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 = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, 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 2643
Rule 3208
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*}
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Mathematica [A] time = 0.06, 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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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \cos \left (d x + c\right )^{m}\right )^{n}, x\right ) \]
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 \[ \int \left (b \cos \left (d x + c\right )^{m}\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int \left (b \left (\cos ^{m}\left (d x +c \right )\right )\right )^{n}\, dx \]
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 \[ \int \left (b \cos \left (d x + c\right )^{m}\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,{\cos \left (c+d\,x\right )}^m\right )}^n \,d x \]
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 \[ \int \left (b \cos ^{m}{\left (c + d x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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