Optimal. Leaf size=128 \[ -\frac {f^a \sqrt {\pi } \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-2 d} f^a \sqrt {\pi } \text {Erf}\left (x \sqrt {2 f-c \log (f)}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d} f^a \sqrt {\pi } \text {Erfi}\left (x \sqrt {2 f+c \log (f)}\right )}{8 \sqrt {2 f+c \log (f)}} \]
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Rubi [A]
time = 0.16, 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 = {5623, 2235,
2325, 2236} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2236
Rule 2325
Rule 5623
Rubi steps
\begin {align*} \int f^{a+c x^2} \sinh ^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 = 0.38, size = 179, normalized size = 1.40 \begin {gather*} \frac {f^a \sqrt {\pi } \left (\text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right ) \left (8 f^2-2 c^2 \log ^2(f)\right )+\sqrt {c} \sqrt {\log (f)} \left (\text {Erf}\left (x \sqrt {2 f-c \log (f)}\right ) \sqrt {2 f-c \log (f)} (2 f+c \log (f)) (-\cosh (2 d)+\sinh (2 d))-\text {Erfi}\left (x \sqrt {2 f+c \log (f)}\right ) (2 f-c \log (f)) \sqrt {2 f+c \log (f)} (\cosh (2 d)+\sinh (2 d))\right )\right )}{8 \sqrt {c} \sqrt {\log (f)} \left (-4 f^2+c^2 \log ^2(f)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.43, size = 101, normalized size = 0.79
method | result | size |
risch | \(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-2 d} \erf \left (x \sqrt {2 f -c \ln \left (f \right )}\right )}{8 \sqrt {2 f -c \ln \left (f \right )}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{2 d} \erf \left (\sqrt {-c \ln \left (f \right )-2 f}\, x \right )}{8 \sqrt {-c \ln \left (f \right )-2 f}}-\frac {f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-c \ln \left (f \right )}\, x \right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 100, normalized size = 0.78 \begin {gather*} \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 2 \, f} x\right ) e^{\left (2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 2 \, f} x\right ) e^{\left (-2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} \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. 254 vs.
\(2 (98) = 196\).
time = 0.35, size = 254, normalized size = 1.98 \begin {gather*} -\frac {{\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \cosh \left (a \log \left (f\right ) - 2 \, d\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \sinh \left (a \log \left (f\right ) - 2 \, d\right )\right )} \sqrt {-c \log \left (f\right ) + 2 \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 2 \, f} x\right ) + {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \cosh \left (a \log \left (f\right ) + 2 \, d\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \sinh \left (a \log \left (f\right ) + 2 \, d\right )\right )} \sqrt {-c \log \left (f\right ) - 2 \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 2 \, f} x\right ) - 2 \, {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \cosh \left (a \log \left (f\right )\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \sinh \left (a \log \left (f\right )\right )\right )} \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )}{8 \, {\left (c^{3} \log \left (f\right )^{3} - 4 \, c f^{2} \log \left (f\right )\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 + c x^{2}} \sinh ^{2}{\left (d + f x^{2} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 107, normalized size = 0.84 \begin {gather*} \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) - 2 \, f} x\right ) e^{\left (a \log \left (f\right ) + 2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) + 2 \, f} x\right ) e^{\left (a \log \left (f\right ) - 2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} \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 = -0.01 \begin {gather*} \int f^{c\,x^2+a}\,{\mathrm {sinh}\left (f\,x^2+d\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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