Optimal. Leaf size=69 \[ -\frac {(d \cos (a+b x))^{1+n} \, _2F_1\left (-\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(a+b x)\right ) \sin (a+b x)}{b d (1+n) \sqrt {\sin ^2(a+b x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2656}
\begin {gather*} -\frac {\sin (a+b x) (d \cos (a+b x))^{n+1} \, _2F_1\left (-\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1) \sqrt {\sin ^2(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2656
Rubi steps
\begin {align*} \int (d \cos (a+b x))^n \sin ^2(a+b x) \, dx &=-\frac {(d \cos (a+b x))^{1+n} \, _2F_1\left (-\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(a+b x)\right ) \sin (a+b x)}{b d (1+n) \sqrt {\sin ^2(a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 68, normalized size = 0.99 \begin {gather*} -\frac {(d \cos (a+b x))^n \, _2F_1\left (-\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(a+b x)\right ) \sin (2 (a+b x))}{2 b (1+n) \sqrt {\sin ^2(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.37, size = 0, normalized size = 0.00 \[\int \left (d \cos \left (b x +a \right )\right )^{n} \left (\sin ^{2}\left (b x +a \right )\right )\, dx\]
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 \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 (d \cos {\left (a + b x \right )}\right )^{n} \sin ^{2}{\left (a + b x \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 {\sin \left (a+b\,x\right )}^2\,{\left (d\,\cos \left (a+b\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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