Optimal. Leaf size=69 \[ -\frac{\sin (a+b x) (d \cos (a+b x))^{n+1} \, _2F_1\left (-\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1) \sqrt{\sin ^2(a+b x)}} \]
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Rubi [A] time = 0.0396596, antiderivative size = 69, normalized size of antiderivative = 1., 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 = {2576} \[ -\frac{\sin (a+b x) (d \cos (a+b x))^{n+1} \, _2F_1\left (-\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1) \sqrt{\sin ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2576
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*}
Mathematica [A] time = 0.0834605, size = 68, normalized size = 0.99 \[ -\frac{\sin (2 (a+b x)) (d \cos (a+b x))^n \, _2F_1\left (-\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{2 b (n+1) \sqrt{\sin ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.992, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( bx+a \right ) \right ) ^{n} \left ( \sin \left ( bx+a \right ) \right ) ^{2}\, 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 \left (d \cos \left (b x + a\right )\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 )} \left (d \cos \left (b x + a\right )\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos{\left (a + b x \right )}\right )^{n} \sin ^{2}{\left (a + b x \right )}\, dx \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 \left (d \cos \left (b x + a\right )\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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