Optimal. Leaf size=142 \[ \frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n}{d (2 n+5)}-\frac {2 (A (2 n+5)+C (2 n+3)) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n+3);\frac {1}{4} (2 n+7);\cos ^2(c+d x)\right )}{d (2 n+3) (2 n+5) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.11, antiderivative size = 132, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {20, 3014, 2643} \[ \frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n}{d (2 n+5)}-\frac {2 \left (\frac {A}{2 n+3}+\frac {C}{2 n+5}\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n+3);\frac {1}{4} (2 n+7);\cos ^2(c+d x)\right )}{d \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 3014
Rubi steps
\begin {align*} \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx &=\left (\cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{\frac {1}{2}+n}(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \sin (c+d x)}{d (5+2 n)}+\frac {\left (\left (C \left (\frac {3}{2}+n\right )+A \left (\frac {5}{2}+n\right )\right ) \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{\frac {1}{2}+n}(c+d x) \, dx}{\frac {5}{2}+n}\\ &=\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^n \sin (c+d x)}{d (5+2 n)}-\frac {2 (C (3+2 n)+A (5+2 n)) \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) (5+2 n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 140, normalized size = 0.99 \[ -\frac {2 \sqrt {\sin ^2(c+d x)} \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) (b \cos (c+d x))^n \left (A (2 n+7) \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n+3);\frac {1}{4} (2 n+7);\cos ^2(c+d x)\right )+C (2 n+3) \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n+7);\frac {1}{4} (2 n+11);\cos ^2(c+d x)\right )\right )}{d (2 n+3) (2 n+7)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \left (b \cos \left (d x +c \right )\right )^{n} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
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 \[ \int \sqrt {\cos \left (c+d\,x\right )}\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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