Optimal. Leaf size=318 \[ -\frac{a b e^{i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i f (c+d x)}{d}\right )}{f}-\frac{a b e^{-i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i f (c+d x)}{d}\right )}{f}+\frac{i b^2 2^{-m-3} e^{2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i f (c+d x)}{d}\right )}{f}-\frac{i b^2 2^{-m-3} e^{-2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i f (c+d x)}{d}\right )}{f}+\frac{a^2 (c+d x)^{m+1}}{d (m+1)}+\frac{b^2 (c+d x)^{m+1}}{2 d (m+1)} \]
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Rubi [A] time = 0.392043, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3317, 3308, 2181, 3312, 3307} \[ -\frac{a b e^{i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i f (c+d x)}{d}\right )}{f}-\frac{a b e^{-i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i f (c+d x)}{d}\right )}{f}+\frac{i b^2 2^{-m-3} e^{2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i f (c+d x)}{d}\right )}{f}-\frac{i b^2 2^{-m-3} e^{-2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i f (c+d x)}{d}\right )}{f}+\frac{a^2 (c+d x)^{m+1}}{d (m+1)}+\frac{b^2 (c+d x)^{m+1}}{2 d (m+1)} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3308
Rule 2181
Rule 3312
Rule 3307
Rubi steps
\begin{align*} \int (c+d x)^m (a+b \sin (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^m+2 a b (c+d x)^m \sin (e+f x)+b^2 (c+d x)^m \sin ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^{1+m}}{d (1+m)}+(2 a b) \int (c+d x)^m \sin (e+f x) \, dx+b^2 \int (c+d x)^m \sin ^2(e+f x) \, dx\\ &=\frac{a^2 (c+d x)^{1+m}}{d (1+m)}+(i a b) \int e^{-i (e+f x)} (c+d x)^m \, dx-(i a b) \int e^{i (e+f x)} (c+d x)^m \, dx+b^2 \int \left (\frac{1}{2} (c+d x)^m-\frac{1}{2} (c+d x)^m \cos (2 e+2 f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^{1+m}}{d (1+m)}+\frac{b^2 (c+d x)^{1+m}}{2 d (1+m)}-\frac{a b e^{i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac{i f (c+d x)}{d}\right )}{f}-\frac{a b e^{-i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac{i f (c+d x)}{d}\right )}{f}-\frac{1}{2} b^2 \int (c+d x)^m \cos (2 e+2 f x) \, dx\\ &=\frac{a^2 (c+d x)^{1+m}}{d (1+m)}+\frac{b^2 (c+d x)^{1+m}}{2 d (1+m)}-\frac{a b e^{i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac{i f (c+d x)}{d}\right )}{f}-\frac{a b e^{-i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac{i f (c+d x)}{d}\right )}{f}-\frac{1}{4} b^2 \int e^{-i (2 e+2 f x)} (c+d x)^m \, dx-\frac{1}{4} b^2 \int e^{i (2 e+2 f x)} (c+d x)^m \, dx\\ &=\frac{a^2 (c+d x)^{1+m}}{d (1+m)}+\frac{b^2 (c+d x)^{1+m}}{2 d (1+m)}-\frac{a b e^{i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac{i f (c+d x)}{d}\right )}{f}-\frac{a b e^{-i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac{i f (c+d x)}{d}\right )}{f}+\frac{i 2^{-3-m} b^2 e^{2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac{2 i f (c+d x)}{d}\right )}{f}-\frac{i 2^{-3-m} b^2 e^{-2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac{2 i f (c+d x)}{d}\right )}{f}\\ \end{align*}
Mathematica [A] time = 3.93237, size = 268, normalized size = 0.84 \[ -\frac{(c+d x)^m \left (8 a b e^{i \left (e-\frac{c f}{d}\right )} \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i f (c+d x)}{d}\right )+8 a b e^{-i \left (e-\frac{c f}{d}\right )} \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i f (c+d x)}{d}\right )-i b^2 2^{-m} e^{2 i \left (e-\frac{c f}{d}\right )} \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i f (c+d x)}{d}\right )+i b^2 2^{-m} e^{-2 i \left (e-\frac{c f}{d}\right )} \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i f (c+d x)}{d}\right )-\frac{4 f \left (2 a^2+b^2\right ) (c+d x)}{d (m+1)}\right )}{8 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.198, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( a+b\sin \left ( fx+e \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97231, size = 662, normalized size = 2.08 \begin{align*} \frac{{\left (-i \, b^{2} d m - i \, b^{2} d\right )} e^{\left (-\frac{d m \log \left (\frac{2 i \, f}{d}\right ) + 2 i \, d e - 2 i \, c f}{d}\right )} \Gamma \left (m + 1, \frac{2 i \, d f x + 2 i \, c f}{d}\right ) - 8 \,{\left (a b d m + a b d\right )} e^{\left (-\frac{d m \log \left (\frac{i \, f}{d}\right ) + i \, d e - i \, c f}{d}\right )} \Gamma \left (m + 1, \frac{i \, d f x + i \, c f}{d}\right ) - 8 \,{\left (a b d m + a b d\right )} e^{\left (-\frac{d m \log \left (-\frac{i \, f}{d}\right ) - i \, d e + i \, c f}{d}\right )} \Gamma \left (m + 1, \frac{-i \, d f x - i \, c f}{d}\right ) +{\left (i \, b^{2} d m + i \, b^{2} d\right )} e^{\left (-\frac{d m \log \left (-\frac{2 i \, f}{d}\right ) - 2 i \, d e + 2 i \, c f}{d}\right )} \Gamma \left (m + 1, \frac{-2 i \, d f x - 2 i \, c f}{d}\right ) + 4 \,{\left ({\left (2 \, a^{2} + b^{2}\right )} d f x +{\left (2 \, a^{2} + b^{2}\right )} c f\right )}{\left (d x + c\right )}^{m}}{8 \,{\left (d f m + d f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sin{\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{m}\, dx \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 (b \sin \left (f x + e\right ) + a\right )}^{2}{\left (d x + c\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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