Optimal. Leaf size=110 \[ \frac{e^{a-\frac{b c}{d}} (c+d x)^m \left (-\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{b (c+d x)}{d}\right )}{2 b}+\frac{e^{\frac{b c}{d}-a} (c+d x)^m \left (\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{b (c+d x)}{d}\right )}{2 b} \]
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Rubi [A] time = 0.0930162, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3308, 2181} \[ \frac{e^{a-\frac{b c}{d}} (c+d x)^m \left (-\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{b (c+d x)}{d}\right )}{2 b}+\frac{e^{\frac{b c}{d}-a} (c+d x)^m \left (\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{b (c+d x)}{d}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int (c+d x)^m \sinh (a+b x) \, dx &=\frac{1}{2} \int e^{-i (i a+i b x)} (c+d x)^m \, dx-\frac{1}{2} \int e^{i (i a+i b x)} (c+d x)^m \, dx\\ &=\frac{e^{a-\frac{b c}{d}} (c+d x)^m \left (-\frac{b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac{b (c+d x)}{d}\right )}{2 b}+\frac{e^{-a+\frac{b c}{d}} (c+d x)^m \left (\frac{b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac{b (c+d x)}{d}\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0546071, size = 101, normalized size = 0.92 \[ \frac{e^{-a-\frac{b c}{d}} (c+d x)^m \left (e^{2 a} \left (-\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{b (c+d x)}{d}\right )+e^{\frac{2 b c}{d}} \left (b \left (\frac{c}{d}+x\right )\right )^{-m} \text{Gamma}\left (m+1,\frac{b (c+d x)}{d}\right )\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m}\sinh \left ( bx+a \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.28157, size = 107, normalized size = 0.97 \begin{align*} \frac{{\left (d x + c\right )}^{m + 1} e^{\left (-a + \frac{b c}{d}\right )} E_{-m}\left (\frac{{\left (d x + c\right )} b}{d}\right )}{2 \, d} - \frac{{\left (d x + c\right )}^{m + 1} e^{\left (a - \frac{b c}{d}\right )} E_{-m}\left (-\frac{{\left (d x + c\right )} b}{d}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.77917, size = 377, normalized size = 3.43 \begin{align*} \frac{\cosh \left (\frac{d m \log \left (\frac{b}{d}\right ) - b c + a d}{d}\right ) \Gamma \left (m + 1, \frac{b d x + b c}{d}\right ) + \cosh \left (\frac{d m \log \left (-\frac{b}{d}\right ) + b c - a d}{d}\right ) \Gamma \left (m + 1, -\frac{b d x + b c}{d}\right ) - \Gamma \left (m + 1, \frac{b d x + b c}{d}\right ) \sinh \left (\frac{d m \log \left (\frac{b}{d}\right ) - b c + a d}{d}\right ) - \Gamma \left (m + 1, -\frac{b d x + b c}{d}\right ) \sinh \left (\frac{d m \log \left (-\frac{b}{d}\right ) + b c - a d}{d}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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