Integrand size = 12, antiderivative size = 79 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{7/2}} \, dx=-\frac {9 E\left (x\left |\frac {5}{4}\right .\right )}{40}-\frac {\operatorname {EllipticF}\left (x,\frac {5}{4}\right )}{40}+\frac {\cos (x) \sin (x)}{4 \left (4-5 \sin ^2(x)\right )^{5/2}}-\frac {\cos (x) \sin (x)}{4 \left (4-5 \sin ^2(x)\right )^{3/2}}+\frac {9 \cos (x) \sin (x)}{16 \sqrt {4-5 \sin ^2(x)}} \] Output:
-9/40*EllipticE(sin(x),1/2*5^(1/2))-1/40*InverseJacobiAM(x,1/2*5^(1/2))+1/ 4*cos(x)*sin(x)/(4-5*sin(x)^2)^(5/2)-1/4*cos(x)*sin(x)/(4-5*sin(x)^2)^(3/2 )+9/16*cos(x)*sin(x)/(4-5*sin(x)^2)^(1/2)
Time = 0.42 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{7/2}} \, dx=\frac {1}{640} \left (-144 E\left (x\left |\frac {5}{4}\right .\right )-16 \operatorname {EllipticF}\left (x,\frac {5}{4}\right )+\frac {5 \sqrt {2} (517 \sin (2 x)+460 \sin (4 x)+225 \sin (6 x))}{(3+5 \cos (2 x))^{5/2}}\right ) \] Input:
Integrate[(4 - 5*Sin[x]^2)^(-7/2),x]
Output:
(-144*EllipticE[x, 5/4] - 16*EllipticF[x, 5/4] + (5*Sqrt[2]*(517*Sin[2*x] + 460*Sin[4*x] + 225*Sin[6*x]))/(3 + 5*Cos[2*x])^(5/2))/640
Time = 0.62 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {3042, 3663, 27, 3042, 3652, 27, 3042, 3652, 3042, 3651, 3042, 3656, 3661}
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.
| \(\displaystyle \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (4-5 \sin (x)^2\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3663 |
| \(\displaystyle \frac {1}{20} \int -\frac {15 \sin ^2(x)}{\left (4-5 \sin ^2(x)\right )^{5/2}}dx+\frac {\sin (x) \cos (x)}{4 \left (4-5 \sin ^2(x)\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin (x) \cos (x)}{4 \left (4-5 \sin ^2(x)\right )^{5/2}}-\frac {3}{4} \int \frac {\sin ^2(x)}{\left (4-5 \sin ^2(x)\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (x) \cos (x)}{4 \left (4-5 \sin ^2(x)\right )^{5/2}}-\frac {3}{4} \int \frac {\sin (x)^2}{\left (4-5 \sin (x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {\sin (x) \cos (x)}{4 \left (4-5 \sin ^2(x)\right )^{5/2}}-\frac {3}{4} \left (\frac {\sin (x) \cos (x)}{3 \left (4-5 \sin ^2(x)\right )^{3/2}}-\frac {1}{12} \int \frac {4 \left (\sin ^2(x)+1\right )}{\left (4-5 \sin ^2(x)\right )^{3/2}}dx\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin (x) \cos (x)}{4 \left (4-5 \sin ^2(x)\right )^{5/2}}-\frac {3}{4} \left (\frac {\sin (x) \cos (x)}{3 \left (4-5 \sin ^2(x)\right )^{3/2}}-\frac {1}{3} \int \frac {\sin ^2(x)+1}{\left (4-5 \sin ^2(x)\right )^{3/2}}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (x) \cos (x)}{4 \left (4-5 \sin ^2(x)\right )^{5/2}}-\frac {3}{4} \left (\frac {\sin (x) \cos (x)}{3 \left (4-5 \sin ^2(x)\right )^{3/2}}-\frac {1}{3} \int \frac {\sin (x)^2+1}{\left (4-5 \sin (x)^2\right )^{3/2}}dx\right )\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {\sin (x) \cos (x)}{4 \left (4-5 \sin ^2(x)\right )^{5/2}}-\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{4} \int \frac {8-9 \sin ^2(x)}{\sqrt {4-5 \sin ^2(x)}}dx-\frac {9 \sin (x) \cos (x)}{4 \sqrt {4-5 \sin ^2(x)}}\right )+\frac {\sin (x) \cos (x)}{3 \left (4-5 \sin ^2(x)\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (x) \cos (x)}{4 \left (4-5 \sin ^2(x)\right )^{5/2}}-\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{4} \int \frac {8-9 \sin (x)^2}{\sqrt {4-5 \sin (x)^2}}dx-\frac {9 \sin (x) \cos (x)}{4 \sqrt {4-5 \sin ^2(x)}}\right )+\frac {\sin (x) \cos (x)}{3 \left (4-5 \sin ^2(x)\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 3651 |
