Optimal. Leaf size=163 \[ -\frac {2 B \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n+9);\frac {1}{4} (2 n+13);\cos ^2(c+d x)\right )}{d (2 n+9) \sqrt {\sin ^2(c+d x)}}-\frac {2 C \sin (c+d x) \cos ^{\frac {11}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n+11);\frac {1}{4} (2 n+15);\cos ^2(c+d x)\right )}{d (2 n+11) \sqrt {\sin ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, 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, 3010, 2748, 2643} \[ -\frac {2 B \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n+9);\frac {1}{4} (2 n+13);\cos ^2(c+d x)\right )}{d (2 n+9) \sqrt {\sin ^2(c+d x)}}-\frac {2 C \sin (c+d x) \cos ^{\frac {11}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n+11);\frac {1}{4} (2 n+15);\cos ^2(c+d x)\right )}{d (2 n+11) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 20
Rule 2643
Rule 2748
Rule 3010
Rubi steps
\begin {align*} \int \cos ^{\frac {5}{2}}(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 {5}{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 {7}{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 {7}{2}+n}(c+d x) \, dx+\left (C \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{\frac {9}{2}+n}(c+d x) \, dx\\ &=-\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)}}-\frac {2 C \cos ^{\frac {11}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (11+2 n);\frac {1}{4} (15+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (11+2 n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.28, size = 138, normalized size = 0.85 \[ -\frac {2 \sqrt {\sin ^2(c+d x)} \cos ^{\frac {9}{2}}(c+d x) \csc (c+d x) (b \cos (c+d x))^n \left (B (2 n+11) \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n+9);\frac {1}{4} (2 n+13);\cos ^2(c+d x)\right )+C (2 n+9) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n+11);\frac {1}{4} (2 n+15);\cos ^2(c+d x)\right )\right )}{d (2 n+9) (2 n+11)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{4} + B \cos \left (d x + c\right )^{3}\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (b \cos \left (d x +c \right )\right )^{n} \left (B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^{5/2}\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________