Integrand size = 25, antiderivative size = 246 \[ \int \frac {\tan ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\frac {2 (2 a+b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {a \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (2 a+b) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{3 (a+b)^2 f}+\frac {\sec ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{3 (a+b) f} \]
2/3*(2*a+b)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(cos(f*x+e)^2)^( 1/2)*(a+b*sin(f*x+e)^2)^(1/2)/(a+b)^2/f/(1+b*sin(f*x+e)^2/a)^(1/2)-1/3*a*E llipticF(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(cos(f*x+e)^2)^(1/2)*(1+b*sin (f*x+e)^2/a)^(1/2)/(a+b)/f/(a+b*sin(f*x+e)^2)^(1/2)-2/3*(2*a+b)*(a+b*sin(f *x+e)^2)^(1/2)*tan(f*x+e)/(a+b)^2/f+1/3*sec(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1 /2)*tan(f*x+e)/(a+b)/f
Time = 2.56 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.76 \[ \int \frac {\tan ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\frac {4 a (2 a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )-2 a (a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )-\frac {\left (2 \left (4 a^2+3 a b+b^2\right ) \cos (2 (e+f x))+(2 a+b) (2 a-b-b \cos (4 (e+f x)))\right ) \sec ^2(e+f x) \tan (e+f x)}{\sqrt {2}}}{6 (a+b)^2 f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
(4*a*(2*a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, - (b/a)] - 2*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] - ((2*(4*a^2 + 3*a*b + b^2)*Cos[2*(e + f*x)] + (2*a + b)*(2*a - b - b*Cos[4*(e + f*x)]))*Sec[e + f*x]^2*Tan[e + f*x])/Sqrt[2])/(6*(a + b)^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])
Time = 0.46 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3675, 372, 402, 25, 399, 323, 321, 330, 327}
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.
| \(\displaystyle \int \frac {\tan ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (e+f x)^4}{\sqrt {a+b \sin (e+f x)^2}}dx\) |
| \(\Big \downarrow \) 3675 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\sin ^4(e+f x)}{\left (1-\sin ^2(e+f x)\right )^{5/2} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\int \frac {(3 a+2 b) \sin ^2(e+f x)+a}{\left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{3 (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\int -\frac {2 b (2 a+b) \sin ^2(e+f x)+a (3 a+b)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a+b}+\frac {2 (2 a+b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{(a+b) \sqrt {1-\sin ^2(e+f x)}}}{3 (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a+b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{(a+b) \sqrt {1-\sin ^2(e+f x)}}-\frac {\int \frac {2 b (2 a+b) \sin ^2(e+f x)+a (3 a+b)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a+b}}{3 (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a+b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{(a+b) \sqrt {1-\sin ^2(e+f x)}}-\frac {2 (2 a+b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-a (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a+b}}{3 (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a+b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{(a+b) \sqrt {1-\sin ^2(e+f x)}}-\frac {2 (2 a+b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}}{a+b}}{3 (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a+b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{(a+b) \sqrt {1-\sin ^2(e+f x)}}-\frac {2 (2 a+b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a+b}}{3 (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a+b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{(a+b) \sqrt {1-\sin ^2(e+f x)}}-\frac {\frac {2 (2 a+b) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a+b}}{3 (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 (a+b) \left (1-\sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (2 a+b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{(a+b) \sqrt {1-\sin ^2(e+f x)}}-\frac {\frac {2 (2 a+b) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a+b}}{3 (a+b)}\right )}{f}\) |
(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*((Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^ 2])/(3*(a + b)*(1 - Sin[e + f*x]^2)^(3/2)) - ((2*(2*a + b)*Sin[e + f*x]*Sq rt[a + b*Sin[e + f*x]^2])/((a + b)*Sqrt[1 - Sin[e + f*x]^2]) - ((2*(2*a + b)*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/Sqr t[1 + (b*Sin[e + f*x]^2)/a] - (a*(a + b)*EllipticF[ArcSin[Sin[e + f*x]], - (b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/Sqrt[a + b*Sin[e + f*x]^2])/(a + b) )/(3*(a + b))))/f
3.6.16.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff^(m + 1 )*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[x^m*((a + b*ff^2*x^2) ^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b , e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Time = 3.80 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.53
| method | result | size |
