Optimal. Leaf size=122 \[ a x+\frac {i b e^{i c} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g n}-\frac {i b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )}{2 g n} \]
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Rubi [A]
time = 0.04, antiderivative size = 122, normalized size of antiderivative = 1.00, 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 = {3446, 2239}
\begin {gather*} \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 \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 3446
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*}
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Mathematica [A]
time = 0.18, size = 126, normalized size = 1.03 \begin {gather*} a x-\frac {i b (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 ) (\cos (c)-i \sin (c))}{2 g n}+\frac {i b (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 ) (\cos (c)+i \sin (c))}{2 g n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int a +b \sin \left (c +d \left (g x +f \right )^{n}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 (a + b \sin {\left (c + d \left (f + g x\right )^{n} \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 a+b\,\sin \left (c+d\,{\left (f+g\,x\right )}^n\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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