Integrand size = 23, antiderivative size = 538 \[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\frac {7 a \sec (e+f x)}{9 f (a+a \sin (e+f x))^{4/3}}+\frac {95 \sec (e+f x) (a-a \sin (e+f x))}{99 f (a+a \sin (e+f x))^{4/3}}-\frac {973 \sec (e+f x) (1-\sin (e+f x))}{495 f \sqrt [3]{a+a \sin (e+f x)}}+\frac {973 \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sec (e+f x) (a+a \sin (e+f x))^{2/3} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+a \sin (e+f x)}\right ) \sqrt {\frac {2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+a \sin (e+f x)}+(a+a \sin (e+f x))^{2/3}}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}\right )^2}}}{495 \sqrt [3]{2} \sqrt [4]{3} a^{4/3} f \sqrt {-\frac {\sqrt [3]{a+a \sin (e+f x)} \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+a \sin (e+f x)}\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sin (e+f x)}\right )^2}}}-\frac {8 a^2 \sin (e+f x) \tan (e+f x)}{3 f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}}+\frac {3 a^2 \sin ^2(e+f x) \tan (e+f x)}{f (a-a \sin (e+f x)) (a+a \sin (e+f x))^{4/3}} \] Output:
7/9*a*sec(f*x+e)/f/(a+a*sin(f*x+e))^(4/3)+95/99*sec(f*x+e)*(a-a*sin(f*x+e) )/f/(a+a*sin(f*x+e))^(4/3)-973/495*sec(f*x+e)*(1-sin(f*x+e))/f/(a+a*sin(f* x+e))^(1/3)+973/2970*InverseJacobiAM(arccos((2^(1/3)*a^(1/3)-(1-3^(1/2))*( a+a*sin(f*x+e))^(1/3))/(2^(1/3)*a^(1/3)-(1+3^(1/2))*(a+a*sin(f*x+e))^(1/3) )),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)-(1+3^(1/2))*(a+a*sin( f*x+e))^(1/3))^2)^(1/2)*2^(2/3)*3^(3/4)/a^(4/3)/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)-(1+3^(1/2))*(a+ a*sin(f*x+e))^(1/3))^2)^(1/2)-8/3*a^2*sin(f*x+e)*tan(f*x+e)/f/(a-a*sin(f*x +e))/(a+a*sin(f*x+e))^(4/3)+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))^(4/3)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.52 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.24 \[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\frac {973 \sqrt {2} \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sin ^2\left (\frac {1}{4} (2 e+\pi +2 f x)\right )\right )+\sec ^3(e+f x) \sqrt {1-\sin (e+f x)} (-49-64 \cos (2 (e+f x))+22 \sin (e+f x)-128 \sin (3 (e+f x)))}{495 f \sqrt {1-\sin (e+f x)} \sqrt [3]{a (1+\sin (e+f x))}} \] Input:
Integrate[Tan[e + f*x]^4/(a + a*Sin[e + f*x])^(1/3),x]
Output:
(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))
Time = 0.49 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 3198, 111, 27, 170, 27, 161, 61, 61, 73, 766}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\tan ^4(e+f x)}{\sqrt [3]{a \sin (e+f x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (e+f x)^4}{\sqrt [3]{a \sin (e+f x)+a}}dx\) |
| \(\Big \downarrow \) 3198 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \int \frac {a^4 \sin ^4(e+f x)}{(a-a \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^{17/6}}d(a \sin (e+f x))}{a f}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (3 \int -\frac {a^3 \sin ^2(e+f x) (9 a-a \sin (e+f x))}{3 (a-a \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^{17/6}}d(a \sin (e+f x))+\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}\right )}{a f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}-a \int \frac {a^2 \sin ^2(e+f x) (9 a-a \sin (e+f x))}{(a-a \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^{17/6}}d(a \sin (e+f x))\right )}{a f}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}-a \left (\frac {3}{4} \int \frac {a^2 \sin (e+f x) (35 \sin (e+f x) a+6 a)}{3 (a-a \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^{17/6}}d(a \sin (e+f x))-\frac {3 a^2 \sin ^2(e+f x)}{4 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}\right )\right )}{a f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}-a \left (\frac {1}{4} a \int \frac {a \sin (e+f x) (35 \sin (e+f x) a+6 a)}{(a-a \sin (e+f x))^{5/2} (\sin (e+f x) a+a)^{17/6}}d(a \sin (e+f x))-\frac {3 a^2 \sin ^2(e+f x)}{4 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}\right )\right )}{a f}\) |
