Optimal. Leaf size=219 \[ -\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {Erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 d-\frac {(2 e-b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {2 e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{2 d-\frac {(2 e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {2 e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{8 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A]
time = 0.28, 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 = {5623, 2266,
2235, 2325} \begin {gather*} -\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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2266
Rule 2325
Rule 5623
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 = 0.39, size = 183, normalized size = 0.84 \begin {gather*} \frac {e^{-\frac {e (e+b \log (f))}{c \log (f)}} f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \left (-2 e^{\frac {e (e+b \log (f))}{c \log (f)}} \text {Erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )+e^{\frac {2 b e}{c}} \text {Erfi}\left (\frac {-2 e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (2 d)-\sinh (2 d))+\text {Erfi}\left (\frac {2 e+(b+2 c x) \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right ) (\cosh (2 d)+\sinh (2 d))\right )}{8 \sqrt {c} \sqrt {\log (f)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.46, size = 211, normalized size = 0.96
method | result | size |
risch | \(-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-4 \ln \left (f \right ) b e +8 d \ln \left (f \right ) c +4 e^{2}}{4 \ln \left (f \right ) c}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )-2 e}{2 \sqrt {-c \ln \left (f \right )}}\right )}{8 \sqrt {-c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+4 \ln \left (f \right ) b e -8 d \ln \left (f \right ) c +4 e^{2}}{4 \ln \left (f \right ) c}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {2 e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{8 \sqrt {-c \ln \left (f \right )}}+\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(211\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 189, normalized size = 0.86 \begin {gather*} \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right ) + 2 \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (2 \, d - \frac {{\left (b \log \left (f\right ) + 2 \, e\right )}^{2}}{4 \, c \log \left (f\right )}\right )}}{8 \, \sqrt {-c \log \left (f\right )}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right ) - 2 \, e}{2 \, \sqrt {-c \log \left (f\right )}}\right ) e^{\left (-2 \, d - \frac {{\left (b \log \left (f\right ) - 2 \, e\right )}^{2}}{4 \, c \log \left (f\right )}\right )}}{8 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right )}}\right )}{4 \, \sqrt {-c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 429 vs.
\(2 (171) = 342\).
time = 0.38, size = 429, normalized size = 1.96 \begin {gather*} \frac {2 \, \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )}{4 \, c}\right ) + \sqrt {\pi } \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )}}{2 \, c}\right ) - \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + 4 \, \cosh \left (1\right )^{2} - 4 \, {\left (2 \, c d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \log \left (f\right ) + 8 \, \cosh \left (1\right ) \sinh \left (1\right ) + 4 \, \sinh \left (1\right )^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + 4 \, \cosh \left (1\right )^{2} - 4 \, {\left (2 \, c d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \log \left (f\right ) + 8 \, \cosh \left (1\right ) \sinh \left (1\right ) + 4 \, \sinh \left (1\right )^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) + 2 \, \cosh \left (1\right ) + 2 \, \sinh \left (1\right )\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right ) - \sqrt {-c \log \left (f\right )} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + 4 \, \cosh \left (1\right )^{2} + 4 \, {\left (2 \, c d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \log \left (f\right ) + 8 \, \cosh \left (1\right ) \sinh \left (1\right ) + 4 \, \sinh \left (1\right )^{2}}{4 \, c \log \left (f\right )}\right ) + \sqrt {\pi } \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} + 4 \, \cosh \left (1\right )^{2} + 4 \, {\left (2 \, c d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \log \left (f\right ) + 8 \, \cosh \left (1\right ) \sinh \left (1\right ) + 4 \, \sinh \left (1\right )^{2}}{4 \, c \log \left (f\right )}\right )\right )} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) - 2 \, \cosh \left (1\right ) - 2 \, \sinh \left (1\right )\right )} \sqrt {-c \log \left (f\right )}}{2 \, c \log \left (f\right )}\right )}{8 \, c \log \left (f\right )} \end {gather*}
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 \begin {gather*} \int f^{a + b x + c x^{2}} \sinh ^{2}{\left (d + e x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 223, normalized size = 1.02 \begin {gather*} \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{4 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b \log \left (f\right ) - 2 \, e}{c \log \left (f\right )}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 8 \, c d \log \left (f\right ) - 4 \, b e \log \left (f\right ) + 4 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{8 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b \log \left (f\right ) + 2 \, e}{c \log \left (f\right )}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 8 \, c d \log \left (f\right ) + 4 \, b e \log \left (f\right ) + 4 \, e^{2}}{4 \, c \log \left (f\right )}\right )}}{8 \, \sqrt {-c \log \left (f\right )}} \end {gather*}
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 \begin {gather*} \int f^{c\,x^2+b\,x+a}\,{\mathrm {sinh}\left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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