Optimal. Leaf size=100 \[ \frac {e^{2 i a} 2^{-\frac {1}{n}-2} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 i b x^n\right )}{n}+\frac {e^{-2 i a} 2^{-\frac {1}{n}-2} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 i b x^n\right )}{n}+\frac {x}{2} \]
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Rubi [A] time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3367, 3366, 2208} \[ \frac {e^{2 i a} 2^{-\frac {1}{n}-2} x \left (-i b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-2 i b x^n\right )}{n}+\frac {e^{-2 i a} 2^{-\frac {1}{n}-2} x \left (i b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},2 i b x^n\right )}{n}+\frac {x}{2} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 3366
Rule 3367
Rubi steps
\begin {align*} \int \sin ^2\left (a+b x^n\right ) \, dx &=\int \left (\frac {1}{2}-\frac {1}{2} \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac {x}{2}-\frac {1}{2} \int \cos \left (2 a+2 b x^n\right ) \, dx\\ &=\frac {x}{2}-\frac {1}{4} \int e^{-2 i a-2 i b x^n} \, dx-\frac {1}{4} \int e^{2 i a+2 i b x^n} \, dx\\ &=\frac {x}{2}+\frac {2^{-2-\frac {1}{n}} e^{2 i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 i b x^n\right )}{n}+\frac {2^{-2-\frac {1}{n}} e^{-2 i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 i b x^n\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 94, normalized size = 0.94 \[ \frac {x \left (e^{2 i a} 2^{-1/n} \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 i b x^n\right )+e^{-2 i a} 2^{-1/n} \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 i b x^n\right )+2 n\right )}{4 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\cos \left (b x^{n} + a\right )^{2} + 1, x\right ) \]
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 \[ \int \sin \left (b x^{n} + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \sin ^{2}\left (a +b \,x^{n}\right )\, dx \]
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 \[ \frac {1}{2} \, x - \frac {1}{2} \, \int \cos \left (2 \, b x^{n} + 2 \, a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (a+b\,x^n\right )}^2 \,d x \]
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 \[ \int \sin ^{2}{\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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