\(\int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx\) [1516]

Optimal result

Integrand size = 35, antiderivative size = 458 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=-\frac {9 \left (64 a b c d-26 a^2 d^2-b^2 \left (18 c^2-13 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{2080 d^3 f}-\frac {9 b (3 b c-2 a d) \cos (e+f x) \sin (e+f x) (c+d \sin (e+f x))^{7/3}}{208 d^2 f}+\frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 (c+d)^2 \left (208 a^2 c d^2-64 a b d \left (3 c^2-5 d^2\right )+b^2 c \left (54 c^2+d^2\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {7}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)}}{1040 \sqrt {2} d^4 f \sqrt {1+\sin (e+f x)} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}-\frac {3 (c-d) (c+d)^2 \left (192 a b c d-208 a^2 d^2-b^2 \left (54 c^2+91 d^2\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)}}{1040 \sqrt {2} d^4 f \sqrt {1+\sin (e+f x)} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}} \]

[Out]

-9/2080*(64*a*b*c*d-26*a^2*d^2-b^2*(18*c^2-13*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(7/3)/d^3/f-9/208*b*(-2*a*d+3*
b*c)*cos(f*x+e)*sin(f*x+e)*(c+d*sin(f*x+e))^(7/3)/d^2/f+3/16*cos(f*x+e)*(a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(7
/3)/d/f-3/2080*(c+d)^2*(208*a^2*c*d^2-64*a*b*d*(3*c^2-5*d^2)+b^2*c*(54*c^2+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^4/f/((c+d*sin(f*x+e))/(c+d))^(1/3)
*2^(1/2)/(1+sin(f*x+e))^(1/2)-3/2080*(c-d)*(c+d)^2*(192*a*b*c*d-208*a^2*d^2-b^2*(54*c^2+91*d^2))*AppellF1(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^4/f/((c+d*sin(f*x+
e))/(c+d))^(1/3)*2^(1/2)/(1+sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3001, 3129, 3112, 3102, 2835, 2744, 144, 143} \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=-\frac {3 (c+d)^2 \left (208 a^2 c d^2-64 a b d \left (3 c^2-5 d^2\right )+b^2 c \left (54 c^2+d^2\right )\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {7}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{1040 \sqrt {2} d^4 f \sqrt {\sin (e+f x)+1} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}-\frac {3 (c-d) (c+d)^2 \left (-208 a^2 d^2+192 a b c d-\left (b^2 \left (54 c^2+91 d^2\right )\right )\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{1040 \sqrt {2} d^4 f \sqrt {\sin (e+f x)+1} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}-\frac {9 \left (-26 a^2 d^2+64 a b c d-\left (b^2 \left (18 c^2-13 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{2080 d^3 f}-\frac {9 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{208 d^2 f}+\frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f} \]

[In]

Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(4/3),x]

[Out]

(-9*(64*a*b*c*d - 26*a^2*d^2 - b^2*(18*c^2 - 13*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(2080*d^3*f) -
(9*b*(3*b*c - 2*a*d)*Cos[e + f*x]*Sin[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(208*d^2*f) + (3*Cos[e + f*x]*(a +
b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(7/3))/(16*d*f) - (3*(c + d)^2*(208*a^2*c*d^2 - 64*a*b*d*(3*c^2 - 5*d^2
) + b^2*c*(54*c^2 + 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))/(1040*Sqrt[2]*d^4*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c +
d))^(1/3)) - (3*(c - d)*(c + d)^2*(192*a*b*c*d - 208*a^2*d^2 - b^2*(54*c^2 + 91*d^2))*AppellF1[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))/(1040*Sq
rt[2]*d^4*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^(1/3))

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(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] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 144

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((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*
(b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 2744

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[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]

Rule 2835

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(b*
c - a*d)/b, Int[(a + b*Sin[e + f*x])^m, x], x] + Dist[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]

Rule 3001

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])

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[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]

Rule 3112

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (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] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*
c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e
+ f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

Rule 3129

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)
*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^
(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e +
f*x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a,
0] && NeQ[c, 0])))

