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

Optimal. Leaf size=982 \[ \text{result too large to display} \]

[Out]

(-361*Sec[e + f*x]*(a + a*Sin[e + f*x])^(1/3))/(126*f) + (361*Sec[e + f*x]*(1 - Sin[e + f*x])*(a + a*Sin[e + f
*x])^(1/3))/(63*f) - (Sec[e + f*x]*(65*a^2 - 142*a^2*Sin[e + f*x]))/(42*f*(a - a*Sin[e + f*x])*(a + a*Sin[e +
f*x])^(2/3)) + (361*(1 + Sqrt[3])*Sec[e + f*x]*(1 - Sin[e + f*x])*(a + a*Sin[e + f*x])^(2/3))/(63*f*(2^(1/3)*a
^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))) - (361*2^(1/3)*EllipticE[ArcCos[(2^(1/3)*a^(1/3) - (1 - Sq
rt[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])/(21*3^(3/4)*a^(2/3)*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)]) - (361*(1 - Sqr
t[3])*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 + Sq
rt[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])/(63*2^(2/3)*3^(1/4)*a^(2/3)
*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])/(2*f*(a - a*Sin[e + f*x])*(a + a
*Sin[e + f*x])^(2/3)) - (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])^(2/3)
)

________________________________________________________________________________________

Rubi [A]  time = 1.2706, antiderivative size = 982, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {2719, 100, 153, 144, 51, 63, 308, 225, 1881} \[ -\frac{3 \sin ^2(e+f x) \tan (e+f x) a^2}{f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}+\frac{3 \sin (e+f x) \tan (e+f x) a^2}{2 f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}-\frac{\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}-\frac{361 \sec (e+f x) \sqrt [3]{\sin (e+f x) a+a}}{126 f}+\frac{361 \sec (e+f x) (1-\sin (e+f x)) \sqrt [3]{\sin (e+f x) a+a}}{63 f}+\frac{361 \left (1+\sqrt{3}\right ) \sec (e+f x) (1-\sin (e+f x)) (\sin (e+f x) a+a)^{2/3}}{63 f \left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )}-\frac{361 \sqrt [3]{2} E\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 ) \sec (e+f x) (\sin (e+f x) a+a)^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{\sin (e+f x) a+a}\right ) \sqrt{\frac{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{\sin (e+f x) a+a} \sqrt [3]{a}+(\sin (e+f x) a+a)^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )^2}}}{21\ 3^{3/4} f \sqrt{-\frac{\sqrt [3]{\sin (e+f x) a+a} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{\sin (e+f x) a+a}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )^2}} a^{2/3}}-\frac{361 \left (1-\sqrt{3}\right ) 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 ) \sec (e+f x) (\sin (e+f x) a+a)^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{\sin (e+f x) a+a}\right ) \sqrt{\frac{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{\sin (e+f x) a+a} \sqrt [3]{a}+(\sin (e+f x) a+a)^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )^2}}}{63\ 2^{2/3} \sqrt [4]{3} f \sqrt{-\frac{\sqrt [3]{\sin (e+f x) a+a} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{\sin (e+f x) a+a}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sin (e+f x) a+a}\right )^2}} a^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-361*Sec[e + f*x]*(a + a*Sin[e + f*x])^(1/3))/(126*f) + (361*Sec[e + f*x]*(1 - Sin[e + f*x])*(a + a*Sin[e + f
*x])^(1/3))/(63*f) - (Sec[e + f*x]*(65*a^2 - 142*a^2*Sin[e + f*x]))/(42*f*(a - a*Sin[e + f*x])*(a + a*Sin[e +
f*x])^(2/3)) + (361*(1 + Sqrt[3])*Sec[e + f*x]*(1 - Sin[e + f*x])*(a + a*Sin[e + f*x])^(2/3))/(63*f*(2^(1/3)*a
^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))) - (361*2^(1/3)*EllipticE[ArcCos[(2^(1/3)*a^(1/3) - (1 - Sq
rt[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])/(21*3^(3/4)*a^(2/3)*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)]) - (361*(1 - Sqr
t[3])*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 + Sq
rt[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])/(63*2^(2/3)*3^(1/4)*a^(2/3)
*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])/(2*f*(a - a*Sin[e + f*x])*(a + a
*Sin[e + f*x])^(2/3)) - (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])^(2/3)
)

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 308

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

Rule 1881

Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/
a, 3]]}, Simp[((1 + Sqrt[3])*d*s^3*x*Sqrt[a + b*x^6])/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2)), x] - Simp[(3^(1/4)*
d*s*x*(s + r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticE[ArcCos[(s + (1 - Sqrt[
3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*
x^2)^2]*Sqrt[a + b*x^6]), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]