| \(\displaystyle \frac {\sin (x) \cos (x)}{4 \left (4-5 \sin ^2(x)\right )^{5/2}}-\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {4}{5} \int \frac {1}{\sqrt {4-5 \sin ^2(x)}}dx+\frac {9}{5} \int \sqrt {4-5 \sin ^2(x)}dx\right )-\frac {9 \sin (x) \cos (x)}{4 \sqrt {4-5 \sin ^2(x)}}\right )+\frac {\sin (x) \cos (x)}{3 \left (4-5 \sin ^2(x)\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (x) \cos (x)}{4 \left (4-5 \sin ^2(x)\right )^{5/2}}-\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {4}{5} \int \frac {1}{\sqrt {4-5 \sin (x)^2}}dx+\frac {9}{5} \int \sqrt {4-5 \sin (x)^2}dx\right )-\frac {9 \sin (x) \cos (x)}{4 \sqrt {4-5 \sin ^2(x)}}\right )+\frac {\sin (x) \cos (x)}{3 \left (4-5 \sin ^2(x)\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 3656 |
| \(\displaystyle \frac {\sin (x) \cos (x)}{4 \left (4-5 \sin ^2(x)\right )^{5/2}}-\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {4}{5} \int \frac {1}{\sqrt {4-5 \sin (x)^2}}dx+\frac {18 E\left (x\left |\frac {5}{4}\right .\right )}{5}\right )-\frac {9 \sin (x) \cos (x)}{4 \sqrt {4-5 \sin ^2(x)}}\right )+\frac {\sin (x) \cos (x)}{3 \left (4-5 \sin ^2(x)\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 3661 |
| \(\displaystyle \frac {\sin (x) \cos (x)}{4 \left (4-5 \sin ^2(x)\right )^{5/2}}-\frac {3}{4} \left (\frac {\sin (x) \cos (x)}{3 \left (4-5 \sin ^2(x)\right )^{3/2}}+\frac {1}{3} \left (\frac {1}{4} \left (\frac {2 \operatorname {EllipticF}\left (x,\frac {5}{4}\right )}{5}+\frac {18 E\left (x\left |\frac {5}{4}\right .\right )}{5}\right )-\frac {9 \sin (x) \cos (x)}{4 \sqrt {4-5 \sin ^2(x)}}\right )\right )\) |
Input:
Int[(4 - 5*Sin[x]^2)^(-7/2),x]
Output:
(Cos[x]*Sin[x])/(4*(4 - 5*Sin[x]^2)^(5/2)) - (3*((Cos[x]*Sin[x])/(3*(4 - 5 *Sin[x]^2)^(3/2)) + (((18*EllipticE[x, 5/4])/5 + (2*EllipticF[x, 5/4])/5)/ 4 - (9*Cos[x]*Sin[x])/(4*Sqrt[4 - 5*Sin[x]^2]))/3))/4
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_.)*sin[(e_.) + (f_.)*(x_)]^2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[B/b Int[Sqrt[a + b*Sin[e + f*x]^2], x] , x] + Simp[(A*b - a*B)/b Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /; Fre eQ[{a, b, e, f, A, B}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b - a*B))*Cos[e + f*x]*Sin[e + f*x ]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(a + b)*(p + 1))), x] - Simp[1/(2* a*(a + b)*(p + 1)) Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[a*B - A*(2*a*( p + 1) + b*(2*p + 3)) + 2*(A*b - a*B)*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a ]/f)*EllipticE[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1/(S qrt[a]*f))*EllipticF[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(p + 1)*(a + b))), x] + Simp[1/(2*a*(p + 1)*(a + b)) Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && LtQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(203\) vs. \(2(68)=136\).
Time = 1.68 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.58
| method | result | size |
| default | \(\frac {\sqrt {-\left (-4+5 \sin \left (x \right )^{2}\right ) \cos \left (x \right )^{2}}\, \left (-\frac {\sin \left (x \right ) \sqrt {5 \cos \left (x \right )^{4}-\cos \left (x \right )^{2}}}{500 \left (\sin \left (x \right )^{2}-\frac {4}{5}\right )^{3}}-\frac {\sin \left (x \right ) \sqrt {5 \cos \left (x \right )^{4}-\cos \left (x \right )^{2}}}{100 \left (\sin \left (x \right )^{2}-\frac {4}{5}\right )^{2}}+\frac {9 \cos \left (x \right )^{2} \sin \left (x \right )}{16 \sqrt {-\left (-4+5 \sin \left (x \right )^{2}\right ) \cos \left (x \right )^{2}}}-\frac {\sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {4-5 \sin \left (x \right )^{2}}\, \operatorname {EllipticF}\left (\sin \left (x \right ), \frac {\sqrt {5}}{2}\right )}{4 \sqrt {5 \cos \left (x \right )^{4}-\cos \left (x \right )^{2}}}+\frac {9 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {4-5 \sin \left (x \right )^{2}}\, \left (\operatorname {EllipticF}\left (\sin \left (x \right ), \frac {\sqrt {5}}{2}\right )-\operatorname {EllipticE}\left (\sin \left (x \right ), \frac {\sqrt {5}}{2}\right )\right )}{40 \sqrt {5 \cos \left (x \right )^{4}-\cos \left (x \right )^{2}}}\right )}{\cos \left (x \right ) \sqrt {4-5 \sin \left (x \right )^{2}}}\) | \(204\) |
Input:
int(1/(4-5*sin(x)^2)^(7/2),x,method=_RETURNVERBOSE)
Output:
(-(-4+5*sin(x)^2)*cos(x)^2)^(1/2)*(-1/500*sin(x)*(5*cos(x)^4-cos(x)^2)^(1/ 2)/(sin(x)^2-4/5)^3-1/100*sin(x)*(5*cos(x)^4-cos(x)^2)^(1/2)/(sin(x)^2-4/5 )^2+9/16*cos(x)^2*sin(x)/(-(-4+5*sin(x)^2)*cos(x)^2)^(1/2)-1/4*(cos(x)^2)^ (1/2)*(4-5*sin(x)^2)^(1/2)/(5*cos(x)^4-cos(x)^2)^(1/2)*EllipticF(sin(x),1/ 2*5^(1/2))+9/40*(cos(x)^2)^(1/2)*(4-5*sin(x)^2)^(1/2)/(5*cos(x)^4-cos(x)^2 )^(1/2)*(EllipticF(sin(x),1/2*5^(1/2))-EllipticE(sin(x),1/2*5^(1/2))))/cos (x)/(4-5*sin(x)^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 264, normalized size of antiderivative = 3.34 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{7/2}} \, dx =\text {Too large to display} \] Input:
integrate(1/(4-5*sin(x)^2)^(7/2),x, algorithm="fricas")
Output:
-1/800*(9*sqrt(4/5*I - 3/5)*(-(375*I + 500)*sqrt(5)*cos(x)^6 + (225*I + 30 0)*sqrt(5)*cos(x)^4 - (45*I + 60)*sqrt(5)*cos(x)^2 + (3*I + 4)*sqrt(5))*el liptic_e(arcsin(sqrt(4/5*I - 3/5)*(cos(x) + I*sin(x))), 24/25*I - 7/25) + 9*sqrt(-4/5*I - 3/5)*((375*I - 500)*sqrt(5)*cos(x)^6 - (225*I - 300)*sqrt( 5)*cos(x)^4 + (45*I - 60)*sqrt(5)*cos(x)^2 - (3*I - 4)*sqrt(5))*elliptic_e (arcsin(sqrt(-4/5*I - 3/5)*(cos(x) - I*sin(x))), -24/25*I - 7/25) + 8*sqrt (4/5*I - 3/5)*((750*I + 125)*sqrt(5)*cos(x)^6 - (450*I + 75)*sqrt(5)*cos(x )^4 + (90*I + 15)*sqrt(5)*cos(x)^2 - (6*I + 1)*sqrt(5))*elliptic_f(arcsin( sqrt(4/5*I - 3/5)*(cos(x) + I*sin(x))), 24/25*I - 7/25) + 8*sqrt(-4/5*I - 3/5)*(-(750*I - 125)*sqrt(5)*cos(x)^6 + (450*I - 75)*sqrt(5)*cos(x)^4 - (9 0*I - 15)*sqrt(5)*cos(x)^2 + (6*I - 1)*sqrt(5))*elliptic_f(arcsin(sqrt(-4/ 5*I - 3/5)*(cos(x) - I*sin(x))), -24/25*I - 7/25) - 50*(225*cos(x)^5 - 110 *cos(x)^3 + 17*cos(x))*sqrt(5*cos(x)^2 - 1)*sin(x))/(125*cos(x)^6 - 75*cos (x)^4 + 15*cos(x)^2 - 1)
Timed out. \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(4-5*sin(x)**2)**(7/2),x)
Output:
Timed out
\[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{7/2}} \, dx=\int { \frac {1}{{\left (-5 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(4-5*sin(x)^2)^(7/2),x, algorithm="maxima")
Output:
integrate((-5*sin(x)^2 + 4)^(-7/2), x)
\[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{7/2}} \, dx=\int { \frac {1}{{\left (-5 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(4-5*sin(x)^2)^(7/2),x, algorithm="giac")
Output:
integrate((-5*sin(x)^2 + 4)^(-7/2), x)
Timed out. \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{7/2}} \, dx=\int \frac {1}{{\left (4-5\,{\sin \left (x\right )}^2\right )}^{7/2}} \,d x \] Input:
int(1/(4 - 5*sin(x)^2)^(7/2),x)
Output:
int(1/(4 - 5*sin(x)^2)^(7/2), x)
\[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{7/2}} \, dx=\int \frac {\sqrt {-5 \sin \left (x \right )^{2}+4}}{625 \sin \left (x \right )^{8}-2000 \sin \left (x \right )^{6}+2400 \sin \left (x \right )^{4}-1280 \sin \left (x \right )^{2}+256}d x \] Input:
int(1/(4-5*sin(x)^2)^(7/2),x)
Output:
int(sqrt( - 5*sin(x)**2 + 4)/(625*sin(x)**8 - 2000*sin(x)**6 + 2400*sin(x) **4 - 1280*sin(x)**2 + 256),x)