Integrand size = 25, antiderivative size = 288 \[ \int \frac {\sec ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {(3 a-b) b \cos (e+f x) \sin (e+f x)}{3 a (a+b)^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {b \left (3 a^2-7 a b-2 b^2\right ) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b)^3 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2-7 a b-2 b^2\right ) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a (a+b)^3 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(3 a-b) \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\tan (e+f x)}{(a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
-1/3*(3*a-b)*b*cos(f*x+e)*sin(f*x+e)/a/(a+b)^2/f/(a+b*sin(f*x+e)^2)^(3/2)- 1/3*b*(3*a^2-7*a*b-2*b^2)*cos(f*x+e)*sin(f*x+e)/a^2/(a+b)^3/f/(a+b*sin(f*x +e)^2)^(1/2)-1/3*(3*a^2-7*a*b-2*b^2)*(cos(f*x+e)^2)^(1/2)/cos(f*x+e)*Ellip ticE(sin(f*x+e),(-b/a)^(1/2))*(1+b*sin(f*x+e)^2/a)^(1/2)/a/(a+b)^3/f/(a+b* sin(f*x+e)^2)^(1/2)+1/3*(3*a-b)*(cos(f*x+e)^2)^(1/2)/cos(f*x+e)*EllipticF( sin(f*x+e),(-b/a)^(1/2))*(1+b*sin(f*x+e)^2/a)^(1/2)/a/(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 = 4.01 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.85 \[ \int \frac {\sec ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {-2 a^2 \left (3 a^2-7 a b-2 b^2\right ) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )+2 a^2 \left (3 a^2+2 a b-b^2\right ) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )+\frac {\left (24 a^4+24 a^3 b+41 a^2 b^2+19 a b^3+2 b^4-4 a b \left (6 a^2-5 a b-3 b^2\right ) \cos (2 (e+f x))+b^2 \left (3 a^2-7 a b-2 b^2\right ) \cos (4 (e+f x))\right ) \tan (e+f x)}{\sqrt {2}}}{6 a^2 (a+b)^3 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \]
(-2*a^2*(3*a^2 - 7*a*b - 2*b^2)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*E llipticE[e + f*x, -(b/a)] + 2*a^2*(3*a^2 + 2*a*b - b^2)*((2*a + b - b*Cos[ 2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] + ((24*a^4 + 24*a^3*b + 41*a^2*b^2 + 19*a*b^3 + 2*b^4 - 4*a*b*(6*a^2 - 5*a*b - 3*b^2)*Cos[2*(e + f *x)] + b^2*(3*a^2 - 7*a*b - 2*b^2)*Cos[4*(e + f*x)])*Tan[e + f*x])/Sqrt[2] )/(6*a^2*(a + b)^3*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))
Time = 0.55 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.22, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 3671, 316, 27, 402, 25, 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.
| \(\displaystyle \int \frac {\sec ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (e+f x)^2 \left (a+b \sin (e+f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3671 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {1}{\left (1-\sin ^2(e+f x)\right )^{3/2} \left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\int \frac {b \left (3 \sin ^2(e+f x)+1\right )}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin (e+f x)}{a+b}+\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {b \int \frac {3 \sin ^2(e+f x)+1}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin (e+f x)}{a+b}+\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {b \left (-\frac {\int -\frac {(3 a-b) \sin ^2(e+f x)+2 (3 a+b)}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a (a+b)}-\frac {(3 a-b) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}+\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {b \left (\frac {\int \frac {(3 a-b) \sin ^2(e+f x)+2 (3 a+b)}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a (a+b)}-\frac {(3 a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}+\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {b \left (\frac {-\frac {\int -\frac {a (9 a+b)-\left (3 a^2-7 b a-2 b^2\right ) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}-\frac {\left (3 a^2-7 a b-2 b^2\right ) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {(3 a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}+\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {b \left (\frac {\frac {\int \frac {a (9 a+b)-\left (3 a^2-7 b a-2 b^2\right ) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}-\frac {\left (3 a^2-7 a b-2 b^2\right ) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {(3 a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}+\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {b \left (\frac {\frac {\frac {a (3 a-b) (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{b}-\frac {\left (3 a^2-7 a b-2 b^2\right ) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}}{a (a+b)}-\frac {\left (3 a^2-7 a b-2 b^2\right ) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {(3 a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}+\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {b \left (\frac {\frac {\frac {a (3 a-b) (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2-7 a b-2 b^2\right ) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}}{a (a+b)}-\frac {\left (3 a^2-7 a b-2 b^2\right ) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {(3 a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}+\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {b \left (\frac {\frac {\frac {a (3 a-b) (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2-7 a b-2 b^2\right ) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}}{a (a+b)}-\frac {\left (3 a^2-7 a b-2 b^2\right ) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {(3 a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}+\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {b \left (\frac {\frac {\frac {a (3 a-b) (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2-7 a b-2 b^2\right ) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}}{a (a+b)}-\frac {\left (3 a^2-7 a b-2 b^2\right ) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {(3 a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}+\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {b \left (\frac {\frac {\frac {a (3 a-b) (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2-7 a b-2 b^2\right ) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}}{a (a+b)}-\frac {\left (3 a^2-7 a b-2 b^2\right ) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}-\frac {(3 a-b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{a+b}+\frac {\sin (e+f x)}{(a+b) \sqrt {1-\sin ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
(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)) + (b*(-1/3*((3*a - b)*Sin[e + f*x]* Sqrt[1 - Sin[e + f*x]^2])/(a*(a + b)*(a + b*Sin[e + f*x]^2)^(3/2)) + (-((( 3*a^2 - 7*a*b - 2*b^2)*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2])/(a*(a + b)*S qrt[a + b*Sin[e + f*x]^2])) + (-(((3*a^2 - 7*a*b - 2*b^2)*EllipticE[ArcSin [Sin[e + f*x]], -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(b*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (a*(3*a - b)*(a + b)*EllipticF[ArcSin[Sin[e + f*x]], -(b/a )]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*Sqrt[a + b*Sin[e + f*x]^2]))/(a*(a + b)))/(3*a*(a + b))))/(a + b)))/f
3.4.71.3.1 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, 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_) + (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[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff*(Sqrt[ Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, 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. \(1081\) vs. \(2(308)=616\).
