized size = 1.03 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f)}{4 \, c}\right )}}{4 \, \sqrt {-c \log \relax (f)}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b \log \relax (f) - 2 \, e}{c \log \relax (f)}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)^{2} + 8 \, c d \log \relax (f) - 4 \, b e \log \relax (f) + 4 \, e^{2}}{4 \, c \log \relax (f)}\right )}}{8 \, \sqrt {-c \log \relax (f)}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b \log \relax (f) + 2 \, e}{c \log \relax (f)}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)^{2} - 8 \, c d \log \relax (f) + 4 \, b e \log \relax (f) + 4 \, e^{2}}{4 \, c \log \relax (f)}\right )}}{8 \, \sqrt {-c \log \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 211, normalized size = 0.96 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-4 \ln \relax (f ) b e +8 d \ln \relax (f ) c +4 e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {b \ln \relax (f )-2 e}{2 \sqrt {-c \ln \relax (f )}}\right )}{8 \sqrt {-c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+4 \ln \relax (f ) b e -8 d \ln \relax (f ) c +4 e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {2 e +b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}\right )}{8 \sqrt {-c \ln \relax (f )}}+\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}\right )}{4 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 185, normalized size = 0.84 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f) + 2 \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (2 \, d - \frac {{\left (b \log \relax (f) + 2 \, e\right )}^{2}}{4 \, c \log \relax (f)}\right )}}{8 \, \sqrt {-c \log \relax (f)}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f) - 2 \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (-2 \, d - \frac {{\left (b \log \relax (f) - 2 \, e\right )}^{2}}{4 \, c \log \relax (f)}\right )}}{8 \, \sqrt {-c \log \relax (f)}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f)}}\right )}{4 \, \sqrt {-c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} \]
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 \[ \int f^{c\,x^2+b\,x+a}\,{\mathrm {sinh}\left (d+e\,x\right )}^2 \,d x \]
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 \[ \int f^{a + b x + c x^{2}} \sinh ^{2}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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