Integrand size = 33, antiderivative size = 142 \[ \int \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 C \cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^n \sin (c+d x)}{d (9+2 n)}-\frac {2 (C (7+2 n)+A (9+2 n)) \cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (7+2 n),\frac {1}{4} (11+2 n),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (7+2 n) (9+2 n) \sqrt {\sin ^2(c+d x)}} \]
2*C*cos(d*x+c)^(7/2)*(b*cos(d*x+c))^n*sin(d*x+c)/d/(9+2*n)-2*(C*(7+2*n)+A* (9+2*n))*cos(d*x+c)^(7/2)*(b*cos(d*x+c))^n*hypergeom([1/2, 7/4+1/2*n],[11/ 4+1/2*n],cos(d*x+c)^2)*sin(d*x+c)/d/(4*n^2+32*n+63)/(sin(d*x+c)^2)^(1/2)
Time = 0.17 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.99 \[ \int \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^n \csc (c+d x) \left (A (11+2 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (7+2 n),\frac {1}{4} (11+2 n),\cos ^2(c+d x)\right )+C (7+2 n) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (11+2 n),\frac {1}{4} (15+2 n),\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (7+2 n) (11+2 n)} \]
(-2*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(11 + 2*n)*Hyper geometric2F1[1/2, (7 + 2*n)/4, (11 + 2*n)/4, Cos[c + d*x]^2] + C*(7 + 2*n) *Cos[c + d*x]^2*Hypergeometric2F1[1/2, (11 + 2*n)/4, (15 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(7 + 2*n)*(11 + 2*n))
Time = 0.46 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2034, 3042, 3493, 3042, 3122}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cos ^{\frac {5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right ) (b \cos (c+d x))^n \, dx\) |
\(\Big \downarrow \) 2034 |
\(\displaystyle \cos ^{-n}(c+d x) (b \cos (c+d x))^n \int \cos ^{n+\frac {5}{2}}(c+d x) \left (C \cos ^2(c+d x)+A\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \cos ^{-n}(c+d x) (b \cos (c+d x))^n \int \sin \left (c+d x+\frac {\pi }{2}\right )^{n+\frac {5}{2}} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx\) |
\(\Big \downarrow \) 3493 |
\(\displaystyle \cos ^{-n}(c+d x) (b \cos (c+d x))^n \left (\frac {(A (2 n+9)+C (2 n+7)) \int \cos ^{n+\frac {5}{2}}(c+d x)dx}{2 n+9}+\frac {2 C \sin (c+d x) \cos ^{n+\frac {7}{2}}(c+d x)}{d (2 n+9)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \cos ^{-n}(c+d x) (b \cos (c+d x))^n \left (\frac {(A (2 n+9)+C (2 n+7)) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{n+\frac {5}{2}}dx}{2 n+9}+\frac {2 C \sin (c+d x) \cos ^{n+\frac {7}{2}}(c+d x)}{d (2 n+9)}\right )\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle \cos ^{-n}(c+d x) (b \cos (c+d x))^n \left (\frac {2 C \sin (c+d x) \cos ^{n+\frac {7}{2}}(c+d x)}{d (2 n+9)}-\frac {2 (A (2 n+9)+C (2 n+7)) \sin (c+d x) \cos ^{n+\frac {7}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2 n+7),\frac {1}{4} (2 n+11),\cos ^2(c+d x)\right )}{d (2 n+7) (2 n+9) \sqrt {\sin ^2(c+d x)}}\right )\) |
((b*Cos[c + d*x])^n*((2*C*Cos[c + d*x]^(7/2 + n)*Sin[c + d*x])/(d*(9 + 2*n )) - (2*(C*(7 + 2*n) + A*(9 + 2*n))*Cos[c + d*x]^(7/2 + n)*Hypergeometric2 F1[1/2, (7 + 2*n)/4, (11 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 2 *n)*(9 + 2*n)*Sqrt[Sin[c + d*x]^2])))/Cos[c + d*x]^n
3.2.90.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[b^IntPart [n]*((b*v)^FracPart[n]/(a^IntPart[n]*(a*v)^FracPart[n])) Int[(a*v)^(m + n )*Fx, x], x] /; FreeQ[{a, b, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[m + n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*( x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f *(m + 2))), x] + Simp[(A*(m + 2) + C*(m + 1))/(m + 2) Int[(b*Sin[e + f*x] )^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && !LtQ[m, -1]
\[\int \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (\cos \left (d x +c \right ) b \right )^{n} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )d x\]
\[ \int \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
\[ \int \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n \,d x \]