Integrand size = 33, antiderivative size = 341 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \, dx=-\frac {3 (6 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{130 d^2 f}+\frac {3 b \cos (e+f x) \sin (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}+\frac {3 (c+d)^2 \left (6 b c^2-13 a c d-10 b d^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {7}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)}}{65 \sqrt {2} d^3 f \sqrt {1+\sin (e+f x)} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}-\frac {3 (c-d) (c+d)^2 (6 b c-13 a d) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)}}{65 \sqrt {2} d^3 f \sqrt {1+\sin (e+f x)} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}} \]
-3/130*(-13*a*d+6*b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(7/3)/d^2/f+3/13*b*cos( f*x+e)*sin(f*x+e)*(c+d*sin(f*x+e))^(7/3)/d/f+3/130*(c+d)^2*(-13*a*c*d+6*b* c^2-10*b*d^2)*AppellF1(1/2,-7/3,1/2,3/2,d*(1-sin(f*x+e))/(c+d),1/2-1/2*sin (f*x+e))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/3)/d^3/f/((c+d*sin(f*x+e))/(c+d))^ (1/3)*2^(1/2)/(1+sin(f*x+e))^(1/2)-3/130*(c-d)*(c+d)^2*(-13*a*d+6*b*c)*App ellF1(1/2,-4/3,1/2,3/2,d*(1-sin(f*x+e))/(c+d),1/2-1/2*sin(f*x+e))*cos(f*x+ e)*(c+d*sin(f*x+e))^(1/3)/d^3/f/((c+d*sin(f*x+e))/(c+d))^(1/3)*2^(1/2)/(1+ sin(f*x+e))^(1/2)
Time = 4.28 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.17 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \, dx=\frac {3 \sec (e+f x) \sqrt [3]{c+d \sin (e+f x)} \left (12 \left (-c^2+d^2\right ) \left (-24 b c^3+52 a c^2 d+68 b c d^2+91 a d^3\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sin (e+f x))}{c-d}}+3 \left (-24 b c^4+52 a c^3 d+84 b c^2 d^2+663 a c d^3+160 b d^4\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sin (e+f x))}{c-d}} (c+d \sin (e+f x))-4 d^2 \cos ^2(e+f x) \left (24 b c^3-52 a c^2 d+128 b c d^2+91 a d^3+14 d^2 (14 b c+13 a d) \cos (2 (e+f x))-2 d \left (8 b c^2+286 a c d+45 b d^2\right ) \sin (e+f x)+70 b d^3 \sin (3 (e+f x))\right )\right )}{14560 d^4 f} \]
(3*Sec[e + f*x]*(c + d*Sin[e + f*x])^(1/3)*(12*(-c^2 + d^2)*(-24*b*c^3 + 5 2*a*c^2*d + 68*b*c*d^2 + 91*a*d^3)*AppellF1[1/3, 1/2, 1/2, 4/3, (c + d*Sin [e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[-((d*(1 + Sin[e + f*x]))/(c - d))] + 3*(-24*b*c^4 + 52*a*c^3*d + 84*b*c^2*d^2 + 663*a*c*d^3 + 160*b*d^4)*AppellF1[4/3, 1/2, 1/ 2, 7/3, (c + d*Sin[e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*Sqrt[- ((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[-((d*(1 + Sin[e + f*x]))/(c - d))] *(c + d*Sin[e + f*x]) - 4*d^2*Cos[e + f*x]^2*(24*b*c^3 - 52*a*c^2*d + 128* b*c*d^2 + 91*a*d^3 + 14*d^2*(14*b*c + 13*a*d)*Cos[2*(e + f*x)] - 2*d*(8*b* c^2 + 286*a*c*d + 45*b*d^2)*Sin[e + f*x] + 70*b*d^3*Sin[3*(e + f*x)])))/(1 4560*d^4*f)
Time = 1.12 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {3042, 3401, 3042, 3513, 27, 3042, 3502, 25, 3042, 3235, 3042, 3144, 156, 155}
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.
