Integrand size = 25, antiderivative size = 292 \[ \int \frac {\tan ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {4 b \cos (e+f x) \sin (e+f x)}{3 (a+b)^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {(7 a-b) b \cos (e+f x) \sin (e+f x)}{3 a (a+b)^3 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(7 a-b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a (a+b)^3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {4 \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\tan (e+f x)}{(a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
-4/3*b*cos(f*x+e)*sin(f*x+e)/(a+b)^2/f/(a+b*sin(f*x+e)^2)^(3/2)-1/3*(7*a-b )*b*cos(f*x+e)*sin(f*x+e)/a/(a+b)^3/f/(a+b*sin(f*x+e)^2)^(1/2)-1/3*(7*a-b) *EllipticE(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(cos(f*x+e)^2)^(1/2)*(a+b*s in(f*x+e)^2)^(1/2)/a/(a+b)^3/f/(1+b*sin(f*x+e)^2/a)^(1/2)+4/3*EllipticF(si n(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)^2/f/(a+b*sin(f*x+e)^2)^(1/2)+tan(f*x+e)/(a+b)/f/(a+b*sin(f*x +e)^2)^(3/2)
Time = 3.22 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.68 \[ \int \frac {\tan ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {-2 a^2 (7 a-b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )+8 a^2 (a+b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )+\frac {\left (24 a^3+4 a^2 b+5 a b^2+b^3-4 a b (11 a+3 b) \cos (2 (e+f x))+(7 a-b) b^2 \cos (4 (e+f x))\right ) \tan (e+f x)}{\sqrt {2}}}{6 a (a+b)^3 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \]
(-2*a^2*(7*a - b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f *x, -(b/a)] + 8*a^2*(a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*Ellip ticF[e + f*x, -(b/a)] + ((24*a^3 + 4*a^2*b + 5*a*b^2 + b^3 - 4*a*b*(11*a + 3*b)*Cos[2*(e + f*x)] + (7*a - b)*b^2*Cos[4*(e + f*x)])*Tan[e + f*x])/Sqr t[2])/(6*a*(a + b)^3*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))
Time = 0.52 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 3675, 373, 402, 25, 27, 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.
| \(\displaystyle \int \frac {\tan ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (e+f x)^2}{\left (a+b \sin (e+f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3675 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\sin ^2(e+f x)}{\left (1-\sin ^2(e+f x)\right )^{3/2} \left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 373 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\int \frac {a-3 b \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin (e+f x)}{a+b}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {4 b \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\int -\frac {a \left (-4 b \sin ^2(e+f x)+3 a-b\right )}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a (a+b)}}{a+b}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\int \frac {a \left (-4 b \sin ^2(e+f x)+3 a-b\right )}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a (a+b)}+\frac {4 b \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{a+b}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\int \frac {-4 b \sin ^2(e+f x)+3 a-b}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 (a+b)}+\frac {4 b \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{a+b}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {b (7 a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\int -\frac {(7 a-b) b \sin ^2(e+f x)+a (3 a-5 b)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}}{3 (a+b)}+\frac {4 b \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{a+b}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {\int \frac {(7 a-b) b \sin ^2(e+f x)+a (3 a-5 b)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}+\frac {b (7 a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 (a+b)}+\frac {4 b \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{a+b}\right )}{f}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {(7 a-b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-4 a (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}+\frac {b (7 a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 (a+b)}+\frac {4 b \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{a+b}\right )}{f}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {(7 a-b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}}{a (a+b)}+\frac {b (7 a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 (a+b)}+\frac {4 b \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{a+b}\right )}{f}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {(7 a-b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a (a+b)}+\frac {b (7 a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 (a+b)}+\frac {4 b \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{a+b}\right )}{f}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {\frac {(7 a-b) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a (a+b)}+\frac {b (7 a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 (a+b)}+\frac {4 b \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{a+b}\right )}{f}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {\frac {\frac {(7 a-b) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a (a+b)}+\frac {b (7 a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 (a+b)}+\frac {4 b \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}}{a+b}\right )}{f}\) |
(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*(Sin[e + f*x]/((a + b)*Sqrt[1 - Sin[e + f*x]^2]*(a + b*Sin[e + f*x]^2)^(3/2)) - ((4*b*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2])/(3*(a + b)*(a + b*Sin[e + f*x]^2)^(3/2)) + (((7*a - b)*b*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2])/(a*(a + b)*Sqrt[a + b*Sin[e + f*x]^2]) + (((7*a - b)*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sqrt[a + b*Sin[e + f* x]^2])/Sqrt[1 + (b*Sin[e + f*x]^2)/a] - (4*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*(a + b)))/(3*(a + b)))/(a + b)))/f
3.6.39.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(850\) vs. \(2(268)=536\).
