\(\int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx\) [1518]

Optimal result

Integrand size = 23, antiderivative size = 125 \[ \int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx=\frac {3 \operatorname {AppellF1}\left (\frac {7}{3},-\frac {1}{2},-\frac {1}{2},\frac {10}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{7 d f \sqrt {1-\frac {c+d \sin (e+f x)}{c-d}} \sqrt {1-\frac {c+d \sin (e+f x)}{c+d}}} \]

[Out]

3/7*AppellF1(7/3,-1/2,-1/2,10/3,(c+d*sin(f*x+e))/(c-d),(c+d*sin(f*x+e))/(c+d))*cos(f*x+e)*(c+d*sin(f*x+e))^(7/
3)/d/f/(1+(-c-d*sin(f*x+e))/(c-d))^(1/2)/(1+(-c-d*sin(f*x+e))/(c+d))^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2783, 143} \[ \int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx=\frac {3 \cos (e+f x) (c+d \sin (e+f x))^{7/3} \operatorname {AppellF1}\left (\frac {7}{3},-\frac {1}{2},-\frac {1}{2},\frac {10}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right )}{7 d f \sqrt {1-\frac {c+d \sin (e+f x)}{c-d}} \sqrt {1-\frac {c+d \sin (e+f x)}{c+d}}} \]

[In]

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

[Out]

(3*AppellF1[7/3, -1/2, -1/2, 10/3, (c + d*Sin[e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*Cos[e + f*x]*(c
+ d*Sin[e + f*x])^(7/3))/(7*d*f*Sqrt[1 - (c + d*Sin[e + f*x])/(c - d)]*Sqrt[1 - (c + d*Sin[e + f*x])/(c + d)]
)

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 2783

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[g*((g*
Cos[e + f*x])^(p - 1)/(f*(1 - (a + b*Sin[e + f*x])/(a - b))^((p - 1)/2)*(1 - (a + b*Sin[e + f*x])/(a + b))^((p
- 1)/2))), Subst[Int[(-b/(a - b) - b*(x/(a - b)))^((p - 1)/2)*(b/(a + b) - b*(x/(a + b)))^((p - 1)/2)*(a + b*
x)^m, x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] &&  !IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) \text {Subst}\left (\int (c+d x)^{4/3} \sqrt {-\frac {d}{c-d}-\frac {d x}{c-d}} \sqrt {\frac {d}{c+d}-\frac {d x}{c+d}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\frac {c+d \sin (e+f x)}{c-d}} \sqrt {1-\frac {c+d \sin (e+f x)}{c+d}}} \\ & = \frac {3 \operatorname {AppellF1}\left (\frac {7}{3},-\frac {1}{2},-\frac {1}{2},\frac {10}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{7 d f \sqrt {1-\frac {c+d \sin (e+f x)}{c-d}} \sqrt {1-\frac {c+d \sin (e+f x)}{c+d}}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(301\) vs. \(2(125)=250\).

Time = 1.56 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.41 \[ \int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx=-\frac {3 \sec (e+f x) \sqrt [3]{c+d \sin (e+f x)} \left (12 \left (4 c^4+3 c^2 d^2-7 d^4\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sin (e+f x))}{c-d}}-3 c \left (4 c^2+51 d^2\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sin (e+f x))}{c-d}} (c+d \sin (e+f x))+4 d^2 \cos ^2(e+f x) \left (-4 c^2+7 d^2+14 d^2 \cos (2 (e+f x))-44 c d \sin (e+f x)\right )\right )}{1120 d^3 f} \]

[In]

Integrate[Cos[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*(4*c^4 + 3*c^2*d^2 - 7*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*c*(4*c^2 + 51*d^2)*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*(-4*c^2 + 7*d^2 + 14*d^2*Cos[2*(e + f*x)] - 44*c*d*Sin[e + f*x])))/
(1120*d^3*f)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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