Integrand size = 33, antiderivative size = 146 \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {3 C \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \sin (c+d x)}{d (8+3 m)}-\frac {3 (C (5+3 m)+A (8+3 m)) \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (5+3 m),\frac {1}{6} (11+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+3 m) (8+3 m) \sqrt {\sin ^2(c+d x)}} \]
3*C*cos(d*x+c)^(1+m)*(b*cos(d*x+c))^(2/3)*sin(d*x+c)/d/(8+3*m)-3*(C*(5+3*m )+A*(8+3*m))*cos(d*x+c)^(1+m)*(b*cos(d*x+c))^(2/3)*hypergeom([1/2, 5/6+1/2 *m],[11/6+1/2*m],cos(d*x+c)^2)*sin(d*x+c)/d/(9*m^2+39*m+40)/(sin(d*x+c)^2) ^(1/2)
Time = 0.17 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.97 \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {3 \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \csc (c+d x) \left (A (11+3 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (5+3 m),\frac {1}{6} (11+3 m),\cos ^2(c+d x)\right )+C (5+3 m) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (11+3 m),\frac {1}{6} (17+3 m),\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (5+3 m) (11+3 m)} \]
(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*(A*(11 + 3*m) *Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2] + C*(5 + 3*m)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (11 + 3*m)/6, (17 + 3*m)/6, C os[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(5 + 3*m)*(11 + 3*m))
Time = 0.44 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.02, 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.
\(\displaystyle \int (b \cos (c+d x))^{2/3} \cos ^m(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 2034 |
\(\displaystyle \frac {(b \cos (c+d x))^{2/3} \int \cos ^{m+\frac {2}{3}}(c+d x) \left (C \cos ^2(c+d x)+A\right )dx}{\cos ^{\frac {2}{3}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(b \cos (c+d x))^{2/3} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{m+\frac {2}{3}} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx}{\cos ^{\frac {2}{3}}(c+d x)}\) |
\(\Big \downarrow \) 3493 |
\(\displaystyle \frac {(b \cos (c+d x))^{2/3} \left (\frac {(A (3 m+8)+C (3 m+5)) \int \cos ^{m+\frac {2}{3}}(c+d x)dx}{3 m+8}+\frac {3 C \sin (c+d x) \cos ^{m+\frac {5}{3}}(c+d x)}{d (3 m+8)}\right )}{\cos ^{\frac {2}{3}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(b \cos (c+d x))^{2/3} \left (\frac {(A (3 m+8)+C (3 m+5)) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{m+\frac {2}{3}}dx}{3 m+8}+\frac {3 C \sin (c+d x) \cos ^{m+\frac {5}{3}}(c+d x)}{d (3 m+8)}\right )}{\cos ^{\frac {2}{3}}(c+d x)}\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle \frac {(b \cos (c+d x))^{2/3} \left (\frac {3 C \sin (c+d x) \cos ^{m+\frac {5}{3}}(c+d x)}{d (3 m+8)}-\frac {3 (A (3 m+8)+C (3 m+5)) \sin (c+d x) \cos ^{m+\frac {5}{3}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (3 m+5),\frac {1}{6} (3 m+11),\cos ^2(c+d x)\right )}{d (3 m+5) (3 m+8) \sqrt {\sin ^2(c+d x)}}\right )}{\cos ^{\frac {2}{3}}(c+d x)}\) |
((b*Cos[c + d*x])^(2/3)*((3*C*Cos[c + d*x]^(5/3 + m)*Sin[c + d*x])/(d*(8 + 3*m)) - (3*(C*(5 + 3*m) + A*(8 + 3*m))*Cos[c + d*x]^(5/3 + m)*Hypergeomet ric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 3*m)*(8 + 3*m)*Sqrt[Sin[c + d*x]^2])))/Cos[c + d*x]^(2/3)
3.2.77.3.1 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]
\[\int \left (\cos ^{m}\left (d x +c \right )\right ) \left (\cos \left (d x +c \right ) b \right )^{\frac {2}{3}} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )d x\]
\[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{m} \,d x } \]
Timed out. \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{m} \,d x } \]
\[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{m} \,d x } \]
Timed out. \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^m\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3} \,d x \]