| default | \(-\frac {2 \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, b \left (2 a +b \right ) \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (4 a^{2}+7 a b +3 b^{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, a \left (F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -4 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a -2 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (a^{2}+2 a b +b^{2}\right ) \sin \left (f x +e \right )}{3 \left (1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-1\right ) \left (a +b \right )^{2} \sqrt {-\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right ) \left (\sin \left (f x +e \right )-1\right ) \left (1+\sin \left (f x +e \right )\right )}\, \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(377\) |
-1/3*(2*(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*b*(2*a+b)*cos(f*x+e)^4* sin(f*x+e)-(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(4*a^2+7*a*b+3*b^2)* cos(f*x+e)^2*sin(f*x+e)-(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(-b/a*c os(f*x+e)^2+(a+b)/a)^(1/2)*(cos(f*x+e)^2)^(1/2)*a*(EllipticF(sin(f*x+e),(- 1/a*b)^(1/2))*a+EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b-4*EllipticE(sin(f*x +e),(-1/a*b)^(1/2))*a-2*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*b)*cos(f*x+e) ^2+(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(a^2+2*a*b+b^2)*sin(f*x+e))/ (1+sin(f*x+e))/(sin(f*x+e)-1)/(a+b)^2/(-(a+b*sin(f*x+e)^2)*(sin(f*x+e)-1)* (1+sin(f*x+e)))^(1/2)/cos(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2)/f
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 845, normalized size of antiderivative = 3.43 \[ \int \frac {\tan ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\frac {{\left (2 \, {\left (2 i \, a b^{2} + i \, b^{3}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \cos \left (f x + e\right )^{3} - {\left (-4 i \, a^{2} b - 4 i \, a b^{2} - i \, b^{3}\right )} \sqrt {-b} \cos \left (f x + e\right )^{3}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left (-2 i \, a b^{2} - i \, b^{3}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \cos \left (f x + e\right )^{3} - {\left (4 i \, a^{2} b + 4 i \, a b^{2} + i \, b^{3}\right )} \sqrt {-b} \cos \left (f x + e\right )^{3}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left (-3 i \, a^{2} b - 5 i \, a b^{2} - 2 i \, b^{3}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \cos \left (f x + e\right )^{3} - {\left (-6 i \, a^{3} - 5 i \, a^{2} b - i \, a b^{2}\right )} \sqrt {-b} \cos \left (f x + e\right )^{3}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left (3 i \, a^{2} b + 5 i \, a b^{2} + 2 i \, b^{3}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \cos \left (f x + e\right )^{3} - {\left (6 i \, a^{3} + 5 i \, a^{2} b + i \, a b^{2}\right )} \sqrt {-b} \cos \left (f x + e\right )^{3}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (a b^{2} + b^{3} - 2 \, {\left (2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{3 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} f \cos \left (f x + e\right )^{3}} \]
1/3*((2*(2*I*a*b^2 + I*b^3)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*cos(f*x + e)^3 - (-4*I*a^2*b - 4*I*a*b^2 - I*b^3)*sqrt(-b)*cos(f*x + e)^3)*sqrt((2*b*sqrt ((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a* b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + e))), (8*a^2 + 8*a*b + b ^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (2*(-2*I*a*b^2 - I*b^3) *sqrt(-b)*sqrt((a^2 + a*b)/b^2)*cos(f*x + e)^3 - (4*I*a^2*b + 4*I*a*b^2 + I*b^3)*sqrt(-b)*cos(f*x + e)^3)*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b) /b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f *x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a ^2 + a*b)/b^2))/b^2) + (2*(-3*I*a^2*b - 5*I*a*b^2 - 2*I*b^3)*sqrt(-b)*sqrt ((a^2 + a*b)/b^2)*cos(f*x + e)^3 - (-6*I*a^3 - 5*I*a^2*b - I*a*b^2)*sqrt(- b)*cos(f*x + e)^3)*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_ f(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*s in(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2 ))/b^2) + (2*(3*I*a^2*b + 5*I*a*b^2 + 2*I*b^3)*sqrt(-b)*sqrt((a^2 + a*b)/b ^2)*cos(f*x + e)^3 - (6*I*a^3 + 5*I*a^2*b + I*a*b^2)*sqrt(-b)*cos(f*x + e) ^3)*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_f(arcsin(sqrt(( 2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (a*b^ 2 + b^3 - 2*(2*a*b^2 + b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a ...
\[ \int \frac {\tan ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {\tan ^{4}{\left (e + f x \right )}}{\sqrt {a + b \sin ^{2}{\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\tan ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{4}}{\sqrt {b \sin \left (f x + e\right )^{2} + a}} \,d x } \]
\[ \int \frac {\tan ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{4}}{\sqrt {b \sin \left (f x + e\right )^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {\tan ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {{\mathrm {tan}\left (e+f\,x\right )}^4}{\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}} \,d x \]