Integrand size = 12, antiderivative size = 30 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{3/2}} \, dx=-\frac {E\left (x\left |\frac {5}{4}\right .\right )}{2}+\frac {5 \cos (x) \sin (x)}{4 \sqrt {4-5 \sin ^2(x)}} \] Output:
-1/2*EllipticE(sin(x),1/2*5^(1/2))+5/4*cos(x)*sin(x)/(4-5*sin(x)^2)^(1/2)
Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{3/2}} \, dx=-\frac {E\left (x\left |\frac {5}{4}\right .\right )}{2}+\frac {5 \sin (2 x)}{4 \sqrt {2} \sqrt {3+5 \cos (2 x)}} \] Input:
Integrate[(4 - 5*Sin[x]^2)^(-3/2),x]
Output:
-1/2*EllipticE[x, 5/4] + (5*Sin[2*x])/(4*Sqrt[2]*Sqrt[3 + 5*Cos[2*x]])
Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3042, 3663, 25, 3042, 3656}
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.
\(\displaystyle \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (4-5 \sin (x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3663 |
\(\displaystyle \frac {1}{4} \int -\sqrt {4-5 \sin ^2(x)}dx+\frac {5 \sin (x) \cos (x)}{4 \sqrt {4-5 \sin ^2(x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {5 \sin (x) \cos (x)}{4 \sqrt {4-5 \sin ^2(x)}}-\frac {1}{4} \int \sqrt {4-5 \sin ^2(x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5 \sin (x) \cos (x)}{4 \sqrt {4-5 \sin ^2(x)}}-\frac {1}{4} \int \sqrt {4-5 \sin (x)^2}dx\) |
\(\Big \downarrow \) 3656 |
\(\displaystyle \frac {5 \sin (x) \cos (x)}{4 \sqrt {4-5 \sin ^2(x)}}-\frac {E\left (x\left |\frac {5}{4}\right .\right )}{2}\) |
Input:
Int[(4 - 5*Sin[x]^2)^(-3/2),x]
Output:
-1/2*EllipticE[x, 5/4] + (5*Cos[x]*Sin[x])/(4*Sqrt[4 - 5*Sin[x]^2])
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[((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]
Time = 0.73 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.73
method | result | size |
default | \(-\frac {2 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {5 \cos \left (x \right )^{2}-1}\, \operatorname {EllipticE}\left (\sin \left (x \right ), \frac {\sqrt {5}}{2}\right )-5 \cos \left (x \right )^{2} \sin \left (x \right )}{4 \cos \left (x \right ) \sqrt {4-5 \sin \left (x \right )^{2}}}\) | \(52\) |
Input:
int(1/(4-5*sin(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/4*(2*(cos(x)^2)^(1/2)*(5*cos(x)^2-1)^(1/2)*EllipticE(sin(x),1/2*5^(1/2) )-5*cos(x)^2*sin(x))/cos(x)/(4-5*sin(x)^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 165, normalized size of antiderivative = 5.50 \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{3/2}} \, dx=-\frac {5 \, \sqrt {\frac {4}{5} i - \frac {3}{5}} {\left (-\left (15 i + 20\right ) \, \sqrt {5} \cos \left (x\right )^{2} + \left (3 i + 4\right ) \, \sqrt {5}\right )} E(\arcsin \left (\sqrt {\frac {4}{5} i - \frac {3}{5}} {\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right )}\right )\,|\,\frac {24}{25} i - \frac {7}{25}) + 5 \, \sqrt {-\frac {4}{5} i - \frac {3}{5}} {\left (\left (15 i - 20\right ) \, \sqrt {5} \cos \left (x\right )^{2} - \left (3 i - 4\right ) \, \sqrt {5}\right )} E(\arcsin \left (\sqrt {-\frac {4}{5} i - \frac {3}{5}} {\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right )}\right )\,|\,-\frac {24}{25} i - \frac {7}{25}) + 8 \, \sqrt {\frac {4}{5} i - \frac {3}{5}} {\left (\left (15 i + 5\right ) \, \sqrt {5} \cos \left (x\right )^{2} - \left (3 i + 1\right ) \, \sqrt {5}\right )} F(\arcsin \left (\sqrt {\frac {4}{5} i - \frac {3}{5}} {\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right )}\right )\,|\,\frac {24}{25} i - \frac {7}{25}) + 8 \, \sqrt {-\frac {4}{5} i - \frac {3}{5}} {\left (-\left (15 i - 5\right ) \, \sqrt {5} \cos \left (x\right )^{2} + \left (3 i - 1\right ) \, \sqrt {5}\right )} F(\arcsin \left (\sqrt {-\frac {4}{5} i - \frac {3}{5}} {\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right )}\right )\,|\,-\frac {24}{25} i - \frac {7}{25}) - 250 \, \sqrt {5 \, \cos \left (x\right )^{2} - 1} \cos \left (x\right ) \sin \left (x\right )}{200 \, {\left (5 \, \cos \left (x\right )^{2} - 1\right )}} \] Input:
integrate(1/(4-5*sin(x)^2)^(3/2),x, algorithm="fricas")
Output:
-1/200*(5*sqrt(4/5*I - 3/5)*(-(15*I + 20)*sqrt(5)*cos(x)^2 + (3*I + 4)*sqr t(5))*elliptic_e(arcsin(sqrt(4/5*I - 3/5)*(cos(x) + I*sin(x))), 24/25*I - 7/25) + 5*sqrt(-4/5*I - 3/5)*((15*I - 20)*sqrt(5)*cos(x)^2 - (3*I - 4)*sqr t(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)*((15*I + 5)*sqrt(5)*cos(x)^2 - (3*I + 1)*sqr t(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)*(-(15*I - 5)*sqrt(5)*cos(x)^2 + (3*I - 1)*sqr t(5))*elliptic_f(arcsin(sqrt(-4/5*I - 3/5)*(cos(x) - I*sin(x))), -24/25*I - 7/25) - 250*sqrt(5*cos(x)^2 - 1)*cos(x)*sin(x))/(5*cos(x)^2 - 1)
Timed out. \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(4-5*sin(x)**2)**(3/2),x)
Output:
Timed out
\[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-5 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(4-5*sin(x)^2)^(3/2),x, algorithm="maxima")
Output:
integrate((-5*sin(x)^2 + 4)^(-3/2), x)
\[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-5 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(4-5*sin(x)^2)^(3/2),x, algorithm="giac")
Output:
integrate((-5*sin(x)^2 + 4)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (4-5\,{\sin \left (x\right )}^2\right )}^{3/2}} \,d x \] Input:
int(1/(4 - 5*sin(x)^2)^(3/2),x)
Output:
int(1/(4 - 5*sin(x)^2)^(3/2), x)
\[ \int \frac {1}{\left (4-5 \sin ^2(x)\right )^{3/2}} \, dx=\int \frac {\sqrt {-5 \sin \left (x \right )^{2}+4}}{25 \sin \left (x \right )^{4}-40 \sin \left (x \right )^{2}+16}d x \] Input:
int(1/(4-5*sin(x)^2)^(3/2),x)
Output:
int(sqrt( - 5*sin(x)**2 + 4)/(25*sin(x)**4 - 40*sin(x)**2 + 16),x)