Time = 4.79 (sec) , antiderivative size = 1082, normalized size of antiderivative = 3.76
1/3*((-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*b^2*(3*a^2-7*a*b-2*b^2)*co s(f*x+e)^4*sin(f*x+e)-2*(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*b*(3*a^ 3-a^2*b-5*a*b^2-b^3)*cos(f*x+e)^2*sin(f*x+e)-(cos(f*x+e)^2)^(1/2)*(-b*cos( f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*a*b*( 3*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a^2+2*EllipticF(sin(f*x+e),(-1/a*b) ^(1/2))*a*b-EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b^2-3*EllipticE(sin(f*x+e ),(-1/a*b)^(1/2))*a^2+7*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a*b+2*Ellipti cE(sin(f*x+e),(-1/a*b)^(1/2))*b^2)*cos(f*x+e)^2+3*(-b*cos(f*x+e)^4+(a+b)*c os(f*x+e)^2)^(1/2)*a^2*(a^2+2*a*b+b^2)*sin(f*x+e)+3*(-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^4+5*(-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)*Ellip ticF(sin(f*x+e),(-1/a*b)^(1/2))*a^3*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)*EllipticF(si n(f*x+e),(-1/a*b)^(1/2))*a^2*b^2-(-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^3-3*(-b*cos(f*x+e)^4+(a+b)*cos(f*x+e)^2)^(1/2)*(co s(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^4+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)*EllipticE(sin(f*x+e),(-1/a...
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 1746, normalized size of antiderivative = 6.06 \[ \int \frac {\sec ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
1/6*((2*((-3*I*a^2*b^3 + 7*I*a*b^4 + 2*I*b^5)*cos(f*x + e)^5 - 2*(-3*I*a^3 *b^2 + 4*I*a^2*b^3 + 9*I*a*b^4 + 2*I*b^5)*cos(f*x + e)^3 + (-3*I*a^4*b + I *a^3*b^2 + 13*I*a^2*b^3 + 11*I*a*b^4 + 2*I*b^5)*cos(f*x + e))*sqrt(-b)*sqr t((a^2 + a*b)/b^2) - ((6*I*a^3*b^2 - 11*I*a^2*b^3 - 11*I*a*b^4 - 2*I*b^5)* cos(f*x + e)^5 + 2*(-6*I*a^4*b + 5*I*a^3*b^2 + 22*I*a^2*b^3 + 13*I*a*b^4 + 2*I*b^5)*cos(f*x + e)^3 + (6*I*a^5 + I*a^4*b - 27*I*a^3*b^2 - 35*I*a^2*b^ 3 - 15*I*a*b^4 - 2*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*((3*I*a^2*b^3 - 7*I*a*b^4 - 2* I*b^5)*cos(f*x + e)^5 - 2*(3*I*a^3*b^2 - 4*I*a^2*b^3 - 9*I*a*b^4 - 2*I*b^5 )*cos(f*x + e)^3 + (3*I*a^4*b - I*a^3*b^2 - 13*I*a^2*b^3 - 11*I*a*b^4 - 2* I*b^5)*cos(f*x + e))*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - ((-6*I*a^3*b^2 + 11* I*a^2*b^3 + 11*I*a*b^4 + 2*I*b^5)*cos(f*x + e)^5 + 2*(6*I*a^4*b - 5*I*a^3* b^2 - 22*I*a^2*b^3 - 13*I*a*b^4 - 2*I*b^5)*cos(f*x + e)^3 + (-6*I*a^5 - I* a^4*b + 27*I*a^3*b^2 + 35*I*a^2*b^3 + 15*I*a*b^4 + 2*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*(4*((3*I*a^2*b^3 + 4*I*a*b^4 + I*b^5)*cos(f*x + e)^5 + 2*(-3*I*a^3*b...
\[ \int \frac {\sec ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\sec ^{2}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\sec ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sec \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sec ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sec \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{{\cos \left (e+f\,x\right )}^2\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]