Integrand size = 31, antiderivative size = 95 \[ \int \cos (c+d x) (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {3 C (b \cos (c+d x))^{10/3} \sin (c+d x)}{13 b^2 d}-\frac {3 (13 A+10 C) (b \cos (c+d x))^{10/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{3},\frac {8}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{130 b^2 d \sqrt {\sin ^2(c+d x)}} \]
3/13*C*(b*cos(d*x+c))^(10/3)*sin(d*x+c)/b^2/d-3/130*(13*A+10*C)*(b*cos(d*x +c))^(10/3)*hypergeom([1/2, 5/3],[8/3],cos(d*x+c)^2)*sin(d*x+c)/b^2/d/(sin (d*x+c)^2)^(1/2)
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.96 \[ \int \cos (c+d x) (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {3 (b \cos (c+d x))^{7/3} \cot (c+d x) \left (8 A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{3},\frac {8}{3},\cos ^2(c+d x)\right )+5 C \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {8}{3},\frac {11}{3},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{80 b d} \]
(-3*(b*Cos[c + d*x])^(7/3)*Cot[c + d*x]*(8*A*Hypergeometric2F1[1/2, 5/3, 8 /3, Cos[c + d*x]^2] + 5*C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 8/3, 11/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(80*b*d)
Time = 0.33 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2030, 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.
\(\displaystyle \int \cos (c+d x) (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle \frac {\int (b \cos (c+d x))^{7/3} \left (C \cos ^2(c+d x)+A\right )dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/3} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx}{b}\) |
\(\Big \downarrow \) 3493 |
\(\displaystyle \frac {\frac {1}{13} (13 A+10 C) \int (b \cos (c+d x))^{7/3}dx+\frac {3 C \sin (c+d x) (b \cos (c+d x))^{10/3}}{13 b d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{13} (13 A+10 C) \int \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/3}dx+\frac {3 C \sin (c+d x) (b \cos (c+d x))^{10/3}}{13 b d}}{b}\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle \frac {\frac {3 C \sin (c+d x) (b \cos (c+d x))^{10/3}}{13 b d}-\frac {3 (13 A+10 C) \sin (c+d x) (b \cos (c+d x))^{10/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{3},\frac {8}{3},\cos ^2(c+d x)\right )}{130 b d \sqrt {\sin ^2(c+d x)}}}{b}\) |
((3*C*(b*Cos[c + d*x])^(10/3)*Sin[c + d*x])/(13*b*d) - (3*(13*A + 10*C)*(b *Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin [c + d*x])/(130*b*d*Sqrt[Sin[c + d*x]^2]))/b
3.2.53.3.1 Defintions of rubi rules used
Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
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]
\[\int \cos \left (d x +c \right ) \left (\cos \left (d x +c \right ) b \right )^{\frac {4}{3}} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )d x\]
\[ \int \cos (c+d x) (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}} \cos \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \cos (c+d x) (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos (c+d x) (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}} \cos \left (d x + c\right ) \,d x } \]
\[ \int \cos (c+d x) (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}} \cos \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \cos (c+d x) (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3} \,d x \]