Optimal. Leaf size=154 \[ \frac {e^{-d+\frac {b^2 \log ^2(f)}{4 f-4 c \log (f)}} f^a \sqrt {\pi } \text {Erf}\left (\frac {b \log (f)-2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{4 \sqrt {f-c \log (f)}}+\frac {e^{d-\frac {b^2 \log ^2(f)}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {b \log (f)+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{4 \sqrt {f+c \log (f)}} \]
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Rubi [A]
time = 0.26, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5623, 2325,
2266, 2236, 2235} \begin {gather*} \frac {\sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{4 f-4 c \log (f)}-d} \text {Erf}\left (\frac {b \log (f)-2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{4 \sqrt {f-c \log (f)}}+\frac {\sqrt {\pi } f^a e^{d-\frac {b^2 \log ^2(f)}{4 (c \log (f)+f)}} \text {Erfi}\left (\frac {b \log (f)+2 x (c \log (f)+f)}{2 \sqrt {c \log (f)+f}}\right )}{4 \sqrt {c \log (f)+f}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2236
Rule 2266
Rule 2325
Rule 5623
Rubi steps
\begin {align*} \int f^{a+b x+c x^2} \sinh \left (d+f x^2\right ) \, dx &=\int \left (-\frac {1}{2} e^{-d-f x^2} f^{a+b x+c x^2}+\frac {1}{2} e^{d+f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int e^{-d-f x^2} f^{a+b x+c x^2} \, dx\right )+\frac {1}{2} \int e^{d+f x^2} f^{a+b x+c x^2} \, dx\\ &=-\left (\frac {1}{2} \int \exp \left (-d+a \log (f)+b x \log (f)-x^2 (f-c \log (f))\right ) \, dx\right )+\frac {1}{2} \int \exp \left (d+a \log (f)+b x \log (f)+x^2 (f+c \log (f))\right ) \, dx\\ &=-\left (\frac {1}{2} \left (e^{-d+\frac {b^2 \log ^2(f)}{4 f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (-f+c \log (f)))^2}{4 (-f+c \log (f))}\right ) \, dx\right )+\frac {1}{2} \left (e^{d-\frac {b^2 \log ^2(f)}{4 (f+c \log (f))}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (f+c \log (f)))^2}{4 (f+c \log (f))}\right ) \, dx\\ &=\frac {e^{-d+\frac {b^2 \log ^2(f)}{4 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {b \log (f)-2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{4 \sqrt {f-c \log (f)}}+\frac {e^{d-\frac {b^2 \log ^2(f)}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {b \log (f)+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{4 \sqrt {f+c \log (f)}}\\ \end {align*}
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Mathematica [A]
time = 0.59, size = 179, normalized size = 1.16 \begin {gather*} \frac {e^{-\frac {b^2 \log ^2(f)}{4 (f+c \log (f))}} f^a \sqrt {\pi } \left (-e^{\frac {b^2 f \log ^2(f)}{2 f^2-2 c^2 \log ^2(f)}} \text {Erf}\left (\frac {2 f x-(b+2 c x) \log (f)}{2 \sqrt {f-c \log (f)}}\right ) \sqrt {f+c \log (f)} (\cosh (d)-\sinh (d))+\text {Erfi}\left (\frac {2 f x+(b+2 c x) \log (f)}{2 \sqrt {f+c \log (f)}}\right ) \sqrt {f-c \log (f)} (\cosh (d)+\sinh (d))\right )}{4 \sqrt {f-c \log (f)} \sqrt {f+c \log (f)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.87, size = 160, normalized size = 1.04
method | result | size |
risch | \(-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-4 d \ln \left (f \right ) c -4 d f}{4 \left (f +c \ln \left (f \right )\right )}} \erf \left (-\sqrt {-c \ln \left (f \right )-f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )-f}}\right )}{4 \sqrt {-c \ln \left (f \right )-f}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+4 d \ln \left (f \right ) c -4 d f}{4 \left (-f +c \ln \left (f \right )\right )}} \erf \left (-x \sqrt {f -c \ln \left (f \right )}+\frac {\ln \left (f \right ) b}{2 \sqrt {f -c \ln \left (f \right )}}\right )}{4 \sqrt {f -c \ln \left (f \right )}}\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 139, normalized size = 0.90 \begin {gather*} \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) - f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, {\left (c \log \left (f\right ) + f\right )}} + d\right )}}{4 \, \sqrt {-c \log \left (f\right ) - f}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) + f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, {\left (c \log \left (f\right ) - f\right )}} - d\right )}}{4 \, \sqrt {-c \log \left (f\right ) + f}} \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. 325 vs.
\(2 (131) = 262\).
time = 0.40, size = 325, normalized size = 2.11 \begin {gather*} \frac {{\left (\sqrt {\pi } {\left (c \log \left (f\right ) + f\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, d f + 4 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right ) + \sqrt {\pi } {\left (c \log \left (f\right ) + f\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, d f + 4 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) + f} \operatorname {erf}\left (-\frac {{\left (2 \, f x - {\left (2 \, c x + b\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) + f}}{2 \, {\left (c \log \left (f\right ) - f\right )}}\right ) - {\left (\sqrt {\pi } {\left (c \log \left (f\right ) - f\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, d f - 4 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right ) + \sqrt {\pi } {\left (c \log \left (f\right ) - f\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, d f - 4 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) - f} \operatorname {erf}\left (\frac {{\left (2 \, f x + {\left (2 \, c x + b\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) - f}}{2 \, {\left (c \log \left (f\right ) + f\right )}}\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} - f^{2}\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 {\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 = 0.43, size = 181, normalized size = 1.18 \begin {gather*} -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - f} {\left (2 \, x + \frac {b \log \left (f\right )}{c \log \left (f\right ) + f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right )}}{4 \, \sqrt {-c \log \left (f\right ) - f}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + f} {\left (2 \, x + \frac {b \log \left (f\right )}{c \log \left (f\right ) - f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) + 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right )}}{4 \, \sqrt {-c \log \left (f\right ) + 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+b\,x+a}\,\mathrm {sinh}\left (f\,x^2+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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