Optimal. Leaf size=80 \[ -\frac {\sin (c+d x) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\cos ^2(c+d x)\right ) \left (a (b \cos (c+d x))^p\right )^n}{d (n p+1) \sqrt {\sin ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3208, 2643} \[ -\frac {\sin (c+d x) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\cos ^2(c+d x)\right ) \left (a (b \cos (c+d x))^p\right )^n}{d (n p+1) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2643
Rule 3208
Rubi steps
\begin {align*} \int \left (a (b \cos (c+d x))^p\right )^n \, dx &=\left ((b \cos (c+d x))^{-n p} \left (a (b \cos (c+d x))^p\right )^n\right ) \int (b \cos (c+d x))^{n p} \, dx\\ &=-\frac {\cos (c+d x) \left (a (b \cos (c+d x))^p\right )^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+n p) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 74, normalized size = 0.92 \[ -\frac {\sqrt {\sin ^2(c+d x)} \cot (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\cos ^2(c+d x)\right ) \left (a (b \cos (c+d x))^p\right )^n}{d (n p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\left (b \cos \left (d x + c\right )\right )^{p} a\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\left (b \cos \left (d x + c\right )\right )^{p} a\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \left (a \left (b \cos \left (d x +c \right )\right )^{p}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\left (b \cos \left (d x + c\right )\right )^{p} a\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,{\left (b\,\cos \left (c+d\,x\right )\right )}^p\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (b \cos {\left (c + d x \right )}\right )^{p}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________