Optimal. Leaf size=80 \[ -\frac {2 \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (3+2 n),\frac {1}{4} (7+2 n),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (3+2 n) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {20, 2722}
\begin {gather*} -\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n+3);\frac {1}{4} (2 n+7);\cos ^2(c+d x)\right )}{d (2 n+3) \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 2722
Rubi steps
\begin {align*} \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^n \, dx &=\left (\cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{\frac {1}{2}+n}(c+d x) \, dx\\ &=-\frac {2 \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (3+2 n);\frac {1}{4} (7+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (3+2 n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 80, normalized size = 1.00 \begin {gather*} -\frac {\cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \csc (c+d x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (\frac {3}{2}+n\right ),\frac {1}{2} \left (\frac {7}{2}+n\right ),\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{d \left (\frac {3}{2}+n\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (b \cos \left (d x +c \right )\right )^{n}\, dx\]
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 \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 (b \cos {\left (c + d x \right )}\right )^{n} \sqrt {\cos {\left (c + d 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 \sqrt {\cos \left (c+d\,x\right )}\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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