Optimal. Leaf size=344 \[ \frac {3 e^{-d+\frac {(e-b \log (f))^2}{4 (f-c \log (f))}} f^a \sqrt {\pi } \text {Erf}\left (\frac {e-b \log (f)+2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{16 \sqrt {f-c \log (f)}}-\frac {e^{-3 d+\frac {(3 e-b \log (f))^2}{12 f-4 c \log (f)}} f^a \sqrt {\pi } \text {Erf}\left (\frac {3 e-b \log (f)+2 x (3 f-c \log (f))}{2 \sqrt {3 f-c \log (f)}}\right )}{16 \sqrt {3 f-c \log (f)}}-\frac {3 e^{d-\frac {(e+b \log (f))^2}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {e+b \log (f)+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{16 \sqrt {f+c \log (f)}}+\frac {e^{3 d-\frac {(3 e+b \log (f))^2}{4 (3 f+c \log (f))}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {3 e+b \log (f)+2 x (3 f+c \log (f))}{2 \sqrt {3 f+c \log (f)}}\right )}{16 \sqrt {3 f+c \log (f)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.65, antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5623, 2325,
2266, 2236, 2235} \begin {gather*} -\frac {\sqrt {\pi } f^a \exp \left (\frac {(3 e-b \log (f))^2}{12 f-4 c \log (f)}-3 d\right ) \text {Erf}\left (\frac {-b \log (f)+2 x (3 f-c \log (f))+3 e}{2 \sqrt {3 f-c \log (f)}}\right )}{16 \sqrt {3 f-c \log (f)}}+\frac {3 \sqrt {\pi } f^a e^{\frac {(e-b \log (f))^2}{4 (f-c \log (f))}-d} \text {Erf}\left (\frac {-b \log (f)+2 x (f-c \log (f))+e}{2 \sqrt {f-c \log (f)}}\right )}{16 \sqrt {f-c \log (f)}}+\frac {\sqrt {\pi } f^a \exp \left (3 d-\frac {(b \log (f)+3 e)^2}{4 (c \log (f)+3 f)}\right ) \text {Erfi}\left (\frac {b \log (f)+2 x (c \log (f)+3 f)+3 e}{2 \sqrt {c \log (f)+3 f}}\right )}{16 \sqrt {c \log (f)+3 f}}-\frac {3 \sqrt {\pi } f^a e^{d-\frac {(b \log (f)+e)^2}{4 (c \log (f)+f)}} \text {Erfi}\left (\frac {b \log (f)+2 x (c \log (f)+f)+e}{2 \sqrt {c \log (f)+f}}\right )}{16 \sqrt {c \log (f)+f}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2235
Rule 2236
Rule 2266
Rule 2325
Rule 5623
Rubi steps
\begin {align*} \int f^{a+b x+c x^2} \sinh ^3\left (d+e x+f x^2\right ) \, dx &=\int \left (-\frac {1}{8} e^{-3 \left (d+e x+f x^2\right )} f^{a+b x+c x^2}+\frac {3}{8} \exp \left (2 d+2 e x+2 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2}-\frac {3}{8} \exp \left (4 d+4 e x+4 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2}+\frac {1}{8} \exp \left (6 d+6 e x+6 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2}\right ) \, dx\\ &=-\left (\frac {1}{8} \int e^{-3 \left (d+e x+f x^2\right )} f^{a+b x+c x^2} \, dx\right )+\frac {1}{8} \int \exp \left (6 d+6 e x+6 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2} \, dx+\frac {3}{8} \int \exp \left (2 d+2 e x+2 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2} \, dx-\frac {3}{8} \int \exp \left (4 d+4 e x+4 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x+c x^2} \, dx\\ &=-\left (\frac {1}{8} \int \exp \left (-3 d+a \log (f)-x (3 e-b \log (f))-x^2 (3 f-c \log (f))\right ) \, dx\right )+\frac {1}{8} \int \exp \left (3 d+a \log (f)+x (3 e+b \log (f))+x^2 (3 f+c \log (f))\right ) \, dx+\frac {3}{8} \int \exp \left (-d+a \log (f)-x (e-b \log (f))-x^2 (f-c \log (f))\right ) \, dx-\frac {3}{8} \int \exp \left (d+a \log (f)+x (e+b \log (f))+x^2 (f+c \log (f))\right ) \, dx\\ &=-\left (\frac {1}{8} \left (\exp \left (-3 d+\frac {(3 e-b \log (f))^2}{12 f-4 c \log (f)}\right ) f^a\right ) \int \exp \left (\frac {(-3 e+b \log (f)+2 x (-3 f+c \log (f)))^2}{4 (-3 f+c \log (f))}\right ) \, dx\right )+\frac {1}{8} \left (3 e^{-d+\frac {(e-b \log (f))^2}{4 (f-c \log (f))}} f^a\right ) \int \exp \left (\frac {(-e+b \log (f)+2 x (-f+c \log (f)))^2}{4 (-f+c \log (f))}\right ) \, dx-\frac {1}{8} \left (3 e^{d-\frac {(e+b \log (f))^2}{4 (f+c \log (f))}} f^a\right ) \int \exp \left (\frac {(e+b \log (f)+2 x (f+c \log (f)))^2}{4 (f+c \log (f))}\right ) \, dx+\frac {1}{8} \left (\exp \left (3 d-\frac {(3 e+b \log (f))^2}{4 (3 f+c \log (f))}\right ) f^a\right ) \int \exp \left (\frac {(3 e+b \log (f)+2 x (3 f+c \log (f)))^2}{4 (3 f+c \log (f))}\right ) \, dx\\ &=\frac {3 e^{-d+\frac {(e-b \log (f))^2}{4 (f-c \log (f))}} f^a \sqrt {\pi } \text {erf}\left (\frac {e-b \log (f)+2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{16 \sqrt {f-c \log (f)}}-\frac {\exp \left (-3 d+\frac {(3 e-b \log (f))^2}{12 f-4 c \log (f)}\right ) f^a \sqrt {\pi } \text {erf}\left (\frac {3 e-b \log (f)+2 x (3 f-c \log (f))}{2 \sqrt {3 f-c \log (f)}}\right )}{16 \sqrt {3 f-c \log (f)}}-\frac {3 e^{d-\frac {(e+b \log (f))^2}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+b \log (f)+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{16 \sqrt {f+c \log (f)}}+\frac {\exp \left (3 d-\frac {(3 e+b \log (f))^2}{4 (3 f+c \log (f))}\right ) f^a \sqrt {\pi } \text {erfi}\left (\frac {3 e+b \log (f)+2 x (3 f+c \log (f))}{2 \sqrt {3 f+c \log (f)}}\right )}{16 \sqrt {3 f+c \log (f)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2991\) vs. \(2(344)=688\).
