Integrand size = 25, antiderivative size = 269 \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {a \cos (e+f x) \sin (e+f x)}{3 b (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (a+2 b) \cos (e+f x) \sin (e+f x)}{3 b (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a+2 b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 b^2 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {(2 a+3 b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 b^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}} \]
1/3*a*cos(f*x+e)*sin(f*x+e)/b/(a+b)/f/(a+b*sin(f*x+e)^2)^(3/2)-2/3*(a+2*b) *cos(f*x+e)*sin(f*x+e)/b/(a+b)^2/f/(a+b*sin(f*x+e)^2)^(1/2)-2/3*(a+2*b)*El lipticE(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)/b^2/(a+b)^2/f/(1+b*sin(f*x+e)^2/a)^(1/2)+1/3*(2*a+3*b)*Ell ipticF(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)/b^2/(a+b)/f/(a+b*sin(f*x+e)^2)^(1/2)
Time = 1.29 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.68 \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {2 a^2 (a+2 b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )-a \left (2 a^2+5 a b+3 b^2\right ) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )-\sqrt {2} b \left (-a^2-4 a b-2 b^2+b (a+2 b) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{3 b^2 (a+b)^2 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \]
-1/3*(2*a^2*(a + 2*b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] - a*(2*a^2 + 5*a*b + 3*b^2)*((2*a + b - b*Cos[2*(e + f*x)] )/a)^(3/2)*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*(-a^2 - 4*a*b - 2*b^2 + b*(a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(b^2*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))
Time = 0.47 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3667, 372, 402, 27, 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.
| \(\displaystyle \int \frac {\sin ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (e+f x)^4}{\left (a+b \sin (e+f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3667 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\sin ^4(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {a \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 b (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\int \frac {a-(2 a+3 b) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 b (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {a \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 b (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (a+2 b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\int \frac {a \left (-2 (a+2 b) \sin ^2(e+f x)+a+3 b\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}}{3 b (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {a \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 b (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (a+2 b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\int \frac {-2 (a+2 b) \sin ^2(e+f x)+a+3 b}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a+b}}{3 b (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {a \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 b (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (a+2 b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\frac {(a+b) (2 a+3 b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{b}-\frac {2 (a+2 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}}{a+b}}{3 b (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {a \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 b (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (a+2 b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\frac {(a+b) (2 a+3 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a+2 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}}{a+b}}{3 b (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {a \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 b (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (a+2 b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\frac {(a+b) (2 a+3 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a+2 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}}{a+b}}{3 b (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {a \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 b (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (a+2 b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\frac {(a+b) (2 a+3 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a+2 b) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}}{a+b}}{3 b (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {a \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{3 b (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\frac {2 (a+2 b) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{(a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\frac {(a+b) (2 a+3 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a+2 b) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}}{a+b}}{3 b (a+b)}\right )}{f}\) |
(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*((a*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^ 2])/(3*b*(a + b)*(a + b*Sin[e + f*x]^2)^(3/2)) - ((2*(a + 2*b)*Sin[e + f*x ]*Sqrt[1 - Sin[e + f*x]^2])/((a + b)*Sqrt[a + b*Sin[e + f*x]^2]) - ((-2*(a + 2*b)*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 + b)*(2*a + 3*b)*EllipticF[ArcS in[Sin[e + f*x]], -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*Sqrt[a + b*Si n[e + f*x]^2]))/(a + b))/(3*b*(a + b))))/f
3.2.67.3.1 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[(-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[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), 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/Sqrt[1 - ff^2*x^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. \(622\) vs. \(2(247)=494\).
Time = 2.80 (sec) , antiderivative size = 623, normalized size of antiderivative = 2.32
| method | result | size |
| default | \(\frac {\left (2 a \,b^{2}+4 b^{3}\right ) \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-a^{2} b -5 a \,b^{2}-4 b^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \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}}\, b \left (2 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+5 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b +3 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}-2 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}-4 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+7 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b +8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}+3 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{3}-2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}-6 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b -4 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}}{3 {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} \left (a +b \right )^{2} b^{2} \cos \left (f x +e \right ) f}\) | \(623\) |
1/3*((2*a*b^2+4*b^3)*cos(f*x+e)^4*sin(f*x+e)+(-a^2*b-5*a*b^2-4*b^3)*cos(f* x+e)^2*sin(f*x+e)-(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*(cos(f*x+e)^2)^(1/2)*b *(2*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a^2+5*EllipticF(sin(f*x+e),(-1/a* b)^(1/2))*a*b+3*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b^2-2*EllipticE(sin(f *x+e),(-1/a*b)^(1/2))*a^2-4*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a*b)*cos( f*x+e)^2+2*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*Elliptic F(sin(f*x+e),(-1/a*b)^(1/2))*a^3+7*(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^2*b+8*(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+3*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*Ellipt icF(sin(f*x+e),(-1/a*b)^(1/2))*b^3-2*(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*(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-4*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*Ellipt icE(sin(f*x+e),(-1/a*b)^(1/2))*a*b^2)/(a+b*sin(f*x+e)^2)^(3/2)/(a+b)^2/b^2 /cos(f*x+e)/f
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 1432, normalized size of antiderivative = 5.32 \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
1/3*((2*((-I*a*b^3 - 2*I*b^4)*cos(f*x + e)^4 - I*a^3*b - 4*I*a^2*b^2 - 5*I *a*b^3 - 2*I*b^4 - 2*(-I*a^2*b^2 - 3*I*a*b^3 - 2*I*b^4)*cos(f*x + e)^2)*sq rt(-b)*sqrt((a^2 + a*b)/b^2) - ((2*I*a^2*b^2 + 5*I*a*b^3 + 2*I*b^4)*cos(f* x + e)^4 + 2*I*a^4 + 9*I*a^3*b + 14*I*a^2*b^2 + 9*I*a*b^3 + 2*I*b^4 + 2*(- 2*I*a^3*b - 7*I*a^2*b^2 - 7*I*a*b^3 - 2*I*b^4)*cos(f*x + e)^2)*sqrt(-b))*s qrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*s qrt((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*((I*a*b ^3 + 2*I*b^4)*cos(f*x + e)^4 + I*a^3*b + 4*I*a^2*b^2 + 5*I*a*b^3 + 2*I*b^4 - 2*(I*a^2*b^2 + 3*I*a*b^3 + 2*I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - ((-2*I*a^2*b^2 - 5*I*a*b^3 - 2*I*b^4)*cos(f*x + e)^4 - 2*I*a ^4 - 9*I*a^3*b - 14*I*a^2*b^2 - 9*I*a*b^3 - 2*I*b^4 + 2*(2*I*a^3*b + 7*I*a ^2*b^2 + 7*I*a*b^3 + 2*I*b^4)*cos(f*x + e)^2)*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*((I*a*b^3 + I*b^4)*cos(f *x + e)^4 + I*a^3*b + 3*I*a^2*b^2 + 3*I*a*b^3 + I*b^4 - 2*(I*a^2*b^2 + 2*I *a*b^3 + I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - ((-2*I*a^ 2*b^2 - 7*I*a*b^3 - 3*I*b^4)*cos(f*x + e)^4 - 2*I*a^4 - 11*I*a^3*b - 19*I* a^2*b^2 - 13*I*a*b^3 - 3*I*b^4 + 2*(2*I*a^3*b + 9*I*a^2*b^2 + 10*I*a*b^...
Timed out. \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sin ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^4}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]