Optimal. Leaf size=315 \[ -\frac {3 \sqrt {\pi } f^a e^{-\frac {(e-b \log (f))^2}{4 c \log (f)}-d} \text {erfi}\left (\frac {-b \log (f)-2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{-\frac {(3 e-b \log (f))^2}{4 c \log (f)}-3 d} \text {erfi}\left (\frac {-b \log (f)-2 c x \log (f)+3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {3 \sqrt {\pi } f^a e^{d-\frac {(b \log (f)+e)^2}{4 c \log (f)}} \text {erfi}\left (\frac {b \log (f)+2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{3 d-\frac {(b \log (f)+3 e)^2}{4 c \log (f)}} \text {erfi}\left (\frac {b \log (f)+2 c x \log (f)+3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.46, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5513, 2287, 2234, 2204} \[ -\frac {3 \sqrt {\pi } f^a e^{-\frac {(e-b \log (f))^2}{4 c \log (f)}-d} \text {Erfi}\left (\frac {-b \log (f)-2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {\sqrt {\pi } f^a e^{-\frac {(3 e-b \log (f))^2}{4 c \log (f)}-3 d} \text {Erfi}\left (\frac {-b \log (f)-2 c x \log (f)+3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {3 \sqrt {\pi } f^a e^{d-\frac {(b \log (f)+e)^2}{4 c \log (f)}} \text {Erfi}\left (\frac {b \log (f)+2 c x \log (f)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {\sqrt {\pi } f^a e^{3 d-\frac {(b \log (f)+3 e)^2}{4 c \log (f)}} \text {Erfi}\left (\frac {b \log (f)+2 c x \log (f)+3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2287
Rule 5513
Rubi steps
\begin {align*} \int f^{a+b x+c x^2} \cosh ^3(d+e x) \, dx &=\int \left (\frac {1}{8} e^{-3 d-3 e x} f^{a+b x+c x^2}+\frac {3}{8} e^{-d-e x} f^{a+b x+c x^2}+\frac {3}{8} e^{d+e x} f^{a+b x+c x^2}+\frac {1}{8} e^{3 d+3 e x} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac {1}{8} \int e^{-3 d-3 e x} f^{a+b x+c x^2} \, dx+\frac {1}{8} \int e^{3 d+3 e x} f^{a+b x+c x^2} \, dx+\frac {3}{8} \int e^{-d-e x} f^{a+b x+c x^2} \, dx+\frac {3}{8} \int e^{d+e x} f^{a+b x+c x^2} \, dx\\ &=\frac {1}{8} \int \exp \left (-3 d+a \log (f)+c x^2 \log (f)-x (3 e-b \log (f))\right ) \, dx+\frac {1}{8} \int \exp \left (3 d+a \log (f)+c x^2 \log (f)+x (3 e+b \log (f))\right ) \, dx+\frac {3}{8} \int \exp \left (-d+a \log (f)+c x^2 \log (f)-x (e-b \log (f))\right ) \, dx+\frac {3}{8} \int \exp \left (d+a \log (f)+c x^2 \log (f)+x (e+b \log (f))\right ) \, dx\\ &=\frac {1}{8} \left (3 e^{-d-\frac {(e-b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(-e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac {1}{8} \left (e^{-3 d-\frac {(3 e-b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(-3 e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx+\frac {1}{8} \left (3 e^{d-\frac {(e+b \log (f))^2}{4 c \log (f)}} f^a\right ) \int e^{\frac {(e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}} \, dx+\frac {1}{8} \left (e^{3 d-\frac {(3 e+b \log (f))^2}{4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(3 e+b \log (f)+2 c x \log (f))^2}{4 c \log (f)}\right ) \, dx\\ &=-\frac {3 e^{-d-\frac {(e-b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-3 d-\frac {(3 e-b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e-b \log (f)-2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {3 e^{d-\frac {(e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+b \log (f)+2 c x \log (f)}{2 \sqrt {c} \sqrt {\log (f)}}\right )}{16 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{3 d-\frac {(3 e+b \log (f))^2}{4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e+b \log (f)+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 = 1.02, size = 262, normalized size = 0.83 \[ \frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} e^{-\frac {3 e (2 b \log (f)+3 e)}{4 c \log (f)}} \left ((\sinh (d)+\cosh (d)) \left (3 (\cosh (2 d)-\sinh (2 d)) e^{\frac {2 e (b \log (f)+e)}{c \log (f)}} \text {erfi}\left (\frac {\log (f) (b+2 c x)-e}{2 \sqrt {c} \sqrt {\log (f)}}\right )+(\sinh (2 d)+\cosh (2 d)) \text {erfi}\left (\frac {\log (f) (b+2 c x)+3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )+3 e^{\frac {e (b \log (f)+2 e)}{c \log (f)}} \text {erfi}\left (\frac {\log (f) (b+2 c x)+e}{2 \sqrt {c} \sqrt {\log (f)}}\right )\right )+e^{\frac {3 b e}{c}} (\cosh (3 d)-\sinh (3 d)) \text {erfi}\left (\frac {\log (f) (b+2 c x)-3 e}{2 \sqrt {c} \sqrt {\log (f)}}\right )\right )}{16 