3.119 \(\int \frac{\tan ^4(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx\)

Optimal. Leaf size=551 \[ \frac{3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac{973 \sec (e+f x) (a \sin (e+f x)+a)^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a \sin (e+f x)+a}\right ) \sqrt{\frac{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a \sin (e+f x)+a}+(a \sin (e+f x)+a)^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{a \sin (e+f x)+a}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt{3}\right ) \sqrt [3]{\sin (e+f x) a+a}}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sin (e+f x) a+a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{495 \sqrt [3]{2} \sqrt [4]{3} a^{4/3} f \sqrt{-\frac{\sqrt [3]{a \sin (e+f x)+a} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a \sin (e+f x)+a}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{a \sin (e+f x)+a}\right )^2}}}-\frac{\sec (e+f x) (356 a \sin (e+f x)+95 a)}{132 f (1-\sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac{973 \sec (e+f x)}{396 f \sqrt [3]{a \sin (e+f x)+a}}-\frac{973 (1-\sin (e+f x)) \sec (e+f x)}{495 f \sqrt [3]{a \sin (e+f x)+a}} \]

[Out]

(973*Sec[e + f*x])/(396*f*(a + a*Sin[e + f*x])^(1/3)) - (973*Sec[e + f*x]*(1 - Sin[e + f*x]))/(495*f*(a + a*Si
n[e + f*x])^(1/3)) - (Sec[e + f*x]*(95*a + 356*a*Sin[e + f*x]))/(132*f*(1 - Sin[e + f*x])*(a + a*Sin[e + f*x])
^(4/3)) + (973*EllipticF[ArcCos[(2^(1/3)*a^(1/3) - (1 - Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))/(2^(1/3)*a^(1/3)
- (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))], (2 + Sqrt[3])/4]*Sec[e + f*x]*(a + a*Sin[e + f*x])^(2/3)*(2^(1/3
)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3))*Sqrt[(2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + a*Sin[e + f*x])^(1/3) + (
a + a*Sin[e + f*x])^(2/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2])/(495*2^(1/3)*3^(1/
4)*a^(4/3)*f*Sqrt[-(((a + a*Sin[e + f*x])^(1/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3)))/(2^(1/3)*a^(1/
3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2)]) + (3*a^2*Sin[e + f*x]*Tan[e + f*x])/(4*f*(a - a*Sin[e + f*
x])*(a + a*Sin[e + f*x])^(4/3)) + (3*a^2*Sin[e + f*x]^2*Tan[e + f*x])/(f*(a - a*Sin[e + f*x])*(a + a*Sin[e + f
*x])^(4/3))

________________________________________________________________________________________

Rubi [A]  time = 0.493573, antiderivative size = 551, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2719, 100, 153, 144, 51, 63, 225} \[ \frac{3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac{973 \sec (e+f x) (a \sin (e+f x)+a)^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a \sin (e+f x)+a}\right ) \sqrt{\frac{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a \sin (e+f x)+a}+(a \sin (e+f x)+a)^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{a \sin (e+f x)+a}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt{3}\right ) \sqrt [3]{\sin (e+f x) a+a}}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sin (e+f x) a+a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{495 \sqrt [3]{2} \sqrt [4]{3} a^{4/3} f \sqrt{-\frac{\sqrt [3]{a \sin (e+f x)+a} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a \sin (e+f x)+a}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{a \sin (e+f x)+a}\right )^2}}}-\frac{\sec (e+f x) (356 a \sin (e+f x)+95 a)}{132 f (1-\sin (e+f x)) (a \sin (e+f x)+a)^{4/3}}+\frac{973 \sec (e+f x)}{396 f \sqrt [3]{a \sin (e+f x)+a}}-\frac{973 (1-\sin (e+f x)) \sec (e+f x)}{495 f \sqrt [3]{a \sin (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Tan[e + f*x]^4/(a + a*Sin[e + f*x])^(1/3),x]

[Out]

(973*Sec[e + f*x])/(396*f*(a + a*Sin[e + f*x])^(1/3)) - (973*Sec[e + f*x]*(1 - Sin[e + f*x]))/(495*f*(a + a*Si
n[e + f*x])^(1/3)) - (Sec[e + f*x]*(95*a + 356*a*Sin[e + f*x]))/(132*f*(1 - Sin[e + f*x])*(a + a*Sin[e + f*x])
^(4/3)) + (973*EllipticF[ArcCos[(2^(1/3)*a^(1/3) - (1 - Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))/(2^(1/3)*a^(1/3)
- (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))], (2 + Sqrt[3])/4]*Sec[e + f*x]*(a + a*Sin[e + f*x])^(2/3)*(2^(1/3
)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3))*Sqrt[(2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + a*Sin[e + f*x])^(1/3) + (
a + a*Sin[e + f*x])^(2/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2])/(495*2^(1/3)*3^(1/
4)*a^(4/3)*f*Sqrt[-(((a + a*Sin[e + f*x])^(1/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3)))/(2^(1/3)*a^(1/
3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2)]) + (3*a^2*Sin[e + f*x]*Tan[e + f*x])/(4*f*(a - a*Sin[e + f*
x])*(a + a*Sin[e + f*x])^(4/3)) + (3*a^2*Sin[e + f*x]^2*Tan[e + f*x])/(f*(a - a*Sin[e + f*x])*(a + a*Sin[e + f
*x])^(4/3))

Rule 2719

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_), x_Symbol] :> Dist[(Sqrt[a + b*Si
n[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])/(b*f*Cos[e + f*x]), Subst[Int[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p
+ 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m] && Inte
gerQ[p/2]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 144

