Optimal. Leaf size=171 \[ \frac {3 e^{-d} f^a \sqrt {\pi } \text {Erf}\left (x \sqrt {f-c \log (f)}\right )}{16 \sqrt {f-c \log (f)}}+\frac {e^{-3 d} f^a \sqrt {\pi } \text {Erf}\left (x \sqrt {3 f-c \log (f)}\right )}{16 \sqrt {3 f-c \log (f)}}+\frac {3 e^d f^a \sqrt {\pi } \text {Erfi}\left (x \sqrt {f+c \log (f)}\right )}{16 \sqrt {f+c \log (f)}}+\frac {e^{3 d} f^a \sqrt {\pi } \text {Erfi}\left (x \sqrt {3 f+c \log (f)}\right )}{16 \sqrt {3 f+c \log (f)}} \]
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Rubi [A]
time = 0.25, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5624, 2325,
2236, 2235} \begin {gather*} \frac {3 \sqrt {\pi } e^{-d} f^a \text {Erf}\left (x \sqrt {f-c \log (f)}\right )}{16 \sqrt {f-c \log (f)}}+\frac {\sqrt {\pi } e^{-3 d} f^a \text {Erf}\left (x \sqrt {3 f-c \log (f)}\right )}{16 \sqrt {3 f-c \log (f)}}+\frac {3 \sqrt {\pi } e^d f^a \text {Erfi}\left (x \sqrt {c \log (f)+f}\right )}{16 \sqrt {c \log (f)+f}}+\frac {\sqrt {\pi } e^{3 d} f^a \text {Erfi}\left (x \sqrt {c \log (f)+3 f}\right )}{16 \sqrt {c \log (f)+3 f}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2236
Rule 2325
Rule 5624
Rubi steps
\begin {align*} \int f^{a+c x^2} \cosh ^3\left (d+f x^2\right ) \, dx &=\int \left (\frac {1}{8} e^{-3 d-3 f x^2} f^{a+c x^2}+\frac {3}{8} e^{-d-f x^2} f^{a+c x^2}+\frac {3}{8} e^{d+f x^2} f^{a+c x^2}+\frac {1}{8} e^{3 d+3 f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac {1}{8} \int e^{-3 d-3 f x^2} f^{a+c x^2} \, dx+\frac {1}{8} \int e^{3 d+3 f x^2} f^{a+c x^2} \, dx+\frac {3}{8} \int e^{-d-f x^2} f^{a+c x^2} \, dx+\frac {3}{8} \int e^{d+f x^2} f^{a+c x^2} \, dx\\ &=\frac {1}{8} \int e^{-3 d+a \log (f)-x^2 (3 f-c \log (f))} \, dx+\frac {1}{8} \int e^{3 d+a \log (f)+x^2 (3 f+c \log (f))} \, dx+\frac {3}{8} \int e^{-d+a \log (f)-x^2 (f-c \log (f))} \, dx+\frac {3}{8} \int e^{d+a \log (f)+x^2 (f+c \log (f))} \, dx\\ &=\frac {3 e^{-d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {f-c \log (f)}\right )}{16 \sqrt {f-c \log (f)}}+\frac {e^{-3 d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {3 f-c \log (f)}\right )}{16 \sqrt {3 f-c \log (f)}}+\frac {3 e^d f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {f+c \log (f)}\right )}{16 \sqrt {f+c \log (f)}}+\frac {e^{3 d} f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {3 f+c \log (f)}\right )}{16 \sqrt {3 f+c \log (f)}}\\ \end {align*}
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Mathematica [A]
time = 0.88, size = 270, normalized size = 1.58 \begin {gather*} \frac {f^a \sqrt {\pi } \left (3 \text {Erf}\left (x \sqrt {f-c \log (f)}\right ) \sqrt {f-c \log (f)} \left (9 f^3+9 c f^2 \log (f)-c^2 f \log ^2(f)-c^3 \log ^3(f)\right ) (\cosh (d)-\sinh (d))+(f-c \log (f)) \left (\text {Erf}\left (x \sqrt {3 f-c \log (f)}\right ) \sqrt {3 f-c \log (f)} \left (3 f^2+4 c f \log (f)+c^2 \log ^2(f)\right ) (\cosh (3 d)-\sinh (3 d))+(3 f-c \log (f)) \left (3 \text {Erfi}\left (x \sqrt {f+c \log (f)}\right ) \sqrt {f+c \log (f)} (3 f+c \log (f)) (\cosh (d)+\sinh (d))+\text {Erfi}\left (x \sqrt {3 f+c \log (f)}\right ) (f+c \log (f)) \sqrt {3 f+c \log (f)} (\cosh (3 d)+\sinh (3 d))\right )\right )\right )}{16 \left (9 f^4-10 c^2 f^2 \log ^2(f)+c^4 \log ^4(f)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.46, size = 144, normalized size = 0.84
method | result | size |
risch | \(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-3 d} \erf \left (x \sqrt {3 f -c \ln \left (f \right )}\right )}{16 \sqrt {3 f -c \ln \left (f \right )}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{3 d} \erf \left (\sqrt {-c \ln \left (f \right )-3 f}\, x \right )}{16 \sqrt {-c \ln \left (f \right )-3 f}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-d} \erf \left (x \sqrt {f -c \ln \left (f \right )}\right )}{16 \sqrt {f -c \ln \left (f \right )}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{d} \erf \left (\sqrt {-c \ln \left (f \right )-f}\, x \right )}{16 \sqrt {-c \ln \left (f \right )-f}}\) | \(144\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 143, normalized size = 0.84 \begin {gather*} \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 3 \, f} x\right ) e^{\left (3 \, d\right )}}{16 \, \sqrt {-c \log \left (f\right ) - 3 \, f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + f} x\right ) e^{\left (-d\right )}}{16 \, \sqrt {-c \log \left (f\right ) + f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 3 \, f} x\right ) e^{\left (-3 \, d\right )}}{16 \, \sqrt {-c \log \left (f\right ) + 3 \, f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - f} x\right ) e^{d}}{16 \, \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. 491 vs.
\(2 (135) = 270\).
time = 0.53, size = 491, normalized size = 2.87 \begin {gather*} -\frac {{\left (\sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + 3 \, c^{2} f \log \left (f\right )^{2} - c f^{2} \log \left (f\right ) - 3 \, f^{3}\right )} \cosh \left (a \log \left (f\right ) - 3 \, d\right ) + \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + 3 \, c^{2} f \log \left (f\right )^{2} - c f^{2} \log \left (f\right ) - 3 \, f^{3}\right )} \sinh \left (a \log \left (f\right ) - 3 \, d\right )\right )} \sqrt {-c \log \left (f\right ) + 3 \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 3 \, f} x\right ) + 3 \, {\left (\sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + c^{2} f \log \left (f\right )^{2} - 9 \, c f^{2} \log \left (f\right ) - 9 \, f^{3}\right )} \cosh \left (a \log \left (f\right ) - d\right ) + \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + c^{2} f \log \left (f\right )^{2} - 9 \, c f^{2} \log \left (f\right ) - 9 \, f^{3}\right )} \sinh \left (a \log \left (f\right ) - d\right )\right )} \sqrt {-c \log \left (f\right ) + f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + f} x\right ) + 3 \, {\left (\sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - c^{2} f \log \left (f\right )^{2} - 9 \, c f^{2} \log \left (f\right ) + 9 \, f^{3}\right )} \cosh \left (a \log \left (f\right ) + d\right ) + \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - c^{2} f \log \left (f\right )^{2} - 9 \, c f^{2} \log \left (f\right ) + 9 \, f^{3}\right )} \sinh \left (a \log \left (f\right ) + d\right )\right )} \sqrt {-c \log \left (f\right ) - f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - f} x\right ) + {\left (\sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - 3 \, c^{2} f \log \left (f\right )^{2} - c f^{2} \log \left (f\right ) + 3 \, f^{3}\right )} \cosh \left (a \log \left (f\right ) + 3 \, d\right ) + \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - 3 \, c^{2} f \log \left (f\right )^{2} - c f^{2} \log \left (f\right ) + 3 \, f^{3}\right )} \sinh \left (a \log \left (f\right ) + 3 \, d\right )\right )} \sqrt {-c \log \left (f\right ) - 3 \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 3 \, f} x\right )}{16 \, {\left (c^{4} \log \left (f\right )^{4} - 10 \, c^{2} f^{2} \log \left (f\right )^{2} + 9 \, f^{4}\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}} \cosh ^{3}{\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 = 0.41, size = 155, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) - 3 \, f} x\right ) e^{\left (a \log \left (f\right ) + 3 \, d\right )}}{16 \, \sqrt {-c \log \left (f\right ) - 3 \, f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) - f} x\right ) e^{\left (a \log \left (f\right ) + d\right )}}{16 \, \sqrt {-c \log \left (f\right ) - f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) + f} x\right ) e^{\left (a \log \left (f\right ) - d\right )}}{16 \, \sqrt {-c \log \left (f\right ) + f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) + 3 \, f} x\right ) e^{\left (a \log \left (f\right ) - 3 \, d\right )}}{16 \, \sqrt {-c \log \left (f\right ) + 3 \, 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 {cosh}\left (f\,x^2+d\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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