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 526, normalized size = 1.67 \[ -\frac {\sqrt {-c \log \relax (f)} {\left (\sqrt {\pi } \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + 9 \, e^{2} - 6 \, {\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} + 9 \, e^{2} - 6 \, {\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) + 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 {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + e^{2} - 2 \, {\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} + e^{2} - 2 \, {\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) + 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 {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} + e^{2} + 2 \, {\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} + e^{2} + 2 \, {\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) - 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} + 9 \, e^{2} + 6 \, {\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} + 9 \, e^{2} + 6 \, {\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) - 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 = 343, normalized size = 1.09 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b \log \relax (f) - 3 \, e}{c \log \relax (f)}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)^{2} + 12 \, c d \log \relax (f) - 6 \, b e \log \relax (f) + 9 \, e^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b \log \relax (f) - e}{c \log \relax (f)}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)^{2} + 4 \, c d \log \relax (f) - 2 \, b e \log \relax (f) + e^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b \log \relax (f) + e}{c \log \relax (f)}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)^{2} - 4 \, c d \log \relax (f) + 2 \, b e \log \relax (f) + e^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b \log \relax (f) + 3 \, e}{c \log \relax (f)}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)^{2} - 12 \, c d \log \relax (f) + 6 \, b e \log \relax (f) + 9 \, e^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 316, normalized size = 1.00 \[ -\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-6 \ln \relax (f ) b e +12 d \ln \relax (f ) c +9 e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {b \ln \relax (f )-3 e}{2 \sqrt {-c \ln \relax (f )}}\right )}{16 \sqrt {-c \ln \relax (f )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+6 \ln \relax (f ) b e -12 d \ln \relax (f ) c +9 e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {3 e +b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}\right )}{16 \sqrt {-c \ln \relax (f )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}-2 \ln \relax (f ) b e +4 d \ln \relax (f ) c +e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {b \ln \relax (f )-e}{2 \sqrt {-c \ln \relax (f )}}\right )}{16 \sqrt {-c \ln \relax (f )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {\ln \relax (f )^{2} b^{2}+2 \ln \relax (f ) b e -4 d \ln \relax (f ) c +e^{2}}{4 \ln \relax (f ) c}} \erf \left (-\sqrt {-c \ln \relax (f )}\, x +\frac {e +b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}\right )}{16 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 263, normalized size = 0.83 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f) + 3 \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (3 \, d - \frac {{\left (b \log \relax (f) + 3 \, e\right )}^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f) + e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (d - \frac {{\left (b \log \relax (f) + e\right )}^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f) - e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (-d - \frac {{\left (b \log \relax (f) - e\right )}^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f) - 3 \, e}{2 \, \sqrt {-c \log \relax (f)}}\right ) e^{\left (-3 \, d - \frac {{\left (b \log \relax (f) - 3 \, e\right )}^{2}}{4 \, c \log \relax (f)}\right )}}{16 \, \sqrt {-c \log \relax (f)}} \]
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 \[ \int f^{c\,x^2+b\,x+a}\,{\mathrm {cosh}\left (d+e\,x\right )}^3 \,d x \]
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 \[ \int f^{a + b x + c x^{2}} \cosh ^{3}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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