Optimal. Leaf size=128 \[ \frac {\sqrt {\pi } e^{-2 d} f^a \text {erf}\left (x \sqrt {2 f-c \log (f)}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } e^{2 d} f^a \text {erfi}\left (x \sqrt {c \log (f)+2 f}\right )}{8 \sqrt {c \log (f)+2 f}}+\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
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Rubi [A] time = 0.20, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5513, 2204, 2287, 2205} \[ \frac {\sqrt {\pi } e^{-2 d} f^a \text {Erf}\left (x \sqrt {2 f-c \log (f)}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } e^{2 d} f^a \text {Erfi}\left (x \sqrt {c \log (f)+2 f}\right )}{8 \sqrt {c \log (f)+2 f}}+\frac {\sqrt {\pi } f^a \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 2287
Rule 5513
Rubi steps
\begin {align*} \int f^{a+c x^2} \cosh ^2\left (d+f x^2\right ) \, dx &=\int \left (\frac {1}{2} f^{a+c x^2}+\frac {1}{4} e^{-2 d-2 f x^2} f^{a+c x^2}+\frac {1}{4} e^{2 d+2 f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac {1}{4} \int e^{-2 d-2 f x^2} f^{a+c x^2} \, dx+\frac {1}{4} \int e^{2 d+2 f x^2} f^{a+c x^2} \, dx+\frac {1}{2} \int f^{a+c x^2} \, dx\\ &=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {1}{4} \int e^{-2 d+a \log (f)-x^2 (2 f-c \log (f))} \, dx+\frac {1}{4} \int e^{2 d+a \log (f)+x^2 (2 f+c \log (f))} \, dx\\ &=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-2 d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {2 f-c \log (f)}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d} f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {2 f+c \log (f)}\right )}{8 \sqrt {2 f+c \log (f)}}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 179, normalized size = 1.40 \[ \frac {\sqrt {\pi } f^a \left (\left (2 c^2 \log ^2(f)-8 f^2\right ) \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )+\sqrt {c} \sqrt {\log (f)} \left (\sqrt {2 f-c \log (f)} (c \log (f)+2 f) (\sinh (2 d)-\cosh (2 d)) \text {erf}\left (x \sqrt {2 f-c \log (f)}\right )-(2 f-c \log (f)) \sqrt {c \log (f)+2 f} (\sinh (2 d)+\cosh (2 d)) \text {erfi}\left (x \sqrt {c \log (f)+2 f}\right )\right )\right )}{8 \sqrt {c} \sqrt {\log (f)} \left (c^2 \log ^2(f)-4 f^2\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 254, normalized size = 1.98 \[ -\frac {{\left (\sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 2 \, c f \log \relax (f)\right )} \cosh \left (a \log \relax (f) - 2 \, d\right ) + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} + 2 \, c f \log \relax (f)\right )} \sinh \left (a \log \relax (f) - 2 \, d\right )\right )} \sqrt {-c \log \relax (f) + 2 \, f} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + 2 \, f} x\right ) + {\left (\sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 2 \, c f \log \relax (f)\right )} \cosh \left (a \log \relax (f) + 2 \, d\right ) + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 2 \, c f \log \relax (f)\right )} \sinh \left (a \log \relax (f) + 2 \, d\right )\right )} \sqrt {-c \log \relax (f) - 2 \, f} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - 2 \, f} x\right ) + 2 \, {\left (\sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 4 \, f^{2}\right )} \cosh \left (a \log \relax (f)\right ) + \sqrt {\pi } {\left (c^{2} \log \relax (f)^{2} - 4 \, f^{2}\right )} \sinh \left (a \log \relax (f)\right )\right )} \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x\right )}{8 \, {\left (c^{3} \log \relax (f)^{3} - 4 \, c f^{2} \log \relax (f)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 107, normalized size = 0.84 \[ -\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-c \log \relax (f)} x\right )}{4 \, \sqrt {-c \log \relax (f)}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f) - 2 \, f} x\right ) e^{\left (a \log \relax (f) + 2 \, d\right )}}{8 \, \sqrt {-c \log \relax (f) - 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f) + 2 \, f} x\right ) e^{\left (a \log \relax (f) - 2 \, d\right )}}{8 \, \sqrt {-c \log \relax (f) + 2 \, f}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 101, normalized size = 0.79 \[ \frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-2 d} \erf \left (x \sqrt {2 f -c \ln \relax (f )}\right )}{8 \sqrt {2 f -c \ln \relax (f )}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{2 d} \erf \left (\sqrt {-c \ln \relax (f )-2 f}\, x \right )}{8 \sqrt {-c \ln \relax (f )-2 f}}+\frac {f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-c \ln \relax (f )}\, x \right )}{4 \sqrt {-c \ln \relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 100, normalized size = 0.78 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - 2 \, f} x\right ) e^{\left (2 \, d\right )}}{8 \, \sqrt {-c \log \relax (f) - 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + 2 \, f} x\right ) e^{\left (-2 \, d\right )}}{8 \, \sqrt {-c \log \relax (f) + 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x\right )}{4 \, \sqrt {-c \log \relax (f)}} \]
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 \[ \int f^{c\,x^2+a}\,{\mathrm {cosh}\left (f\,x^2+d\right )}^2 \,d x \]
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 \[ \int f^{a + c x^{2}} \cosh ^{2}{\left (d + f x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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