Optimal. Leaf size=122 \[ \frac{i b e^{i c} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i d (f+g x)^n\right )}{2 g n}-\frac{i b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i d (f+g x)^n\right )}{2 g n}+a x \]
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Rubi [A] time = 0.0579811, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3365, 2208} \[ \frac{i b e^{i c} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i d (f+g x)^n\right )}{2 g n}-\frac{i b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i d (f+g x)^n\right )}{2 g n}+a x \]
Antiderivative was successfully verified.
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Rule 3365
Rule 2208
Rubi steps
\begin{align*} \int \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx &=a x+b \int \sin \left (c+d (f+g x)^n\right ) \, dx\\ &=a x+\frac{1}{2} (i b) \int e^{-i c-i d (f+g x)^n} \, dx-\frac{1}{2} (i b) \int e^{i c+i d (f+g x)^n} \, dx\\ &=a x+\frac{i b e^{i c} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac{1}{n},-i d (f+g x)^n\right )}{2 g n}-\frac{i b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac{1}{n},i d (f+g x)^n\right )}{2 g n}\\ \end{align*}
Mathematica [A] time = 0.245949, size = 126, normalized size = 1.03 \[ \frac{i b (\cos (c)+i \sin (c)) (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i d (f+g x)^n\right )}{2 g n}-\frac{i b (\cos (c)-i \sin (c)) (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i d (f+g x)^n\right )}{2 g n}+a x \]
Antiderivative was successfully verified.
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Maple [F] time = 0.142, size = 0, normalized size = 0. \begin{align*} \int a+b\sin \left ( c+d \left ( gx+f \right ) ^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} a x + b \int \sin \left ({\left (g x + f\right )}^{n} d + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a, x\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 (c + d \left (f + g x\right )^{n} \right )}\right )\, 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 b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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