Optimal. Leaf size=78 \[ -\frac{\sin (c+d x) \cos (c+d x) \left (b \cos ^m(c+d x)\right )^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m n+1);\frac{1}{2} (m n+3);\cos ^2(c+d x)\right )}{d (m n+1) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0347181, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3208, 2643} \[ -\frac{\sin (c+d x) \cos (c+d x) \left (b \cos ^m(c+d x)\right )^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m n+1);\frac{1}{2} (m n+3);\cos ^2(c+d x)\right )}{d (m n+1) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3208
Rule 2643
Rubi steps
\begin{align*} \int \left (b \cos ^m(c+d x)\right )^n \, dx &=\left (\cos ^{-m n}(c+d x) \left (b \cos ^m(c+d x)\right )^n\right ) \int \cos ^{m n}(c+d x) \, dx\\ &=-\frac{\cos (c+d x) \left (b \cos ^m(c+d x)\right )^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1+m n);\frac{1}{2} (3+m n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+m n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.061062, size = 72, normalized size = 0.92 \[ -\frac{\sqrt{\sin ^2(c+d x)} \cot (c+d x) \left (b \cos ^m(c+d x)\right )^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m n+1);\frac{1}{2} (m n+3);\cos ^2(c+d x)\right )}{d (m n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.274, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ( \cos \left ( dx+c \right ) \right ) ^{m} \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 \left (b \cos \left (d x + c\right )^{m}\right )^{n}\,{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 (b \cos \left (d x + c\right )^{m}\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 (b \cos ^{m}{\left (c + d x \right )}\right )^{n}\, 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 (b \cos \left (d x + c\right )^{m}\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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