Rubi steps \begin{align*} \text {integral}& = \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \left (1-\sin ^2(e+f x)\right ) \, dx \\ & = \frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}+\frac {3 \int (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \left (-2 b c+3 a d+(a c+b d) \sin (e+f x)+(3 b c-2 a d) \sin ^2(e+f x)\right ) \, dx}{16 d} \\ & = -\frac {9 b (3 b c-2 a d) \cos (e+f x) \sin (e+f x) (c+d \sin (e+f x))^{7/3}}{208 d^2 f}+\frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}+\frac {9 \int (c+d \sin (e+f x))^{4/3} \left (\frac {1}{3} \left (9 b^2 c^2-32 a b c d+39 a^2 d^2\right )+\frac {1}{3} d \left (13 a^2 c+4 b^2 c+32 a b d\right ) \sin (e+f x)+\frac {1}{3} \left (64 a b c d-26 a^2 d^2-b^2 \left (18 c^2-13 d^2\right )\right ) \sin ^2(e+f x)\right ) \, dx}{208 d^2} \\ & = -\frac {9 \left (64 a b c d-26 a^2 d^2-b^2 \left (18 c^2-13 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{2080 d^3 f}-\frac {9 b (3 b c-2 a d) \cos (e+f x) \sin (e+f x) (c+d \sin (e+f x))^{7/3}}{208 d^2 f}+\frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}+\frac {27 \int (c+d \sin (e+f x))^{4/3} \left (\frac {1}{9} d \left (128 a b c d+208 a^2 d^2-b^2 \left (36 c^2-91 d^2\right )\right )+\frac {1}{9} \left (208 a^2 c d^2-64 a b d \left (3 c^2-5 d^2\right )+b^2 c \left (54 c^2+d^2\right )\right ) \sin (e+f x)\right ) \, dx}{2080 d^3} \\ & = -\frac {9 \left (64 a b c d-26 a^2 d^2-b^2 \left (18 c^2-13 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{2080 d^3 f}-\frac {9 b (3 b c-2 a d) \cos (e+f x) \sin (e+f x) (c+d \sin (e+f x))^{7/3}}{208 d^2 f}+\frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}+\frac {\left (3 \left (208 a^2 c d^2-64 a b d \left (3 c^2-5 d^2\right )+b^2 c \left (54 c^2+d^2\right )\right )\right ) \int (c+d \sin (e+f x))^{7/3} \, dx}{2080 d^4}+\frac {\left (3 \left (c^2-d^2\right ) \left (192 a b c d-208 a^2 d^2-b^2 \left (54 c^2+91 d^2\right )\right )\right ) \int (c+d \sin (e+f x))^{4/3} \, dx}{2080 d^4} \\ & = -\frac {9 \left (64 a b c d-26 a^2 d^2-b^2 \left (18 c^2-13 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{2080 d^3 f}-\frac {9 b (3 b c-2 a d) \cos (e+f x) \sin (e+f x) (c+d \sin (e+f x))^{7/3}}{208 d^2 f}+\frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}+\frac {\left (3 \left (208 a^2 c d^2-64 a b d \left (3 c^2-5 d^2\right )+b^2 c \left (54 c^2+d^2\right )\right ) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(c+d x)^{7/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (e+f x)\right )}{2080 d^4 f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}+\frac {\left (3 \left (c^2-d^2\right ) \left (192 a b c d-208 a^2 d^2-b^2 \left (54 c^2+91 d^2\right )\right ) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(c+d x)^{4/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (e+f x)\right )}{2080 d^4 f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \\ & = -\frac {9 \left (64 a b c d-26 a^2 d^2-b^2 \left (18 c^2-13 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{2080 d^3 f}-\frac {9 b (3 b c-2 a d) \cos (e+f x) \sin (e+f x) (c+d \sin (e+f x))^{7/3}}{208 d^2 f}+\frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}+\frac {\left (3 (-c-d)^2 \left (208 a^2 c d^2-64 a b d \left (3 c^2-5 d^2\right )+b^2 c \left (54 c^2+d^2\right )\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)}\right ) \text {Subst}\left (\int \frac {\left (-\frac {c}{-c-d}-\frac {d x}{-c-d}\right )^{7/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (e+f x)\right )}{2080 d^4 f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)} \sqrt [3]{-\frac {c+d \sin (e+f x)}{-c-d}}}-\frac {\left (3 (-c-d) \left (c^2-d^2\right ) \left (192 a b c d-208 a^2 d^2-b^2 \left (54 c^2+91 d^2\right )\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)}\right ) \text {Subst}\left (\int \frac {\left (-\frac {c}{-c-d}-\frac {d x}{-c-d}\right )^{4/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (e+f x)\right )}{2080 d^4 f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)} \sqrt [3]{-\frac {c+d \sin (e+f x)}{-c-d}}} \\ & = -\frac {9 \left (64 a b c d-26 a^2 d^2-b^2 \left (18 c^2-13 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{2080 d^3 f}-\frac {9 b (3 b c-2 a d) \cos (e+f x) \sin (e+f x) (c+d \sin (e+f x))^{7/3}}{208 d^2 f}+\frac {3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}-\frac {3 (c+d)^2 \left (208 a^2 c d^2-64 a b d \left (3 c^2-5 d^2\right )+b^2 c \left (54 c^2+d^2\right )\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)}}{1040 \sqrt {2} d^4 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 \left (192 a b c d-208 a^2 d^2-b^2 \left (54 c^2+91 d^2\right )\right ) \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)}}{1040 \sqrt {2} d^4 f \sqrt {1+\sin (e+f x)} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 7.60 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.25 \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=-\frac {3 \sec (e+f x) \sqrt [3]{c+d \sin (e+f x)} \left (12 \left (c^2-d^2\right ) \left (-128 a b c d \left (6 c^2-17 d^2\right )+208 a^2 d^2 \left (4 c^2+7 d^2\right )+b^2 \left (216 c^4-248 c^2 d^2+637 d^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sin (e+f x))}{c-d}}-3 \left (208 a^2 c d^2 \left (4 c^2+51 d^2\right )+128 a b d \left (-6 c^4+21 c^2 d^2+40 d^4\right )+b^2 \left (216 c^5-392 c^3 d^2+3201 c d^4\right )\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sin (e+f x))}{c-d}} (c+d \sin (e+f x))+4 d^2 \cos ^2(e+f x) \left (14 d^2 \left (448 a b c d+208 a^2 d^2+b^2 \left (4 c^2+91 d^2\right )\right ) \cos (2 (e+f x))-455 b^2 d^4 \cos (4 (e+f x))+2 \left (-108 b^2 c^4+384 a b c^3 d-416 a^2 c^2 d^2+152 b^2 c^2 d^2+2048 a b c d^3+728 a^2 d^4+546 b^2 d^4-d \left (4576 a^2 c d^2+32 a b d \left (8 c^2+45 d^2\right )+b^2 \left (-72 c^3+687 c d^2\right )\right ) \sin (e+f x)+35 b d^3 (17 b c+32 a d) \sin (3 (e+f x))\right )\right )\right )}{232960 d^5 f} \]

