Optimal. Leaf size=261 \[ \frac {1}{2} x \left (2 a^2+b^2\right )+\frac {i a b e^{i c} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )}{g n}-\frac {i a b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )}{g n}+\frac {b^2 e^{2 i c} 2^{-\frac {1}{n}-2} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 i d (f+g x)^n\right )}{g n}+\frac {b^2 e^{-2 i c} 2^{-\frac {1}{n}-2} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 i d (f+g x)^n\right )}{g n} \]
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Rubi [A] time = 0.15, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3367, 3366, 2208, 3365} \[ \frac {i a b e^{i c} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-i d (f+g x)^n\right )}{g n}-\frac {i a b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},i d (f+g x)^n\right )}{g n}+\frac {b^2 e^{2 i c} 2^{-\frac {1}{n}-2} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-2 i d (f+g x)^n\right )}{g n}+\frac {b^2 e^{-2 i c} 2^{-\frac {1}{n}-2} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},2 i d (f+g x)^n\right )}{g n}+\frac {1}{2} x \left (2 a^2+b^2\right ) \]
Antiderivative was successfully verified.
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Rule 2208
Rule 3365
Rule 3366
Rule 3367
Rubi steps
\begin {align*} \int \left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2 \, dx &=\int \left (a^2+\frac {b^2}{2}-\frac {1}{2} b^2 \cos \left (2 c+2 d (f+g x)^n\right )+2 a b \sin \left (c+d (f+g x)^n\right )\right ) \, dx\\ &=\frac {1}{2} \left (2 a^2+b^2\right ) x+(2 a b) \int \sin \left (c+d (f+g x)^n\right ) \, dx-\frac {1}{2} b^2 \int \cos \left (2 c+2 d (f+g x)^n\right ) \, dx\\ &=\frac {1}{2} \left (2 a^2+b^2\right ) x+(i a b) \int e^{-i c-i d (f+g x)^n} \, dx-(i a b) \int e^{i c+i d (f+g x)^n} \, dx-\frac {1}{4} b^2 \int e^{-2 i c-2 i d (f+g x)^n} \, dx-\frac {1}{4} b^2 \int e^{2 i c+2 i d (f+g x)^n} \, dx\\ &=\frac {1}{2} \left (2 a^2+b^2\right ) x+\frac {i a 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 )}{g n}-\frac {i a 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 )}{g n}+\frac {2^{-2-\frac {1}{n}} b^2 e^{2 i c} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 i d (f+g x)^n\right )}{g n}+\frac {2^{-2-\frac {1}{n}} b^2 e^{-2 i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 i d (f+g x)^n\right )}{g n}\\ \end {align*}
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Mathematica [A] time = 1.73, size = 381, normalized size = 1.46 \[ \frac {4 a^2 g n x-4 i a b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )+4 i a b (\cos (c)+i \sin (c)) (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )+b^2 e^{2 i c} f 2^{-1/n} \left (i d (f+g x)^n\right )^{\frac {1}{n}} \left (d^2 (f+g x)^{2 n}\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 i d (f+g x)^n\right )+b^2 e^{2 i c} g 2^{-1/n} x \left (i d (f+g x)^n\right )^{\frac {1}{n}} \left (d^2 (f+g x)^{2 n}\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 i d (f+g x)^n\right )+b^2 e^{-2 i c} f 2^{-1/n} \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 i d (f+g x)^n\right )+b^2 e^{-2 i c} g 2^{-1/n} x \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 i d (f+g x)^n\right )+2 b^2 g n x}{4 g n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-b^{2} \cos \left ({\left (g x + f\right )}^{n} d + c\right )^{2} + 2 \, a b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a^{2} + b^{2}, x\right ) \]
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 \[ \int {\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \left (a +b \sin \left (c +d \left (g x +f \right )^{n}\right )\right )^{2}\, dx \]
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 \[ a^{2} x + \frac {1}{2} \, b^{2} x - \frac {1}{2} \, b^{2} \int \cos \left (2 \, {\left (g x + f\right )}^{n} d + 2 \, c\right )\,{d x} + 2 \, a b \int \sin \left ({\left (g x + f\right )}^{n} d + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\sin \left (c+d\,{\left (f+g\,x\right )}^n\right )\right )}^2 \,d x \]
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 \[ \int \left (a + b \sin {\left (c + d \left (f + g x\right )^{n} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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