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :>
Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(
m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m +
1) + d^2*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b*d*(b*c - a*d)^2*(m + 1)*(n + 1)), x] -
Dist[(a^2*d^2*f*h*(2 + 3*n + n^2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h
*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b
*c - a*d)^2*(m + 1)*(n + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, h
}, x] && LtQ[m, -1] && LtQ[n, -1]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 225

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(x*(s
+ r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2
)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[(r*x^2*(s + r*x^2))/(s + (1
+ Sqrt[3])*r*x^2)^2]), x]] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\tan ^4(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx &=\frac{\left (\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x^4}{(a-x)^{5/2} (a+x)^{17/6}} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{\left (3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (-3 a^2+\frac{a x}{3}\right )}{(a-x)^{5/2} (a+x)^{17/6}} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{\left (9 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x \left (-\frac{2 a^3}{3}-\frac{35 a^2 x}{9}\right )}{(a-x)^{5/2} (a+x)^{17/6}} \, dx,x,a \sin (e+f x)\right )}{4 a f}\\ &=-\frac{\sec (e+f x) (95 a+356 a \sin (e+f x))}{132 f (1-\sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{\left (973 a \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^{3/2} (a+x)^{11/6}} \, dx,x,a \sin (e+f x)\right )}{396 f}\\ &=\frac{973 \sec (e+f x)}{396 f \sqrt [3]{a+a \sin (e+f x)}}-\frac{\sec (e+f x) (95 a+356 a \sin (e+f x))}{132 f (1-\sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{\left (973 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-x} (a+x)^{11/6}} \, dx,x,a \sin (e+f x)\right )}{297 f}\\ &=\frac{973 \sec (e+f x)}{396 f \sqrt [3]{a+a \sin (e+f x)}}-\frac{973 \sec (e+f x) (1-\sin (e+f x))}{495 f \sqrt [3]{a+a \sin (e+f x)}}-\frac{\sec (e+f x) (95 a+356 a \sin (e+f x))}{132 f (1-\sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{\left (973 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-x} (a+x)^{5/6}} \, dx,x,a \sin (e+f x)\right )}{1485 a f}\\ &=\frac{973 \sec (e+f x)}{396 f \sqrt [3]{a+a \sin (e+f x)}}-\frac{973 \sec (e+f x) (1-\sin (e+f x))}{495 f \sqrt [3]{a+a \sin (e+f x)}}-\frac{\sec (e+f x) (95 a+356 a \sin (e+f x))}{132 f (1-\sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{\left (1946 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 a-x^6}} \, dx,x,\sqrt [6]{a+a \sin (e+f x)}\right )}{495 a f}\\ &=\frac{973 \sec (e+f x)}{396 f \sqrt [3]{a+a \sin (e+f x)}}-\frac{973 \sec (e+f x) (1-\sin (e+f x))}{495 f \sqrt [3]{a+a \sin (e+f x)}}-\frac{\sec (e+f x) (95 a+356 a \sin (e+f x))}{132 f (1-\sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{973 F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt{3}\right ) \sqrt [3]{a+a \sin (e+f x)}}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{a+a \sin (e+f x)}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) \sec (e+f x) (a+a \sin (e+f x))^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+a \sin (e+f x)}\right ) \sqrt{\frac{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+a \sin (e+f x)}+(a+a \sin (e+f x))^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{a+a \sin (e+f x)}\right )^2}}}{495 \sqrt [3]{2} \sqrt [4]{3} a^{4/3} f \sqrt{-\frac{\sqrt [3]{a+a \sin (e+f x)} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+a \sin (e+f x)}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{a+a \sin (e+f x)}\right )^2}}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{4 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac{3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}\\ \end{align*}

Mathematica [C]  time = 0.796308, size = 128, normalized size = 0.23 \[ \frac{973 \sqrt{2} \cos (e+f x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right )\right )+\sqrt{1-\sin (e+f x)} \sec ^3(e+f x) (22 \sin (e+f x)-128 \sin (3 (e+f x))-64 \cos (2 (e+f x))-49)}{495 f \sqrt{1-\sin (e+f x)} \sqrt [3]{a (\sin (e+f x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[e + f*x]^4/(a + a*Sin[e + f*x])^(1/3),x]

[Out]

(973*Sqrt[2]*Cos[e + f*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*e + Pi + 2*f*x)/4]^2] + Sec[e + f*x]^3*Sqrt[
1 - Sin[e + f*x]]*(-49 - 64*Cos[2*(e + f*x)] + 22*Sin[e + f*x] - 128*Sin[3*(e + f*x)]))/(495*f*Sqrt[1 - Sin[e
+ f*x]]*(a*(1 + Sin[e + f*x]))^(1/3))

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Maple [F]  time = 0.119, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \tan \left ( fx+e \right ) \right ) ^{4}{\frac{1}{\sqrt [3]{a+a\sin \left ( fx+e \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(f*x+e)^4/(a+a*sin(f*x+e))^(1/3),x)

[Out]

int(tan(f*x+e)^4/(a+a*sin(f*x+e))^(1/3),x)

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Maxima [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(tan(f*x+e)^4/(a+a*sin(f*x+e))^(1/3),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\tan \left (f x + e\right )^{4}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{1}{3}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^4/(a+a*sin(f*x+e))^(1/3),x, algorithm="fricas")

[Out]

integral(tan(f*x + e)^4/(a*sin(f*x + e) + a)^(1/3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{4}{\left (e + f x \right )}}{\sqrt [3]{a \left (\sin{\left (e + f x \right )} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)**4/(a+a*sin(f*x+e))**(1/3),x)

[Out]

Integral(tan(e + f*x)**4/(a*(sin(e + f*x) + 1))**(1/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (f x + e\right )^{4}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^4/(a+a*sin(f*x+e))^(1/3),x, algorithm="giac")

[Out]

integrate(tan(f*x + e)^4/(a*sin(f*x + e) + a)^(1/3), x)