Optimal. Leaf size=386 \[ -\frac {i 2^{-n-2} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac {1}{2} \left (-\frac {2 b}{d}-n\right ),-n;\frac {1}{2} \left (-\frac {2 b}{d}-n+2\right );-e^{2 i (c+d x)}\right ) \exp (-i (2 a+c n)-i x (2 b+d n)+i n (c+d x))}{2 b+d n}+\frac {i 2^{-n-2} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac {1}{2} \left (\frac {2 b}{d}-n\right ),-n;\frac {1}{2} \left (\frac {2 b}{d}-n+2\right );-e^{2 i (c+d x)}\right ) \exp (i (2 a-c n)+i x (2 b-d n)+i n (c+d x))}{2 b-d n}+\frac {i 2^{-n-1} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i (c+d x)}\right )^{-n} \, _2F_1\left (-n,-\frac {n}{2};1-\frac {n}{2};-e^{2 i (c+d x)}\right )}{d n} \]
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Rubi [A] time = 0.70, antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {4555, 2282, 2032, 364, 2285, 2253, 2251} \[ -\frac {i 2^{-n-2} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac {1}{2} \left (-\frac {2 b}{d}-n\right ),-n;\frac {1}{2} \left (-\frac {2 b}{d}-n+2\right );-e^{2 i (c+d x)}\right ) \exp (-i (2 a+c n)-i x (2 b+d n)+i n (c+d x))}{2 b+d n}+\frac {i 2^{-n-2} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac {1}{2} \left (\frac {2 b}{d}-n\right ),-n;\frac {1}{2} \left (\frac {2 b}{d}-n+2\right );-e^{2 i (c+d x)}\right ) \exp (i (2 a-c n)+i x (2 b-d n)+i n (c+d x))}{2 b-d n}+\frac {i 2^{-n-1} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i (c+d x)}\right )^{-n} \, _2F_1\left (-n,-\frac {n}{2};1-\frac {n}{2};-e^{2 i (c+d x)}\right )}{d n} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2032
Rule 2251
Rule 2253
Rule 2282
Rule 2285
Rule 4555
Rubi steps
\begin {align*} \int \cos ^n(c+d x) \sin ^2(a+b x) \, dx &=2^{-2-n} \int \left (2 \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n-e^{-2 i a-2 i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n-e^{2 i a+2 i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n\right ) \, dx\\ &=-\left (2^{-2-n} \int e^{-2 i a-2 i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \, dx\right )-2^{-2-n} \int e^{2 i a+2 i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \, dx+2^{-1-n} \int \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \, dx\\ &=-\frac {\left (i 2^{-1-n}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{x}+x\right )^n}{x} \, dx,x,e^{i (c+d x)}\right )}{d}-\left (2^{-2-n} e^{i n (c+d x)} \left (1+e^{2 i c+2 i d x}\right )^{-n} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n\right ) \int e^{-2 i a-2 i b x-i n (c+d x)} \left (1+e^{2 i c+2 i d x}\right )^n \, dx-\left (2^{-2-n} e^{i n (c+d x)} \left (1+e^{2 i c+2 i d x}\right )^{-n} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n\right ) \int e^{2 i a+2 i b x-i n (c+d x)} \left (1+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 (e^{-i (c+d x)}+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 (e^{-i (c+d x)}+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 (e^{-i (c+d x)}+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 (e^{-i (c+d x)}+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 (e^{-i (c+d x)}+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 (e^{-i (c+d x)}+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*}
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Mathematica [A] time = 1.83, size = 242, normalized size = 0.63 \[ -\frac {i 2^{-n-2} \left (e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^{n+1} e^{i (c+d x)-2 i (a+b x)} \left (e^{2 i (a+b x)} (2 b+d n) \left (d n e^{2 i (a+b x)} \, _2F_1\left (1,\frac {b}{d}+\frac {n}{2}+1;\frac {b}{d}-\frac {n}{2}+1;-e^{2 i (c+d x)}\right )+2 (2 b-d n) \, _2F_1\left (1,\frac {n+2}{2};1-\frac {n}{2};-e^{2 i (c+d x)}\right )\right )+d n (d n-2 b) \, _2F_1\left (1,-\frac {b}{d}+\frac {n}{2}+1;-\frac {b}{d}-\frac {n}{2}+1;-e^{2 i (c+d x)}\right )\right )}{d^3 n^3-4 b^2 d n} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (b x + a\right )^{2} - 1\right )} \cos \left (d x + c\right )^{n}, x\right ) \]
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 \[ \int \cos \left (d x + c\right )^{n} \sin \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.43, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{n}\left (d x +c \right )\right ) \left (\sin ^{2}\left (b x +a \right )\right )\, dx \]
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 \[ \int \cos \left (d x + c\right )^{n} \sin \left (b x + a\right )^{2}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^n\,{\sin \left (a+b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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