time = 6.43, size = 2991, normalized size = 8.69 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 2.08, size = 384, normalized size = 1.12
method | result | size |
risch | \(-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+6 \ln \left (f \right ) b e -12 d \ln \left (f \right ) c -36 d f +9 e^{2}}{4 \left (3 f +c \ln \left (f \right )\right )}} \erf \left (-\sqrt {-c \ln \left (f \right )-3 f}\, x +\frac {3 e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )-3 f}}\right )}{16 \sqrt {-c \ln \left (f \right )-3 f}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-6 \ln \left (f \right ) b e +12 d \ln \left (f \right ) c -36 d f +9 e^{2}}{4 \left (-3 f +c \ln \left (f \right )\right )}} \erf \left (-x \sqrt {3 f -c \ln \left (f \right )}+\frac {b \ln \left (f \right )-3 e}{2 \sqrt {3 f -c \ln \left (f \right )}}\right )}{16 \sqrt {3 f -c \ln \left (f \right )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-2 \ln \left (f \right ) b e +4 d \ln \left (f \right ) c -4 d f +e^{2}}{4 \left (-f +c \ln \left (f \right )\right )}} \erf \left (-x \sqrt {f -c \ln \left (f \right )}+\frac {b \ln \left (f \right )-e}{2 \sqrt {f -c \ln \left (f \right )}}\right )}{16 \sqrt {f -c \ln \left (f \right )}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+2 \ln \left (f \right ) b e -4 d \ln \left (f \right ) c -4 d f +e^{2}}{4 \left (f +c \ln \left (f \right )\right )}} \erf \left (-\sqrt {-c \ln \left (f \right )-f}\, x +\frac {e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )-f}}\right )}{16 \sqrt {-c \ln \left (f \right )-f}}\) | \(384\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 323, normalized size = 0.94 \begin {gather*} \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 3 \, f} x - \frac {b \log \left (f\right ) + 3 \, e}{2 \, \sqrt {-c \log \left (f\right ) - 3 \, f}}\right ) e^{\left (-\frac {{\left (b \log \left (f\right ) + 3 \, e\right )}^{2}}{4 \, {\left (c \log \left (f\right ) + 3 \, f\right )}} + 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 - \frac {b \log \left (f\right ) + e}{2 \, \sqrt {-c \log \left (f\right ) - f}}\right ) e^{\left (-\frac {{\left (b \log \left (f\right ) + e\right )}^{2}}{4 \, {\left (c \log \left (f\right ) + f\right )}} + d\right )}}{16 \, \sqrt {-c \log \left (f\right ) - f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + f} x - \frac {b \log \left (f\right ) - e}{2 \, \sqrt {-c \log \left (f\right ) + f}}\right ) e^{\left (-\frac {{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \, {\left (c \log \left (f\right ) - f\right )}} - 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 - \frac {b \log \left (f\right ) - 3 \, e}{2 \, \sqrt {-c \log \left (f\right ) + 3 \, f}}\right ) e^{\left (-\frac {{\left (b \log \left (f\right ) - 3 \, e\right )}^{2}}{4 \, {\left (c \log \left (f\right ) - 3 \, 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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1100 vs.
\(2 (303) = 606\).
time = 0.47, size = 1100, normalized size = 3.20 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.45, size = 427, normalized size = 1.24 \begin {gather*} -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - 3 \, f} {\left (2 \, x + \frac {b \log \left (f\right ) + 3 \, e}{c \log \left (f\right ) + 3 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) + 6 \, b e \log \left (f\right ) - 12 \, a f \log \left (f\right ) + 9 \, e^{2} - 36 \, d f}{4 \, {\left (c \log \left (f\right ) + 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) - 3 \, f}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - f} {\left (2 \, x + \frac {b \log \left (f\right ) + e}{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 ) + 2 \, b e \log \left (f\right ) - 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) - f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + f} {\left (2 \, x + \frac {b \log \left (f\right ) - e}{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 ) - 2 \, b e \log \left (f\right ) + 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) + f}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + 3 \, f} {\left (2 \, x + \frac {b \log \left (f\right ) - 3 \, e}{c \log \left (f\right ) - 3 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) - 6 \, b e \log \left (f\right ) + 12 \, a f \log \left (f\right ) + 9 \, e^{2} - 36 \, d f}{4 \, {\left (c \log \left (f\right ) - 3 \, f\right )}}\right )}}{16 \, \sqrt {-c \log \left (f\right ) + 3 \, f}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int f^{c\,x^2+b\,x+a}\,{\mathrm {sinh}\left (f\,x^2+e\,x+d\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________