| \(\Big \downarrow \) 161 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}-a \left (\frac {1}{4} a \left (\frac {356 a \sin (e+f x)+95 a}{33 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}-\frac {973}{99} \int \frac {1}{(a-a \sin (e+f x))^{3/2} (\sin (e+f x) a+a)^{11/6}}d(a \sin (e+f x))\right )-\frac {3 a^2 \sin ^2(e+f x)}{4 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}\right )\right )}{a f}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}-a \left (\frac {1}{4} a \left (\frac {356 a \sin (e+f x)+95 a}{33 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}-\frac {973}{99} \left (\frac {4 \int \frac {1}{\sqrt {a-a \sin (e+f x)} (\sin (e+f x) a+a)^{11/6}}d(a \sin (e+f x))}{3 a}+\frac {1}{a \sqrt {a-a \sin (e+f x)} (a \sin (e+f x)+a)^{5/6}}\right )\right )-\frac {3 a^2 \sin ^2(e+f x)}{4 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}\right )\right )}{a f}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}-a \left (\frac {1}{4} a \left (\frac {356 a \sin (e+f x)+95 a}{33 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}-\frac {973}{99} \left (\frac {4 \left (\frac {\int \frac {1}{\sqrt {a-a \sin (e+f x)} (\sin (e+f x) a+a)^{5/6}}d(a \sin (e+f x))}{5 a}-\frac {3 \sqrt {a-a \sin (e+f x)}}{5 a (a \sin (e+f x)+a)^{5/6}}\right )}{3 a}+\frac {1}{a \sqrt {a-a \sin (e+f x)} (a \sin (e+f x)+a)^{5/6}}\right )\right )-\frac {3 a^2 \sin ^2(e+f x)}{4 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}\right )\right )}{a f}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}-a \left (\frac {1}{4} a \left (\frac {356 a \sin (e+f x)+95 a}{33 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}-\frac {973}{99} \left (\frac {4 \left (\frac {6 \int \frac {1}{\sqrt {2 a-a^6 \sin ^6(e+f x)}}d\sqrt [6]{\sin (e+f x) a+a}}{5 a}-\frac {3 \sqrt {a-a \sin (e+f x)}}{5 a (a \sin (e+f x)+a)^{5/6}}\right )}{3 a}+\frac {1}{a \sqrt {a-a \sin (e+f x)} (a \sin (e+f x)+a)^{5/6}}\right )\right )-\frac {3 a^2 \sin ^2(e+f x)}{4 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}\right )\right )}{a f}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a} \left (\frac {3 a^3 \sin ^3(e+f x)}{(a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}-a \left (\frac {1}{4} a \left (\frac {356 a \sin (e+f x)+95 a}{33 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}-\frac {973}{99} \left (\frac {4 \left (\frac {3^{3/4} \sin (e+f x) \left (\sqrt [3]{2} \sqrt [3]{a}-a^2 \sin ^2(e+f x)\right ) \sqrt {\frac {\sqrt [3]{2} a^{7/3} \sin ^2(e+f x)+2^{2/3} a^{2/3}+a^4 \sin ^4(e+f x)}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) a^2 \sin ^2(e+f x)\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2} \sqrt [3]{a}-\left (1-\sqrt {3}\right ) a^2 \sin ^2(e+f x)}{\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) a^2 \sin ^2(e+f x)}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt [3]{2} \sqrt [3]{a} \sqrt {-\frac {a^2 \sin ^2(e+f x) \left (\sqrt [3]{2} \sqrt [3]{a}-a^2 \sin ^2(e+f x)\right )}{\left (\sqrt [3]{2} \sqrt [3]{a}-\left (1+\sqrt {3}\right ) a^2 \sin ^2(e+f x)\right )^2}} \sqrt {2 a-a^6 \sin ^6(e+f x)}}-\frac {3 \sqrt {a-a \sin (e+f x)}}{5 a (a \sin (e+f x)+a)^{5/6}}\right )}{3 a}+\frac {1}{a \sqrt {a-a \sin (e+f x)} (a \sin (e+f x)+a)^{5/6}}\right )\right )-\frac {3 a^2 \sin ^2(e+f x)}{4 (a-a \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{11/6}}\right )\right )}{a f}\) |
Input:
