Optimal. Leaf size=163 \[ \frac {2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n-3);\frac {1}{4} (2 n+1);\cos ^2(c+d x)\right )}{d (3-2 n) \sqrt {\sin ^2(c+d x)} \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n-1);\frac {1}{4} (2 n+3);\cos ^2(c+d x)\right )}{d (1-2 n) \sqrt {\sin ^2(c+d x)} \sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {20, 2748, 2643} \[ \frac {2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n-3);\frac {1}{4} (2 n+1);\cos ^2(c+d x)\right )}{d (3-2 n) \sqrt {\sin ^2(c+d x)} \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n-1);\frac {1}{4} (2 n+3);\cos ^2(c+d x)\right )}{d (1-2 n) \sqrt {\sin ^2(c+d x)} \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 2748
Rubi steps
\begin {align*} \int \frac {(b \cos (c+d x))^n (A+B \cos (c+d x))}{\cos ^{\frac {5}{2}}(c+d x)} \, dx &=\left (\cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac {5}{2}+n}(c+d x) (A+B \cos (c+d x)) \, dx\\ &=\left (A \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac {5}{2}+n}(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\\ &=\frac {2 A (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-3+2 n);\frac {1}{4} (1+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (3-2 n) \cos ^{\frac {3}{2}}(c+d x) \sqrt {\sin ^2(c+d x)}}+\frac {2 B (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-1+2 n);\frac {1}{4} (3+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1-2 n) \sqrt {\cos (c+d x)} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 138, normalized size = 0.85 \[ -\frac {2 \sqrt {\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left (A (2 n-1) \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n-3);\frac {1}{4} (2 n+1);\cos ^2(c+d x)\right )+B (2 n-3) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n-1);\frac {1}{4} (2 n+3);\cos ^2(c+d x)\right )\right )}{d (2 n-3) (2 n-1) \cos ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \cos \left (d x +c \right )\right )^{n} \left (A +B \cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^n\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{{\cos \left (c+d\,x\right )}^{5/2}} \,d x \]
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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