Time = 5.92 (sec) , antiderivative size = 851, normalized size of antiderivative = 2.91
1/3*((-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*b^2*(7*a-b)*cos(f*x+e)^4*s in(f*x+e)-(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*b*(11*a^2+10*a*b-b^2) *cos(f*x+e)^2*sin(f*x+e)-(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*(-b*cos(f*x+e)^ 4+(a+b)*cos(f*x+e)^2)^(1/2)*(cos(f*x+e)^2)^(1/2)*a*b*(4*EllipticF(sin(f*x+ e),(-1/a*b)^(1/2))*a+4*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b-7*EllipticE( sin(f*x+e),(-1/a*b)^(1/2))*a+EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*b)*cos(f *x+e)^2+3*(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*a*(a^2+2*a*b+b^2)*sin (f*x+e)+4*(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(cos(f*x+e)^2)^(1/2)* (-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a^3 +8*(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(cos(f*x+e)^2)^(1/2)*(-b/a*c os(f*x+e)^2+(a+b)/a)^(1/2)*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a^2*b+4*(- b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f* x+e)^2+(a+b)/a)^(1/2)*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a*b^2-7*(-b*cos (f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^ 2+(a+b)/a)^(1/2)*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a^3-6*(-b*cos(f*x+e) ^4+(a+b)*cos(f*x+e)^2)^(1/2)*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b) /a)^(1/2)*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a^2*b+(-b*cos(f*x+e)^4+(a+b )*cos(f*x+e)^2)^(1/2)*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/ 2)*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a*b^2)/(a+b*sin(f*x+e)^2)^(3/2)/(- (a+b*sin(f*x+e)^2)*(sin(f*x+e)-1)*(1+sin(f*x+e)))^(1/2)/(a+b)^3/a/cos(f...
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 1632, normalized size of antiderivative = 5.59 \[ \int \frac {\tan ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
1/6*((2*((-7*I*a*b^4 + I*b^5)*cos(f*x + e)^5 - 2*(-7*I*a^2*b^3 - 6*I*a*b^4 + I*b^5)*cos(f*x + e)^3 + (-7*I*a^3*b^2 - 13*I*a^2*b^3 - 5*I*a*b^4 + I*b^ 5)*cos(f*x + e))*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - ((14*I*a^2*b^3 + 5*I*a*b ^4 - I*b^5)*cos(f*x + e)^5 + 2*(-14*I*a^3*b^2 - 19*I*a^2*b^3 - 4*I*a*b^4 + I*b^5)*cos(f*x + e)^3 + (14*I*a^4*b + 33*I*a^3*b^2 + 23*I*a^2*b^3 + 3*I*a *b^4 - I*b^5)*cos(f*x + e))*sqrt(-b))*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)*s qrt((a^2 + a*b)/b^2))/b^2) + (2*((7*I*a*b^4 - I*b^5)*cos(f*x + e)^5 - 2*(7 *I*a^2*b^3 + 6*I*a*b^4 - I*b^5)*cos(f*x + e)^3 + (7*I*a^3*b^2 + 13*I*a^2*b ^3 + 5*I*a*b^4 - I*b^5)*cos(f*x + e))*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - ((- 14*I*a^2*b^3 - 5*I*a*b^4 + I*b^5)*cos(f*x + e)^5 + 2*(14*I*a^3*b^2 + 19*I* a^2*b^3 + 4*I*a*b^4 - I*b^5)*cos(f*x + e)^3 + (-14*I*a^4*b - 33*I*a^3*b^2 - 23*I*a^2*b^3 - 3*I*a*b^4 + I*b^5)*cos(f*x + e))*sqrt(-b))*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*((-3*I*a^2*b^3 - 2 *I*a*b^4 + I*b^5)*cos(f*x + e)^5 + 2*(3*I*a^3*b^2 + 5*I*a^2*b^3 + I*a*b^4 - I*b^5)*cos(f*x + e)^3 + (-3*I*a^4*b - 8*I*a^3*b^2 - 6*I*a^2*b^3 + I*b^5) *cos(f*x + e))*sqrt(-b)*sqrt((a^2 + a*b)/b^2) + ((6*I*a^3*b^2 - 7*I*a^2...
\[ \int \frac {\tan ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\tan ^{2}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\tan ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\tan ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\tan ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {tan}\left (e+f\,x\right )}^2}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]