3.1518 \(\int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx\)

Optimal. Leaf size=125 \[ \frac{3 \cos (e+f x) (c+d \sin (e+f x))^{7/3} F_1\left (\frac{7}{3};-\frac{1}{2},-\frac{1}{2};\frac{10}{3};\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right )}{7 d f \sqrt{1-\frac{c+d \sin (e+f x)}{c-d}} \sqrt{1-\frac{c+d \sin (e+f x)}{c+d}}} \]

[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)]
)

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Rubi [A]  time = 0.129703, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2704, 138} \[ \frac{3 \cos (e+f x) (c+d \sin (e+f x))^{7/3} F_1\left (\frac{7}{3};-\frac{1}{2},-\frac{1}{2};\frac{10}{3};\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right )}{7 d f \sqrt{1-\frac{c+d \sin (e+f x)}{c-d}} \sqrt{1-\frac{c+d \sin (e+f x)}{c+d}}} \]

Antiderivative was successfully verified.

[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 2704

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]

Rule 138

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b*(m + 1
)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p), 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])

Rubi steps

\begin{align*} \int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx &=\frac{\cos (e+f x) \operatorname{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 F_1\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]  time = 2.1045, size = 301, normalized size = 2.41 \[ -\frac{3 \sec (e+f x) \sqrt [3]{c+d \sin (e+f x)} \left (-3 c \left (4 c^2+51 d^2\right ) \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} \sqrt{-\frac{d (\sin (e+f x)+1)}{c-d}} (c+d \sin (e+f x)) F_1\left (\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right )+12 \left (3 c^2 d^2+4 c^4-7 d^4\right ) \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} \sqrt{-\frac{d (\sin (e+f x)+1)}{c-d}} F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right )+4 d^2 \cos ^2(e+f x) \left (-4 c^2-44 c d \sin (e+f x)+14 d^2 \cos (2 (e+f x))+7 d^2\right )\right )}{1120 d^3 f} \]

Warning: Unable to verify antiderivative.

[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)

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Maple [F]  time = 0.167, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( c+d\sin \left ( fx+e \right ) \right ) ^{{\frac{4}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{4}{3}} \cos \left (f x + e\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) + c \cos \left (f x + e\right )^{2}\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{1}{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{4}{3}} \cos \left (f x + e\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)