Optimal. Leaf size=163 \[ -\frac {2 B \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (5+2 n);\frac {1}{4} (9+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+2 n) \sqrt {\sin ^2(c+d x)}}-\frac {2 C \cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (7+2 n);\frac {1}{4} (11+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (7+2 n) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {20, 3089, 2827,
2722} \begin {gather*} -\frac {2 B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n+5);\frac {1}{4} (2 n+9);\cos ^2(c+d x)\right )}{d (2 n+5) \sqrt {\sin ^2(c+d x)}}-\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n+7);\frac {1}{4} (2 n+11);\cos ^2(c+d x)\right )}{d (2 n+7) \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 2722
Rule 2827
Rule 3089
Rubi steps
\begin {align*} \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^n \left (B \cos (c+d x)+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 (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\left (\cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{\frac {3}{2}+n}(c+d x) (B+C \cos (c+d x)) \, dx\\ &=\left (B \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{\frac {3}{2}+n}(c+d x) \, dx+\left (C \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{\frac {5}{2}+n}(c+d x) \, dx\\ &=-\frac {2 B \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (5+2 n);\frac {1}{4} (9+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+2 n) \sqrt {\sin ^2(c+d x)}}-\frac {2 C \cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (7+2 n);\frac {1}{4} (11+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (7+2 n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 138, normalized size = 0.85 \begin {gather*} -\frac {2 \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \csc (c+d x) \left (B (7+2 n) \, _2F_1\left (\frac {1}{2},\frac {1}{4} (5+2 n);\frac {1}{4} (9+2 n);\cos ^2(c+d x)\right )+C (5+2 n) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{4} (7+2 n);\frac {1}{4} (11+2 n);\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (5+2 n) (7+2 n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.28, size = 0, normalized size = 0.00 \[\int \left (b \cos \left (d x +c \right )\right )^{n} \left (B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )\, dx\]
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 \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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