Optimal. Leaf size=64 \[ \frac {4 \sinh \left (\frac {a}{2}+\frac {b x}{2}\right ) e^{n \sinh \left (\frac {a}{2}+\frac {b x}{2}\right )}}{b n}-\frac {4 e^{n \sinh \left (\frac {a}{2}+\frac {b x}{2}\right )}}{b n^2} \]
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Rubi [A] time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {12, 2176, 2194} \[ \frac {4 \sinh \left (\frac {a}{2}+\frac {b x}{2}\right ) e^{n \sinh \left (\frac {a}{2}+\frac {b x}{2}\right )}}{b n}-\frac {4 e^{n \sinh \left (\frac {a}{2}+\frac {b x}{2}\right )}}{b n^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rubi steps
\begin {align*} \int e^{n \sinh \left (\frac {a}{2}+\frac {b x}{2}\right )} \sinh (a+b x) \, dx &=\frac {2 \operatorname {Subst}\left (\int 2 e^{n x} x \, dx,x,\sinh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}\\ &=\frac {4 \operatorname {Subst}\left (\int e^{n x} x \, dx,x,\sinh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}\\ &=\frac {4 e^{n \sinh \left (\frac {a}{2}+\frac {b x}{2}\right )} \sinh \left (\frac {a}{2}+\frac {b x}{2}\right )}{b n}-\frac {4 \operatorname {Subst}\left (\int e^{n x} \, dx,x,\sinh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b n}\\ &=-\frac {4 e^{n \sinh \left (\frac {a}{2}+\frac {b x}{2}\right )}}{b n^2}+\frac {4 e^{n \sinh \left (\frac {a}{2}+\frac {b x}{2}\right )} \sinh \left (\frac {a}{2}+\frac {b x}{2}\right )}{b n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 36, normalized size = 0.56 \[ \frac {4 e^{n \sinh \left (\frac {1}{2} (a+b x)\right )} \left (n \sinh \left (\frac {1}{2} (a+b x)\right )-1\right )}{b n^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 91, normalized size = 1.42 \[ \frac {4 \, {\left ({\left (n \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) - 1\right )} \cosh \left (n \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right ) + {\left (n \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) - 1\right )} \sinh \left (n \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right )\right )}}{b n^{2} \cosh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - b n^{2} \sinh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 255, normalized size = 3.98 \[ \frac {2 \, {\left (n e^{\left (b x + \frac {1}{4} \, {\left (2 \, b x e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )} + n e^{\left (b x + a\right )} - n\right )} e^{\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a\right )} - \frac {1}{4} \, {\left (2 \, b x e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )} - n e^{\left (b x + a\right )} + n\right )} e^{\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a\right )} + a\right )} - n e^{\left (\frac {1}{4} \, {\left (2 \, b x e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )} + n e^{\left (b x + a\right )} - n\right )} e^{\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a\right )} - \frac {1}{4} \, {\left (2 \, b x e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )} - n e^{\left (b x + a\right )} + n\right )} e^{\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a\right )}\right )} - 2 \, e^{\left (\frac {1}{2} \, b x + \frac {1}{4} \, {\left (2 \, b x e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )} + n e^{\left (b x + a\right )} - n\right )} e^{\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a\right )} - \frac {1}{4} \, {\left (2 \, b x e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )} - n e^{\left (b x + a\right )} + n\right )} e^{\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a\right )} + \frac {1}{2} \, a\right )}\right )} e^{\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a\right )}}{b n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 65, normalized size = 1.02 \[ \frac {2 \left (n \,{\mathrm e}^{b x +a}-n -2 \,{\mathrm e}^{\frac {b x}{2}+\frac {a}{2}}\right ) {\mathrm e}^{-\frac {b x}{2}-\frac {a}{2}+\frac {n \,{\mathrm e}^{\frac {b x}{2}+\frac {a}{2}}}{2}-\frac {n \,{\mathrm e}^{-\frac {b x}{2}-\frac {a}{2}}}{2}}}{n^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 117, normalized size = 1.83 \[ \frac {2 \, e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, n e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )} - \frac {1}{2} \, n e^{\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a\right )} + \frac {1}{2} \, a\right )}}{b n} - \frac {2 \, e^{\left (-\frac {1}{2} \, b x + \frac {1}{2} \, n e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )} - \frac {1}{2} \, n e^{\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a\right )} - \frac {1}{2} \, a\right )}}{b n} - \frac {4 \, e^{\left (\frac {1}{2} \, n e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )} - \frac {1}{2} \, n e^{\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a\right )}\right )}}{b n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.79, size = 127, normalized size = 1.98 \[ \frac {2\,{\mathrm {e}}^{-\frac {a}{2}}\,{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^{-\frac {b\,x}{2}}\,{\mathrm {e}}^a\,{\mathrm {e}}^{-\frac {n\,{\mathrm {e}}^{-\frac {a}{2}}\,{\mathrm {e}}^{-\frac {b\,x}{2}}}{2}}\,{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{a/2}\,{\mathrm {e}}^{\frac {b\,x}{2}}}{2}}}{b\,n}-\frac {2\,{\mathrm {e}}^{-\frac {a}{2}}\,{\mathrm {e}}^{-\frac {b\,x}{2}}\,{\mathrm {e}}^{-\frac {n\,{\mathrm {e}}^{-\frac {a}{2}}\,{\mathrm {e}}^{-\frac {b\,x}{2}}}{2}}\,{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{a/2}\,{\mathrm {e}}^{\frac {b\,x}{2}}}{2}}}{b\,n}-\frac {4\,{\mathrm {e}}^{-\frac {n\,{\mathrm {e}}^{-\frac {a}{2}}\,{\mathrm {e}}^{-\frac {b\,x}{2}}}{2}}\,{\mathrm {e}}^{\frac {n\,{\mathrm {e}}^{a/2}\,{\mathrm {e}}^{\frac {b\,x}{2}}}{2}}}{b\,n^2} \]
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 \[ \int e^{n \sinh {\left (\frac {a}{2} + \frac {b x}{2} \right )}} \sinh {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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