Rubi steps

\begin{align*} \int \sqrt [3]{a+a \sin (e+f x)} \tan ^4(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)^{13/6}} \, 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))^{2/3}}-\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)^{13/6}} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}-\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{17 a^2 x}{9}\right )}{(a-x)^{5/2} (a+x)^{13/6}} \, dx,x,a \sin (e+f x)\right )}{2 a f}\\ &=-\frac{\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}-\frac{\left (361 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)^{7/6}} \, dx,x,a \sin (e+f x)\right )}{126 f}\\ &=-\frac{361 \sec (e+f x) \sqrt [3]{a+a \sin (e+f x)}}{126 f}-\frac{\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}-\frac{\left (361 \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)^{7/6}} \, dx,x,a \sin (e+f x)\right )}{189 f}\\ &=-\frac{361 \sec (e+f x) \sqrt [3]{a+a \sin (e+f x)}}{126 f}+\frac{361 \sec (e+f x) (1-\sin (e+f x)) \sqrt [3]{a+a \sin (e+f x)}}{63 f}-\frac{\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}+\frac{\left (361 \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} \sqrt [6]{a+x}} \, dx,x,a \sin (e+f x)\right )}{189 a f}\\ &=-\frac{361 \sec (e+f x) \sqrt [3]{a+a \sin (e+f x)}}{126 f}+\frac{361 \sec (e+f x) (1-\sin (e+f x)) \sqrt [3]{a+a \sin (e+f x)}}{63 f}-\frac{\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}+\frac{\left (722 \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}{\sqrt{2 a-x^6}} \, dx,x,\sqrt [6]{a+a \sin (e+f x)}\right )}{63 a f}\\ &=-\frac{361 \sec (e+f x) \sqrt [3]{a+a \sin (e+f x)}}{126 f}+\frac{361 \sec (e+f x) (1-\sin (e+f x)) \sqrt [3]{a+a \sin (e+f x)}}{63 f}-\frac{\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}+\frac{3 a^2 \sin (e+f x) \tan (e+f x)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}-\frac{\left (361 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{2^{2/3} \left (-1+\sqrt{3}\right ) a^{2/3}-2 x^4}{\sqrt{2 a-x^6}} \, dx,x,\sqrt [6]{a+a \sin (e+f x)}\right )}{63 a f}-\frac{\left (361\ 2^{2/3} \left (1-\sqrt{3}\right ) \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 )}{63 \sqrt [3]{a} f}\\ &=-\frac{361 \sec (e+f x) \sqrt [3]{a+a \sin (e+f x)}}{126 f}+\frac{361 \sec (e+f x) (1-\sin (e+f x)) \sqrt [3]{a+a \sin (e+f x)}}{63 f}-\frac{\sec (e+f x) \left (65 a^2-142 a^2 \sin (e+f x)\right )}{42 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}+\frac{361 \left (1+\sqrt{3}\right ) \sec (e+f x) (1-\sin (e+f x)) (a+a \sin (e+f x))^{2/3}}{63 f \left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt{3}\right ) \sqrt [3]{a+a \sin (e+f x)}\right )}-\frac{361 \sqrt [3]{2} E\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}}}{21\ 3^{3/4} a^{2/3} 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{361 \left (1-\sqrt{3}\right ) 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}}}{63\ 2^{2/3} \sqrt [4]{3} a^{2/3} 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)}{2 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{2/3}}-\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))^{2/3}}\\ \end{align*}

Mathematica [C]  time = 3.273, size = 318, normalized size = 0.32 \[ \frac{\sqrt [3]{a (\sin (e+f x)+1)} \left (3 \left (-172 \tan (e+f x)-3 \sec ^3(e+f x)+86 \sec (e+f x)+24 \tan (e+f x) \sec ^2(e+f x)+361\right )+\frac{\left (\frac{1083}{10}+\frac{1083 i}{10}\right ) (-1)^{3/4} e^{-i (e+f x)} \left (-2 \left (1+i e^{-i (e+f x)}\right )^{2/3} \left (1+e^{2 i (e+f x)}\right ) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right )\right )+5 i \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-i e^{-i (e+f x)}\right ) \sqrt{2-2 \sin (e+f x)}+20 e^{i (e+f x)} \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{2}{3};-i e^{-i (e+f x)}\right ) \sqrt{\cos ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right )}\right )}{\sqrt{2} \left (1+i e^{-i (e+f x)}\right )^{2/3} \sqrt{i e^{-i (e+f x)} \left (e^{i (e+f x)}-i\right )^2}}\right )}{189 f} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a*(1 + Sin[e + f*x]))^(1/3)*(((1083/10 + (1083*I)/10)*(-1)^(3/4)*(20*E^(I*(e + f*x))*Sqrt[Cos[(2*e + Pi + 2*
f*x)/4]^2]*Hypergeometric2F1[-1/3, 1/3, 2/3, (-I)/E^(I*(e + f*x))] - 2*(1 + I/E^(I*(e + f*x)))^(2/3)*(1 + E^((
2*I)*(e + f*x)))*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[(2*e + Pi + 2*f*x)/4]^2] + (5*I)*Hypergeometric2F1[1/3,
2/3, 5/3, (-I)/E^(I*(e + f*x))]*Sqrt[2 - 2*Sin[e + f*x]]))/(Sqrt[2]*E^(I*(e + f*x))*(1 + I/E^(I*(e + f*x)))^(
2/3)*Sqrt[(I*(-I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]) + 3*(361 + 86*Sec[e + f*x] - 3*Sec[e + f*x]^3 - 172*T
an[e + f*x] + 24*Sec[e + f*x]^2*Tan[e + f*x])))/(189*f)

________________________________________________________________________________________

Maple [F]  time = 0.128, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{a+a\sin \left ( fx+e \right ) } \left ( \tan \left ( fx+e \right ) \right ) ^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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