Optimal. Leaf size=410 \[ -\frac{i 2^{-n-2} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \text{Hypergeometric2F1}\left (\frac{1}{2} \left (-\frac{2 b}{d}-n\right ),-n,\frac{1}{2} \left (-\frac{2 b}{d}-n+2\right ),e^{2 i (c+d x)}\right ) \exp (-i (2 a+c n)-i x (2 b+d n)+i n (c+d x))}{2 b+d n}+\frac{i 2^{-n-2} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \text{Hypergeometric2F1}\left (\frac{1}{2} \left (\frac{2 b}{d}-n\right ),-n,\frac{1}{2} \left (\frac{2 b}{d}-n+2\right ),e^{2 i (c+d x)}\right ) \exp (i (2 a-c n)+i x (2 b-d n)+i n (c+d x))}{2 b-d n}+\frac{i 2^{-n-1} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i (c+d x)}\right )^{-n} \text{Hypergeometric2F1}\left (-n,-\frac{n}{2},1-\frac{n}{2},e^{2 i (c+d x)}\right )}{d n} \]
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Rubi [A] time = 0.972974, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.588, Rules used = {4553, 2282, 1980, 2032, 365, 364, 2285, 2253, 2252, 2251} \[ -\frac{i 2^{-n-2} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac{1}{2} \left (-\frac{2 b}{d}-n\right ),-n;\frac{1}{2} \left (-\frac{2 b}{d}-n+2\right );e^{2 i (c+d x)}\right ) \exp (-i (2 a+c n)-i x (2 b+d n)+i n (c+d x))}{2 b+d n}+\frac{i 2^{-n-2} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac{1}{2} \left (\frac{2 b}{d}-n\right ),-n;\frac{1}{2} \left (\frac{2 b}{d}-n+2\right );e^{2 i (c+d x)}\right ) \exp (i (2 a-c n)+i x (2 b-d n)+i n (c+d x))}{2 b-d n}+\frac{i 2^{-n-1} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i (c+d x)}\right )^{-n} \, _2F_1\left (-n,-\frac{n}{2};1-\frac{n}{2};e^{2 i (c+d x)}\right )}{d n} \]
Antiderivative was successfully verified.
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Rule 4553
Rule 2282
Rule 1980
Rule 2032
Rule 365
Rule 364
Rule 2285
Rule 2253
Rule 2252
Rule 2251
Rubi steps
\begin{align*} \int \sin ^2(a+b x) \sin ^n(c+d x) \, dx &=2^{-2-n} \int \left (2 \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n-e^{-2 i a-2 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n-e^{2 i a+2 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \, dx\\ &=-\left (2^{-2-n} \int e^{-2 i a-2 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, dx\right )-2^{-2-n} \int e^{2 i a+2 i b x} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, dx+2^{-1-n} \int \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, dx\\ &=-\frac{\left (i 2^{-1-n}\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{i \left (-1+x^2\right )}{x}\right )^n}{x} \, dx,x,e^{i (c+d x)}\right )}{d}-\left (2^{-2-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-2 i a-2 i b x-i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx-\left (2^{-2-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{2 i a+2 i b x-i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx\\ &=-\frac{\left (i 2^{-1-n}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{i}{x}-i x\right )^n}{x} \, dx,x,e^{i (c+d x)}\right )}{d}-\left (2^{-2-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{i (2 a-c n)+i (2 b-d n) x} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx-\left (2^{-2-n} e^{i n (c+d x)} \left (i-i e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-i (2 a+c n)-i (2 b+d n) x} \left (i-i e^{2 i c+2 i d x}\right )^n \, dx\\ &=-\left (\left (2^{-2-n} e^{i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{i (2 a-c n)+i (2 b-d n) x} \left (1-e^{2 i c+2 i d x}\right )^n \, dx\right )-\left (2^{-2-n} e^{i n (c+d x)} \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n\right ) \int e^{-i (2 a+c n)-i (2 b+d n) x} \left (1-e^{2 i c+2 i d x}\right )^n \, dx-\frac{\left (i 2^{-1-n} \left (e^{i (c+d x)}\right )^n \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (i-i e^{2 i (c+d x)}\right )^{-n}\right ) \operatorname{Subst}\left (\int x^{-1-n} \left (i-i x^2\right )^n \, dx,x,e^{i (c+d x)}\right )}{d}\\ &=-\frac{i 2^{-2-n} \exp (-i (2 a+c n)-i (2 b+d n) x+i n (c+d x)) \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, _2F_1\left (\frac{1}{2} \left (-\frac{2 b}{d}-n\right ),-n;\frac{1}{2} \left (2-\frac{2 b}{d}-n\right );e^{2 i (c+d x)}\right )}{2 b+d n}+\frac{i 2^{-2-n} \exp (i (2 a-c n)+i (2 b-d n) x+i n (c+d x)) \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, _2F_1\left (\frac{1}{2} \left (\frac{2 b}{d}-n\right ),-n;\frac{1}{2} \left (2+\frac{2 b}{d}-n\right );e^{2 i (c+d x)}\right )}{2 b-d n}-\frac{\left (i 2^{-1-n} \left (e^{i (c+d x)}\right )^n \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i (c+d x)}\right )^{-n}\right ) \operatorname{Subst}\left (\int x^{-1-n} \left (1-x^2\right )^n \, dx,x,e^{i (c+d x)}\right )}{d}\\ &=-\frac{i 2^{-2-n} \exp (-i (2 a+c n)-i (2 b+d n) x+i n (c+d x)) \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, _2F_1\left (\frac{1}{2} \left (-\frac{2 b}{d}-n\right ),-n;\frac{1}{2} \left (2-\frac{2 b}{d}-n\right );e^{2 i (c+d x)}\right )}{2 b+d n}+\frac{i 2^{-2-n} \exp (i (2 a-c n)+i (2 b-d n) x+i n (c+d x)) \left (1-e^{2 i c+2 i d x}\right )^{-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \, _2F_1\left (\frac{1}{2} \left (\frac{2 b}{d}-n\right ),-n;\frac{1}{2} \left (2+\frac{2 b}{d}-n\right );e^{2 i (c+d x)}\right )}{2 b-d n}+\frac{i 2^{-1-n} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i (c+d x)}\right )^{-n} \, _2F_1\left (-n,-\frac{n}{2};1-\frac{n}{2};e^{2 i (c+d x)}\right )}{d n}\\ \end{align*}
Mathematica [F] time = 0.375229, size = 0, normalized size = 0. \[ \int \sin ^2(a+b x) \sin ^n(c+d x) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 1.504, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( bx+a \right ) \right ) ^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{n}\, 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*} \int \sin \left (d x + c\right )^{n} \sin \left (b x + a\right )^{2}\,{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 (-{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (d x + c\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \sin \left (d x + c\right )^{n} \sin \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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