| \(\displaystyle \int \cos ^2(e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (e+f x)^2 (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3}dx\) |
| \(\Big \downarrow \) 3401 |
| \(\displaystyle \int \left (1-\sin ^2(e+f x)\right ) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (1-\sin (e+f x)^2\right ) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3}dx\) |
| \(\Big \downarrow \) 3513 |
| \(\displaystyle \frac {3 \int -\frac {1}{3} (c+d \sin (e+f x))^{4/3} \left (-\left ((6 b c-13 a d) \sin ^2(e+f x)\right )-3 b d \sin (e+f x)+3 b c-13 a d\right )dx}{13 d}+\frac {3 b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\int (c+d \sin (e+f x))^{4/3} \left (-\left ((6 b c-13 a d) \sin ^2(e+f x)\right )-3 b d \sin (e+f x)+3 b c-13 a d\right )dx}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\int (c+d \sin (e+f x))^{4/3} \left (-\left ((6 b c-13 a d) \sin (e+f x)^2\right )-3 b d \sin (e+f x)+3 b c-13 a d\right )dx}{13 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {3 b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {3 \int -(c+d \sin (e+f x))^{4/3} \left (d (4 b c+13 a d)-\left (6 b c^2-13 a d c-10 b d^2\right ) \sin (e+f x)\right )dx}{10 d}+\frac {3 (6 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}}{13 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {3 (6 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}-\frac {3 \int (c+d \sin (e+f x))^{4/3} \left (d (4 b c+13 a d)-\left (6 b c^2-13 a d c-10 b d^2\right ) \sin (e+f x)\right )dx}{10 d}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {3 (6 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}-\frac {3 \int (c+d \sin (e+f x))^{4/3} \left (d (4 b c+13 a d)-\left (6 b c^2-13 a d c-10 b d^2\right ) \sin (e+f x)\right )dx}{10 d}}{13 d}\) |
| \(\Big \downarrow \) 3235 |
| \(\displaystyle \frac {3 b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {3 (6 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}-\frac {3 \left (\frac {\left (c^2-d^2\right ) (6 b c-13 a d) \int (c+d \sin (e+f x))^{4/3}dx}{d}-\frac {\left (-13 a c d+6 b c^2-10 b d^2\right ) \int (c+d \sin (e+f x))^{7/3}dx}{d}\right )}{10 d}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {3 (6 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}-\frac {3 \left (\frac {\left (c^2-d^2\right ) (6 b c-13 a d) \int (c+d \sin (e+f x))^{4/3}dx}{d}-\frac {\left (-13 a c d+6 b c^2-10 b d^2\right ) \int (c+d \sin (e+f x))^{7/3}dx}{d}\right )}{10 d}}{13 d}\) |
| \(\Big \downarrow \) 3144 |
| \(\displaystyle \frac {3 b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {3 (6 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}-\frac {3 \left (\frac {\left (c^2-d^2\right ) (6 b c-13 a d) \cos (e+f x) \int \frac {(c+d \sin (e+f x))^{4/3}}{\sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}d\sin (e+f x)}{d f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}-\frac {\left (-13 a c d+6 b c^2-10 b d^2\right ) \cos (e+f x) \int \frac {(c+d \sin (e+f x))^{7/3}}{\sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}d\sin (e+f x)}{d f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}\right )}{10 d}}{13 d}\) |
| \(\Big \downarrow \) 156 |
| \(\displaystyle \frac {3 b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {3 (6 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}-\frac {3 \left (\frac {(c+d) \left (c^2-d^2\right ) (6 b c-13 a d) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)} \int \frac {\left (\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}\right )^{4/3}}{\sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}d\sin (e+f x)}{d f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}-\frac {(c+d)^2 \left (-13 a c d+6 b c^2-10 b d^2\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)} \int \frac {\left (\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}\right )^{7/3}}{\sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}d\sin (e+f x)}{d f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}\right )}{10 d}}{13 d}\) |
| \(\Big \downarrow \) 155 |
| \(\displaystyle \frac {3 b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}-\frac {\frac {3 (6 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{10 d f}-\frac {3 \left (\frac {\sqrt {2} (c+d)^2 \left (-13 a c d+6 b c^2-10 b d^2\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {7}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{d f \sqrt {\sin (e+f x)+1} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\sqrt {2} (c+d) \left (c^2-d^2\right ) (6 b c-13 a d) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{d f \sqrt {\sin (e+f x)+1} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}\right )}{10 d}}{13 d}\) |
(3*b*Cos[e + f*x]*Sin[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(13*d*f) - ((3* (6*b*c - 13*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(10*d*f) - (3*(( Sqrt[2]*(c + d)^2*(6*b*c^2 - 13*a*c*d - 10*b*d^2)*AppellF1[1/2, 1/2, -7/3, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*( c + d*Sin[e + f*x])^(1/3))/(d*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x ])/(c + d))^(1/3)) - (Sqrt[2]*(c + d)*(6*b*c - 13*a*d)*(c^2 - d^2)*AppellF 1[1/2, 1/2, -4/3, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d )]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(d*f*Sqrt[1 + Sin[e + f*x]]*(( c + d*Sin[e + f*x])/(c + d))^(1/3))))/(10*d))/(13*d)
3.16.17.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ (b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] && GtQ[Sim plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] && !(GtQ[Simpl ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d *x, a + b*x]) && !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c - e*d)], 0] && SimplerQ[e + f*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p ]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]) Int[(a + b*x)^m*(c + d*x)^n*Si mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & GtQ[Simplify[b/(b*c - a*d)], 0] && !GtQ[Simplify[b/(b*e - a*f)], 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]/(d*Sqrt[1 + Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]]) Subst[Int[(a + b*x )^n/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && !IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)/b Int[(a + b*Sin[e + f*x])^m, x], x] + Simp[d/b Int[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)* ((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, c , d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[ (-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3) )), x] + Simp[1/(b*(m + 3)) Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c *(m + 3) + b*d*(C*(m + 2) + A*(m + 3))*Sin[e + f*x] - (2*a*C*d - b*c*C*(m + 3))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
\[\int \left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {4}{3}}d x\]
\[ \int \cos ^2(e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2} \,d x } \]
integral(-(b*d*cos(f*x + e)^4 - (b*c + a*d)*cos(f*x + e)^2*sin(f*x + e) - (a*c + b*d)*cos(f*x + e)^2)*(d*sin(f*x + e) + c)^(1/3), x)
\[ \int \cos ^2(e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {4}{3}} \cos ^{2}{\left (e + f x \right )}\, dx \]
\[ \int \cos ^2(e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2} \,d x } \]
\[ \int \cos ^2(e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2} \,d x } \]
Timed out. \[ \int \cos ^2(e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \, dx=\int {\cos \left (e+f\,x\right )}^2\,\left (a+b\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{4/3} \,d x \]