3.2.15 \(\int \sqrt [3]{a+a \sin (e+f x)} \tan ^4(e+f x) \, dx\) [115]

Optimal. Leaf size=982 \[ -\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}} \]

[Out]

-361/126*sec(f*x+e)*(a+a*sin(f*x+e))^(1/3)/f+361/63*sec(f*x+e)*(1-sin(f*x+e))*(a+a*sin(f*x+e))^(1/3)/f-1/42*se
c(f*x+e)*(65*a^2-142*a^2*sin(f*x+e))/f/(a-a*sin(f*x+e))/(a+a*sin(f*x+e))^(2/3)+361/63*sec(f*x+e)*(1-sin(f*x+e)
)*(a+a*sin(f*x+e))^(2/3)*(1+3^(1/2))/f/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1+3^(1/2)))-361/63*2^(1/3)*((2
^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1-3^(1/2)))^2/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1+3^(1/2)))^2)^(
1/2)/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1-3^(1/2)))*(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1+3^(1/2)))
*EllipticE((1-(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1-3^(1/2)))^2/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(
1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*sec(f*x+e)*(a+a*sin(f*x+e))^(2/3)*(2^(1/3)*a^(1/3)-(a+a*sin(f*x+
e))^(1/3))*((2^(2/3)*a^(2/3)+2^(1/3)*a^(1/3)*(a+a*sin(f*x+e))^(1/3)+(a+a*sin(f*x+e))^(2/3))/(2^(1/3)*a^(1/3)-(
a+a*sin(f*x+e))^(1/3)*(1+3^(1/2)))^2)^(1/2)*3^(1/4)/a^(2/3)/f/(-(a+a*sin(f*x+e))^(1/3)*(2^(1/3)*a^(1/3)-(a+a*s
in(f*x+e))^(1/3))/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1+3^(1/2)))^2)^(1/2)-361/378*((2^(1/3)*a^(1/3)-(a+a
*sin(f*x+e))^(1/3)*(1-3^(1/2)))^2/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1+3^(1/2)))^2)^(1/2)/(2^(1/3)*a^(1/
3)-(a+a*sin(f*x+e))^(1/3)*(1-3^(1/2)))*(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1+3^(1/2)))*EllipticF((1-(2^(1
/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1-3^(1/2)))^2/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1+3^(1/2)))^2)^(1/2
),1/4*6^(1/2)+1/4*2^(1/2))*sec(f*x+e)*(a+a*sin(f*x+e))^(2/3)*(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3))*(1-3^(1/
2))*((2^(2/3)*a^(2/3)+2^(1/3)*a^(1/3)*(a+a*sin(f*x+e))^(1/3)+(a+a*sin(f*x+e))^(2/3))/(2^(1/3)*a^(1/3)-(a+a*sin
(f*x+e))^(1/3)*(1+3^(1/2)))^2)^(1/2)*2^(1/3)*3^(3/4)/a^(2/3)/f/(-(a+a*sin(f*x+e))^(1/3)*(2^(1/3)*a^(1/3)-(a+a*
sin(f*x+e))^(1/3))/(2^(1/3)*a^(1/3)-(a+a*sin(f*x+e))^(1/3)*(1+3^(1/2)))^2)^(1/2)+3/2*a^2*sin(f*x+e)*tan(f*x+e)
/f/(a-a*sin(f*x+e))/(a+a*sin(f*x+e))^(2/3)-3*a^2*sin(f*x+e)^2*tan(f*x+e)/f/(a-a*sin(f*x+e))/(a+a*sin(f*x+e))^(
2/3)

________________________________________________________________________________________

Rubi [A]
time = 0.89, antiderivative size = 982, normalized size of antiderivative = 1.00, 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 = {2798, 102, 158, 149, 53, 65, 314, 231, 1895} \begin {gather*} -\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 (\text {ArcCos}\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 (\text {ArcCos}\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}} \end {gather*}

Warning: Unable to verify antiderivative.

[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 53

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 65

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 102

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 149

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)/(b*d*(b*c - a*d)^2*(m + 1)*(n + 1)))*(a + b*x)^(m + 1)*(c + d*x)^(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 158

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 231

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]/(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)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^
2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x]

Rule 314

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 1895

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]/(2*r^2*Sqrt[(r*x^2*(s + r*x^2))/
(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]))*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r
*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]

Rule 2798

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_), x_Symbol] :> Dist[Sqrt[a + b*Sin
[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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 9.63, size = 318, normalized size = 0.32 \begin {gather*} \frac {\sqrt [3]{a (1+\sin (e+f x))} \left (\frac {\left (\frac {1083}{10}+\frac {1083 i}{10}\right ) (-1)^{3/4} e^{-i (e+f x)} \left (20 e^{i (e+f x)} \sqrt {\cos ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-i e^{-i (e+f x)}\right )-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+\pi +2 f x)\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)}\right )}{\sqrt {2} \left (1+i e^{-i (e+f x)}\right )^{2/3} \sqrt {i e^{-i (e+f x)} \left (-i+e^{i (e+f x)}\right )^2}}+3 \left (361+86 \sec (e+f x)-3 \sec ^3(e+f x)-172 \tan (e+f x)+24 \sec ^2(e+f x) \tan (e+f x)\right )\right )}{189 f} \end {gather*}

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)

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{\frac {1}{3}} \left (\tan ^{4}\left (f x +e \right )\right )\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{a \left (\sin {\left (e + f x \right )} + 1\right )} \tan ^{4}{\left (e + f x \right )}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{1/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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