Optimal. Leaf size=163 \[ -\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right )}{d (2 n+7) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);\cos ^2(c+d x)\right )}{d (2 n+9) \sqrt{\sin ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0946512, antiderivative size = 163, normalized size of antiderivative = 1., 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} \[ -\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right )}{d (2 n+7) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);\cos ^2(c+d x)\right )}{d (2 n+9) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 20
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n (A+B \cos (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{7}{2}+n}(c+d x) \, dx\\ &=-\frac{2 A \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)}}-\frac{2 B \cos ^{\frac{9}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (9+2 n);\frac{1}{4} (13+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (9+2 n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.40253, size = 138, normalized size = 0.85 \[ -\frac{2 \sqrt{\sin ^2(c+d x)} \cos ^{\frac{7}{2}}(c+d x) \csc (c+d x) (b \cos (c+d x))^n \left (A (2 n+9) \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right )+B (2 n+7) \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);\cos ^2(c+d x)\right )\right )}{d (2 n+7) (2 n+9)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.356, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}} \left ( b\cos \left ( dx+c \right ) \right ) ^{n} \left ( A+B\cos \left ( dx+c \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B \cos \left (d x + c\right )^{3} + A \cos \left (d x + c\right )^{2}\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sqrt{\cos \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]