Int[Tan[e + f*x]^4/(a + a*Sin[e + f*x])^(1/3),x]
Output:
(Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]]*((3*a^3*Si n[e + f*x]^3)/((a - a*Sin[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(11/6)) - a *((-3*a^2*Sin[e + f*x]^2)/(4*(a - a*Sin[e + f*x])^(3/2)*(a + a*Sin[e + f*x ])^(11/6)) + (a*((95*a + 356*a*Sin[e + f*x])/(33*(a - a*Sin[e + f*x])^(3/2 )*(a + a*Sin[e + f*x])^(11/6)) - (973*(1/(a*Sqrt[a - a*Sin[e + f*x]]*(a + a*Sin[e + f*x])^(5/6)) + (4*((-3*Sqrt[a - a*Sin[e + f*x]])/(5*a*(a + a*Sin [e + f*x])^(5/6)) + (3^(3/4)*EllipticF[ArcCos[(2^(1/3)*a^(1/3) - (1 - Sqrt [3])*a^2*Sin[e + f*x]^2)/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*a^2*Sin[e + f*x] ^2)], (2 + Sqrt[3])/4]*Sin[e + f*x]*(2^(1/3)*a^(1/3) - a^2*Sin[e + f*x]^2) *Sqrt[(2^(2/3)*a^(2/3) + 2^(1/3)*a^(7/3)*Sin[e + f*x]^2 + a^4*Sin[e + f*x] ^4)/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*a^2*Sin[e + f*x]^2)^2])/(5*2^(1/3)*a^ (1/3)*Sqrt[-((a^2*Sin[e + f*x]^2*(2^(1/3)*a^(1/3) - a^2*Sin[e + f*x]^2))/( 2^(1/3)*a^(1/3) - (1 + Sqrt[3])*a^2*Sin[e + f*x]^2)^2)]*Sqrt[2*a - a^6*Sin [e + f*x]^6])))/(3*a)))/99))/4)))/(a*f)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] - Simp[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] && 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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_)) *((g_.) + (h_.)*(x_)), x_] :> 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] - Simp[(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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[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] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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 ]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Simp[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] && IntegerQ[p/2]
\[\int \frac {\tan \left (f x +e \right )^{4}}{\left (a +\sin \left (f x +e \right ) a \right )^{\frac {1}{3}}}d x\]
Input:
int(tan(f*x+e)^4/(a+sin(f*x+e)*a)^(1/3),x)
Output:
int(tan(f*x+e)^4/(a+sin(f*x+e)*a)^(1/3),x)
\[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{4}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(tan(f*x+e)^4/(a+a*sin(f*x+e))^(1/3),x, algorithm="fricas")
Output:
integral(tan(f*x + e)^4/(a*sin(f*x + e) + a)^(1/3), x)
\[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\int \frac {\tan ^{4}{\left (e + f x \right )}}{\sqrt [3]{a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \] Input:
integrate(tan(f*x+e)**4/(a+a*sin(f*x+e))**(1/3),x)
Output:
Integral(tan(e + f*x)**4/(a*(sin(e + f*x) + 1))**(1/3), x)
\[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{4}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(tan(f*x+e)^4/(a+a*sin(f*x+e))^(1/3),x, algorithm="maxima")
Output:
integrate(tan(f*x + e)^4/(a*sin(f*x + e) + a)^(1/3), x)
\[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{4}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(tan(f*x+e)^4/(a+a*sin(f*x+e))^(1/3),x, algorithm="giac")
Output:
integrate(tan(f*x + e)^4/(a*sin(f*x + e) + a)^(1/3), x)
Timed out. \[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\int \frac {{\mathrm {tan}\left (e+f\,x\right )}^4}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{1/3}} \,d x \] Input:
int(tan(e + f*x)^4/(a + a*sin(e + f*x))^(1/3),x)
Output:
int(tan(e + f*x)^4/(a + a*sin(e + f*x))^(1/3), x)
\[ \int \frac {\tan ^4(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx=\frac {\int \frac {\tan \left (f x +e \right )^{4}}{\left (\sin \left (f x +e \right )+1\right )^{\frac {1}{3}}}d x}{a^{\frac {1}{3}}} \] Input:
int(tan(f*x+e)^4/(a+a*sin(f*x+e))^(1/3),x)
Output:
int(tan(e + f*x)**4/(sin(e + f*x) + 1)**(1/3),x)/a**(1/3)