Optimal. Leaf size=128 \[ \frac {e \cosh \left (a+b x+c x^2\right )}{2 c}-\frac {(2 c d-b e) e^{-a+\frac {b^2}{4 c}} \sqrt {\pi } \text {Erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {(2 c d-b e) e^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {Erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {5490, 5482,
2266, 2235, 2236} \begin {gather*} -\frac {\sqrt {\pi } e^{\frac {b^2}{4 c}-a} (2 c d-b e) \text {Erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {Erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {e \cosh \left (a+b x+c x^2\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2236
Rule 2266
Rule 5482
Rule 5490
Rubi steps
\begin {align*} \int (d+e x) \sinh \left (a+b x+c x^2\right ) \, dx &=\frac {e \cosh \left (a+b x+c x^2\right )}{2 c}-\frac {(-2 c d+b e) \int \sinh \left (a+b x+c x^2\right ) \, dx}{2 c}\\ &=\frac {e \cosh \left (a+b x+c x^2\right )}{2 c}-\frac {(2 c d-b e) \int e^{-a-b x-c x^2} \, dx}{4 c}+\frac {(2 c d-b e) \int e^{a+b x+c x^2} \, dx}{4 c}\\ &=\frac {e \cosh \left (a+b x+c x^2\right )}{2 c}+\frac {\left ((2 c d-b e) e^{a-\frac {b^2}{4 c}}\right ) \int e^{\frac {(b+2 c x)^2}{4 c}} \, dx}{4 c}-\frac {\left ((2 c d-b e) e^{-a+\frac {b^2}{4 c}}\right ) \int e^{-\frac {(-b-2 c x)^2}{4 c}} \, dx}{4 c}\\ &=\frac {e \cosh \left (a+b x+c x^2\right )}{2 c}-\frac {(2 c d-b e) e^{-a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {(2 c d-b e) e^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 146, normalized size = 1.14 \begin {gather*} \frac {4 \sqrt {c} e \cosh (a+x (b+c x))+(2 c d-b e) \sqrt {\pi } \text {Erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (-\cosh \left (a-\frac {b^2}{4 c}\right )+\sinh \left (a-\frac {b^2}{4 c}\right )\right )+(2 c d-b e) \sqrt {\pi } \text {Erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\cosh \left (a-\frac {b^2}{4 c}\right )+\sinh \left (a-\frac {b^2}{4 c}\right )\right )}{8 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(210\) vs.
\(2(100)=200\).
time = 0.25, size = 211, normalized size = 1.65
method | result | size |
risch | \(-\frac {d \sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}}{4 c}} \erf \left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e \,{\mathrm e}^{-c \,x^{2}-b x -a}}{4 c}+\frac {e b \sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}}{4 c}} \erf \left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{8 c^{\frac {3}{2}}}-\frac {d \sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}}{4 c}} \erf \left (-x \sqrt {-c}+\frac {b}{2 \sqrt {-c}}\right )}{4 \sqrt {-c}}+\frac {e \,{\mathrm e}^{c \,x^{2}+b x +a}}{4 c}+\frac {e b \sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}}{4 c}} \erf \left (-x \sqrt {-c}+\frac {b}{2 \sqrt {-c}}\right )}{8 c \sqrt {-c}}\) | \(211\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 254 vs.
\(2 (103) = 206\).
time = 0.35, size = 254, normalized size = 1.98 \begin {gather*} \frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {-c} x - \frac {b}{2 \, \sqrt {-c}}\right ) e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {-c}} - \frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {c} x + \frac {b}{2 \, \sqrt {c}}\right ) e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {c}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {2 \, e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\sqrt {c}}\right )} e^{\left (a - \frac {b^{2}}{4 \, c} + 1\right )}}{8 \, \sqrt {c}} + \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac {3}{2}}} + \frac {2 \, c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {3}{2}}}\right )} e^{\left (-a + \frac {b^{2}}{4 \, c} + 1\right )}}{8 \, \sqrt {-c}} \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. 488 vs.
\(2 (103) = 206\).
time = 0.40, size = 488, normalized size = 3.81 \begin {gather*} \frac {2 \, {\left (c \cosh \left (1\right ) + c \sinh \left (1\right )\right )} \cosh \left (c x^{2} + b x + a\right )^{2} - \sqrt {\pi } {\left ({\left (2 \, c d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (2 \, c d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left ({\left (2 \, c d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (2 \, c d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {-c} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, c}\right ) - \sqrt {\pi } {\left ({\left (2 \, c d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - {\left (2 \, c d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left ({\left (2 \, c d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - {\left (2 \, c d - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {c} \operatorname {erf}\left (\frac {2 \, c x + b}{2 \, \sqrt {c}}\right ) + 4 \, {\left (c \cosh \left (1\right ) + c \sinh \left (1\right )\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + 2 \, {\left (c \cosh \left (1\right ) + c \sinh \left (1\right )\right )} \sinh \left (c x^{2} + b x + a\right )^{2} + 2 \, c \cosh \left (1\right ) + 2 \, c \sinh \left (1\right )}{8 \, {\left (c^{2} \cosh \left (c x^{2} + b x + a\right ) + c^{2} \sinh \left (c x^{2} + b x + a\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 \left (d + e x\right ) \sinh {\left (a + b x + c 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 = 139, normalized size = 1.09 \begin {gather*} \frac {\frac {\sqrt {\pi } {\left (2 \, c d - b e\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt {c}} + 2 \, e e^{\left (-c x^{2} - b x - a\right )}}{8 \, c} - \frac {\frac {\sqrt {\pi } {\left (2 \, c d - b e\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt {-c}} - 2 \, e e^{\left (c x^{2} + b x + a\right )}}{8 \, c} \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 \mathrm {sinh}\left (c\,x^2+b\,x+a\right )\,\left (d+e\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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