[In]

Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(4/3),x]

[Out]

(-3*Sec[e + f*x]*(c + d*Sin[e + f*x])^(1/3)*(12*(c^2 - d^2)*(-128*a*b*c*d*(6*c^2 - 17*d^2) + 208*a^2*d^2*(4*c^
2 + 7*d^2) + b^2*(216*c^4 - 248*c^2*d^2 + 637*d^4))*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*(208*a^2*c*d^2*(4*c^2 + 51*d^2) + 128*a*b*d*(-6*c^4 + 21*c^2*d^2 + 40*d^4) + b^2*(216*c^5 - 392*c^3*d^2
+ 3201*c*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*(14*d^2*(448*a*b*c*d + 208*a^2*d^2 + b^2*(4*c^2 + 91*d^2))*Cos[2*(e + f*x)] - 455*b^2*d^4*Cos[4*(e +
f*x)] + 2*(-108*b^2*c^4 + 384*a*b*c^3*d - 416*a^2*c^2*d^2 + 152*b^2*c^2*d^2 + 2048*a*b*c*d^3 + 728*a^2*d^4 + 5
46*b^2*d^4 - d*(4576*a^2*c*d^2 + 32*a*b*d*(8*c^2 + 45*d^2) + b^2*(-72*c^3 + 687*c*d^2))*Sin[e + f*x] + 35*b*d^
3*(17*b*c + 32*a*d)*Sin[3*(e + f*x)]))))/(232960*d^5*f)

Maple [F]

\[\int \left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \sin \left (f x +e \right )\right )^{2} \left (c +d \sin \left (f x +e \right )\right )^{\frac {4}{3}}d x\]

[In]

int(cos(f*x+e)^2*(a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(4/3),x)

[Out]

int(cos(f*x+e)^2*(a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(4/3),x)

Fricas [F]

\[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(4/3),x, algorithm="fricas")

[Out]

integral(-((b^2*c + 2*a*b*d)*cos(f*x + e)^4 - (2*a*b*d + (a^2 + b^2)*c)*cos(f*x + e)^2 + (b^2*d*cos(f*x + e)^4
- (2*a*b*c + (a^2 + b^2)*d)*cos(f*x + e)^2)*sin(f*x + e))*(d*sin(f*x + e) + c)^(1/3), x)

Sympy [F(-1)]

Timed out. \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)**2*(a+b*sin(f*x+e))**2*(c+d*sin(f*x+e))**(4/3),x)

[Out]

Timed out

Maxima [F]

\[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(4/3),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^2*(d*sin(f*x + e) + c)^(4/3)*cos(f*x + e)^2, x)

Giac [F]

\[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(4/3),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^2*(d*sin(f*x + e) + c)^(4/3)*cos(f*x + e)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx=\int {\cos \left (e+f\,x\right )}^2\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{4/3} \,d x \]

[In]

int(cos(e + f*x)^2*(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(4/3),x)

[Out]

int(cos(e + f*x)^2*(a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(4/3), x)