Integrand size = 12, antiderivative size = 79 \[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{7/2}} \, dx=\frac {41 E\left (x\left |\frac {3}{4}\right .\right )}{120}-\frac {\operatorname {EllipticF}\left (x,\frac {3}{4}\right )}{24}-\frac {3 \cos (x) \sin (x)}{20 \left (4-3 \sin ^2(x)\right )^{5/2}}-\frac {\cos (x) \sin (x)}{4 \left (4-3 \sin ^2(x)\right )^{3/2}}-\frac {41 \cos (x) \sin (x)}{80 \sqrt {4-3 \sin ^2(x)}} \] Output:
41/120*EllipticE(sin(x),1/2*3^(1/2))-1/24*InverseJacobiAM(x,1/2*3^(1/2))-3 /20*cos(x)*sin(x)/(4-3*sin(x)^2)^(5/2)-1/4*cos(x)*sin(x)/(4-3*sin(x)^2)^(3 /2)-41/80*cos(x)*sin(x)/(4-3*sin(x)^2)^(1/2)
Time = 0.44 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{7/2}} \, dx=\frac {656 E\left (x\left |\frac {3}{4}\right .\right )-80 \operatorname {EllipticF}\left (x,\frac {3}{4}\right )-\frac {3 \sqrt {2} (5461 \sin (2 x)+2700 \sin (4 x)+369 \sin (6 x))}{(5+3 \cos (2 x))^{5/2}}}{1920} \] Input:
Integrate[(4 - 3*Sin[x]^2)^(-7/2),x]
Output:
(656*EllipticE[x, 3/4] - 80*EllipticF[x, 3/4] - (3*Sqrt[2]*(5461*Sin[2*x] + 2700*Sin[4*x] + 369*Sin[6*x]))/(5 + 3*Cos[2*x])^(5/2))/1920
Time = 0.63 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {3042, 3663, 25, 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.
| \(\displaystyle \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (4-3 \sin (x)^2\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3663 |
| \(\displaystyle -\frac {1}{20} \int -\frac {9 \sin ^2(x)+8}{\left (4-3 \sin ^2(x)\right )^{5/2}}dx-\frac {3 \sin (x) \cos (x)}{20 \left (4-3 \sin ^2(x)\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{20} \int \frac {9 \sin ^2(x)+8}{\left (4-3 \sin ^2(x)\right )^{5/2}}dx-\frac {3 \sin (x) \cos (x)}{20 \left (4-3 \sin ^2(x)\right )^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{20} \int \frac {9 \sin (x)^2+8}{\left (4-3 \sin (x)^2\right )^{5/2}}dx-\frac {3 \sin (x) \cos (x)}{20 \left (4-3 \sin ^2(x)\right )^{5/2}}\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {1}{20} \left (\frac {1}{12} \int \frac {12 \left (5 \sin ^2(x)+7\right )}{\left (4-3 \sin ^2(x)\right )^{3/2}}dx-\frac {5 \sin (x) \cos (x)}{\left (4-3 \sin ^2(x)\right )^{3/2}}\right )-\frac {3 \sin (x) \cos (x)}{20 \left (4-3 \sin ^2(x)\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{20} \left (\int \frac {5 \sin ^2(x)+7}{\left (4-3 \sin ^2(x)\right )^{3/2}}dx-\frac {5 \sin (x) \cos (x)}{\left (4-3 \sin ^2(x)\right )^{3/2}}\right )-\frac {3 \sin (x) \cos (x)}{20 \left (4-3 \sin ^2(x)\right )^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{20} \left (\int \frac {5 \sin (x)^2+7}{\left (4-3 \sin (x)^2\right )^{3/2}}dx-\frac {5 \sin (x) \cos (x)}{\left (4-3 \sin ^2(x)\right )^{3/2}}\right )-\frac {3 \sin (x) \cos (x)}{20 \left (4-3 \sin ^2(x)\right )^{5/2}}\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {1}{20} \left (\frac {1}{4} \int \frac {48-41 \sin ^2(x)}{\sqrt {4-3 \sin ^2(x)}}dx-\frac {41 \sin (x) \cos (x)}{4 \sqrt {4-3 \sin ^2(x)}}-\frac {5 \sin (x) \cos (x)}{\left (4-3 \sin ^2(x)\right )^{3/2}}\right )-\frac {3 \sin (x) \cos (x)}{20 \left (4-3 \sin ^2(x)\right )^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{20} \left (\frac {1}{4} \int \frac {48-41 \sin (x)^2}{\sqrt {4-3 \sin (x)^2}}dx-\frac {41 \sin (x) \cos (x)}{4 \sqrt {4-3 \sin ^2(x)}}-\frac {5 \sin (x) \cos (x)}{\left (4-3 \sin ^2(x)\right )^{3/2}}\right )-\frac {3 \sin (x) \cos (x)}{20 \left (4-3 \sin ^2(x)\right )^{5/2}}\) |
| \(\Big \downarrow \) 3651 |
| \(\displaystyle \frac {1}{20} \left (\frac {1}{4} \left (\frac {41}{3} \int \sqrt {4-3 \sin ^2(x)}dx-\frac {20}{3} \int \frac {1}{\sqrt {4-3 \sin ^2(x)}}dx\right )-\frac {41 \sin (x) \cos (x)}{4 \sqrt {4-3 \sin ^2(x)}}-\frac {5 \sin (x) \cos (x)}{\left (4-3 \sin ^2(x)\right )^{3/2}}\right )-\frac {3 \sin (x) \cos (x)}{20 \left (4-3 \sin ^2(x)\right )^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{20} \left (\frac {1}{4} \left (\frac {41}{3} \int \sqrt {4-3 \sin (x)^2}dx-\frac {20}{3} \int \frac {1}{\sqrt {4-3 \sin (x)^2}}dx\right )-\frac {41 \sin (x) \cos (x)}{4 \sqrt {4-3 \sin ^2(x)}}-\frac {5 \sin (x) \cos (x)}{\left (4-3 \sin ^2(x)\right )^{3/2}}\right )-\frac {3 \sin (x) \cos (x)}{20 \left (4-3 \sin ^2(x)\right )^{5/2}}\) |
| \(\Big \downarrow \) 3656 |
| \(\displaystyle \frac {1}{20} \left (\frac {1}{4} \left (\frac {82 E\left (x\left |\frac {3}{4}\right .\right )}{3}-\frac {20}{3} \int \frac {1}{\sqrt {4-3 \sin (x)^2}}dx\right )-\frac {41 \sin (x) \cos (x)}{4 \sqrt {4-3 \sin ^2(x)}}-\frac {5 \sin (x) \cos (x)}{\left (4-3 \sin ^2(x)\right )^{3/2}}\right )-\frac {3 \sin (x) \cos (x)}{20 \left (4-3 \sin ^2(x)\right )^{5/2}}\) |
| \(\Big \downarrow \) 3661 |
| \(\displaystyle \frac {1}{20} \left (-\frac {41 \sin (x) \cos (x)}{4 \sqrt {4-3 \sin ^2(x)}}-\frac {5 \sin (x) \cos (x)}{\left (4-3 \sin ^2(x)\right )^{3/2}}+\frac {1}{4} \left (\frac {82 E\left (x\left |\frac {3}{4}\right .\right )}{3}-\frac {10 \operatorname {EllipticF}\left (x,\frac {3}{4}\right )}{3}\right )\right )-\frac {3 \sin (x) \cos (x)}{20 \left (4-3 \sin ^2(x)\right )^{5/2}}\) |
Input:
Int[(4 - 3*Sin[x]^2)^(-7/2),x]
Output:
(-3*Cos[x]*Sin[x])/(20*(4 - 3*Sin[x]^2)^(5/2)) + (((82*EllipticE[x, 3/4])/ 3 - (10*EllipticF[x, 3/4])/3)/4 - (5*Cos[x]*Sin[x])/(4 - 3*Sin[x]^2)^(3/2) - (41*Cos[x]*Sin[x])/(4*Sqrt[4 - 3*Sin[x]^2]))/20
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. \(195\) vs. \(2(68)=136\).
Time = 1.66 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.48
| method | result | size |
| default | \(\frac {\sqrt {-\left (-4+3 \sin \left (x \right )^{2}\right ) \cos \left (x \right )^{2}}\, \left (\frac {\sin \left (x \right ) \sqrt {3 \cos \left (x \right )^{4}+\cos \left (x \right )^{2}}}{180 \left (\sin \left (x \right )^{2}-\frac {4}{3}\right )^{3}}-\frac {\sin \left (x \right ) \sqrt {3 \cos \left (x \right )^{4}+\cos \left (x \right )^{2}}}{36 \left (\sin \left (x \right )^{2}-\frac {4}{3}\right )^{2}}-\frac {41 \cos \left (x \right )^{2} \sin \left (x \right )}{80 \sqrt {-\left (-4+3 \sin \left (x \right )^{2}\right ) \cos \left (x \right )^{2}}}+\frac {3 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {4-3 \sin \left (x \right )^{2}}\, \operatorname {EllipticF}\left (\sin \left (x \right ), \frac {\sqrt {3}}{2}\right )}{10 \sqrt {3 \cos \left (x \right )^{4}+\cos \left (x \right )^{2}}}-\frac {41 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {4-3 \sin \left (x \right )^{2}}\, \left (\operatorname {EllipticF}\left (\sin \left (x \right ), \frac {\sqrt {3}}{2}\right )-\operatorname {EllipticE}\left (\sin \left (x \right ), \frac {\sqrt {3}}{2}\right )\right )}{120 \sqrt {3 \cos \left (x \right )^{4}+\cos \left (x \right )^{2}}}\right )}{\cos \left (x \right ) \sqrt {4-3 \sin \left (x \right )^{2}}}\) | \(196\) |
Input:
int(1/(4-3*sin(x)^2)^(7/2),x,method=_RETURNVERBOSE)
Output:
(-(-4+3*sin(x)^2)*cos(x)^2)^(1/2)*(1/180*sin(x)*(3*cos(x)^4+cos(x)^2)^(1/2 )/(sin(x)^2-4/3)^3-1/36*sin(x)*(3*cos(x)^4+cos(x)^2)^(1/2)/(sin(x)^2-4/3)^ 2-41/80*cos(x)^2*sin(x)/(-(-4+3*sin(x)^2)*cos(x)^2)^(1/2)+3/10*(cos(x)^2)^ (1/2)*(4-3*sin(x)^2)^(1/2)/(3*cos(x)^4+cos(x)^2)^(1/2)*EllipticF(sin(x),1/ 2*3^(1/2))-41/120*(cos(x)^2)^(1/2)*(4-3*sin(x)^2)^(1/2)/(3*cos(x)^4+cos(x) ^2)^(1/2)*(EllipticF(sin(x),1/2*3^(1/2))-EllipticE(sin(x),1/2*3^(1/2))))/c os(x)/(4-3*sin(x)^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.73 \[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{7/2}} \, dx=-\frac {18 \, {\left (369 \, \cos \left (x\right )^{5} + 306 \, \cos \left (x\right )^{3} + 73 \, \cos \left (x\right )\right )} \sqrt {3 \, \cos \left (x\right )^{2} + 1} \sin \left (x\right ) - 41 \, {\left (27 \, \cos \left (x\right )^{6} + 27 \, \cos \left (x\right )^{4} + 9 \, \cos \left (x\right )^{2} + 1\right )} E(\arcsin \left (\frac {1}{3} i \, \sqrt {3} \cos \left (x\right ) - \frac {1}{3} \, \sqrt {3} \sin \left (x\right )\right )\,|\,9) - 41 \, {\left (27 \, \cos \left (x\right )^{6} + 27 \, \cos \left (x\right )^{4} + 9 \, \cos \left (x\right )^{2} + 1\right )} E(\arcsin \left (-\frac {1}{3} i \, \sqrt {3} \cos \left (x\right ) - \frac {1}{3} \, \sqrt {3} \sin \left (x\right )\right )\,|\,9) + 536 \, {\left (27 \, \cos \left (x\right )^{6} + 27 \, \cos \left (x\right )^{4} + 9 \, \cos \left (x\right )^{2} + 1\right )} F(\arcsin \left (\frac {1}{3} i \, \sqrt {3} \cos \left (x\right ) - \frac {1}{3} \, \sqrt {3} \sin \left (x\right )\right )\,|\,9) + 536 \, {\left (27 \, \cos \left (x\right )^{6} + 27 \, \cos \left (x\right )^{4} + 9 \, \cos \left (x\right )^{2} + 1\right )} F(\arcsin \left (-\frac {1}{3} i \, \sqrt {3} \cos \left (x\right ) - \frac {1}{3} \, \sqrt {3} \sin \left (x\right )\right )\,|\,9)}{1440 \, {\left (27 \, \cos \left (x\right )^{6} + 27 \, \cos \left (x\right )^{4} + 9 \, \cos \left (x\right )^{2} + 1\right )}} \] Input:
integrate(1/(4-3*sin(x)^2)^(7/2),x, algorithm="fricas")
Output:
-1/1440*(18*(369*cos(x)^5 + 306*cos(x)^3 + 73*cos(x))*sqrt(3*cos(x)^2 + 1) *sin(x) - 41*(27*cos(x)^6 + 27*cos(x)^4 + 9*cos(x)^2 + 1)*elliptic_e(arcsi n(1/3*I*sqrt(3)*cos(x) - 1/3*sqrt(3)*sin(x)), 9) - 41*(27*cos(x)^6 + 27*co s(x)^4 + 9*cos(x)^2 + 1)*elliptic_e(arcsin(-1/3*I*sqrt(3)*cos(x) - 1/3*sqr t(3)*sin(x)), 9) + 536*(27*cos(x)^6 + 27*cos(x)^4 + 9*cos(x)^2 + 1)*ellipt ic_f(arcsin(1/3*I*sqrt(3)*cos(x) - 1/3*sqrt(3)*sin(x)), 9) + 536*(27*cos(x )^6 + 27*cos(x)^4 + 9*cos(x)^2 + 1)*elliptic_f(arcsin(-1/3*I*sqrt(3)*cos(x ) - 1/3*sqrt(3)*sin(x)), 9))/(27*cos(x)^6 + 27*cos(x)^4 + 9*cos(x)^2 + 1)
Timed out. \[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(4-3*sin(x)**2)**(7/2),x)
Output:
Timed out
\[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{7/2}} \, dx=\int { \frac {1}{{\left (-3 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(4-3*sin(x)^2)^(7/2),x, algorithm="maxima")
Output:
integrate((-3*sin(x)^2 + 4)^(-7/2), x)
\[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{7/2}} \, dx=\int { \frac {1}{{\left (-3 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(4-3*sin(x)^2)^(7/2),x, algorithm="giac")
Output:
integrate((-3*sin(x)^2 + 4)^(-7/2), x)
Timed out. \[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{7/2}} \, dx=\int \frac {1}{{\left (4-3\,{\sin \left (x\right )}^2\right )}^{7/2}} \,d x \] Input:
int(1/(4 - 3*sin(x)^2)^(7/2),x)
Output:
int(1/(4 - 3*sin(x)^2)^(7/2), x)
\[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{7/2}} \, dx=\int \frac {\sqrt {-3 \sin \left (x \right )^{2}+4}}{81 \sin \left (x \right )^{8}-432 \sin \left (x \right )^{6}+864 \sin \left (x \right )^{4}-768 \sin \left (x \right )^{2}+256}d x \] Input:
int(1/(4-3*sin(x)^2)^(7/2),x)
Output:
int(sqrt( - 3*sin(x)**2 + 4)/(81*sin(x)**8 - 432*sin(x)**6 + 864*sin(x)**4 - 768*sin(x)**2 + 256),x)