Integrand size = 12, antiderivative size = 30 \[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{3/2}} \, dx=\frac {E\left (x\left |\frac {3}{4}\right .\right )}{2}-\frac {3 \cos (x) \sin (x)}{4 \sqrt {4-3 \sin ^2(x)}} \] Output:
1/2*EllipticE(sin(x),1/2*3^(1/2))-3/4*cos(x)*sin(x)/(4-3*sin(x)^2)^(1/2)
Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{3/2}} \, dx=\frac {E\left (x\left |\frac {3}{4}\right .\right )}{2}-\frac {3 \sin (2 x)}{4 \sqrt {2} \sqrt {5+3 \cos (2 x)}} \] Input:
Integrate[(4 - 3*Sin[x]^2)^(-3/2),x]
Output:
EllipticE[x, 3/4]/2 - (3*Sin[2*x])/(4*Sqrt[2]*Sqrt[5 + 3*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.
\(\displaystyle \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (4-3 \sin (x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3663 |
\(\displaystyle -\frac {1}{4} \int -\sqrt {4-3 \sin ^2(x)}dx-\frac {3 \sin (x) \cos (x)}{4 \sqrt {4-3 \sin ^2(x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{4} \int \sqrt {4-3 \sin ^2(x)}dx-\frac {3 \sin (x) \cos (x)}{4 \sqrt {4-3 \sin ^2(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \int \sqrt {4-3 \sin (x)^2}dx-\frac {3 \sin (x) \cos (x)}{4 \sqrt {4-3 \sin ^2(x)}}\) |
\(\Big \downarrow \) 3656 |
\(\displaystyle \frac {E\left (x\left |\frac {3}{4}\right .\right )}{2}-\frac {3 \sin (x) \cos (x)}{4 \sqrt {4-3 \sin ^2(x)}}\) |
Input:
Int[(4 - 3*Sin[x]^2)^(-3/2),x]
Output:
EllipticE[x, 3/4]/2 - (3*Cos[x]*Sin[x])/(4*Sqrt[4 - 3*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 = 0.70 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.73
method | result | size |
default | \(\frac {2 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {3 \cos \left (x \right )^{2}+1}\, \operatorname {EllipticE}\left (\sin \left (x \right ), \frac {\sqrt {3}}{2}\right )-3 \cos \left (x \right )^{2} \sin \left (x \right )}{4 \cos \left (x \right ) \sqrt {4-3 \sin \left (x \right )^{2}}}\) | \(52\) |
Input:
int(1/(4-3*sin(x)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/4*(2*(cos(x)^2)^(1/2)*(3*cos(x)^2+1)^(1/2)*EllipticE(sin(x),1/2*3^(1/2)) -3*cos(x)^2*sin(x))/cos(x)/(4-3*sin(x)^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 4.70 \[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{3/2}} \, dx=-\frac {18 \, \sqrt {3 \, \cos \left (x\right )^{2} + 1} \cos \left (x\right ) \sin \left (x\right ) - {\left (3 \, \cos \left (x\right )^{2} + 1\right )} E(\arcsin \left (\frac {1}{3} i \, \sqrt {3} \cos \left (x\right ) - \frac {1}{3} \, \sqrt {3} \sin \left (x\right )\right )\,|\,9) - {\left (3 \, \cos \left (x\right )^{2} + 1\right )} E(\arcsin \left (-\frac {1}{3} i \, \sqrt {3} \cos \left (x\right ) - \frac {1}{3} \, \sqrt {3} \sin \left (x\right )\right )\,|\,9) + 16 \, {\left (3 \, \cos \left (x\right )^{2} + 1\right )} F(\arcsin \left (\frac {1}{3} i \, \sqrt {3} \cos \left (x\right ) - \frac {1}{3} \, \sqrt {3} \sin \left (x\right )\right )\,|\,9) + 16 \, {\left (3 \, \cos \left (x\right )^{2} + 1\right )} F(\arcsin \left (-\frac {1}{3} i \, \sqrt {3} \cos \left (x\right ) - \frac {1}{3} \, \sqrt {3} \sin \left (x\right )\right )\,|\,9)}{24 \, {\left (3 \, \cos \left (x\right )^{2} + 1\right )}} \] Input:
integrate(1/(4-3*sin(x)^2)^(3/2),x, algorithm="fricas")
Output:
-1/24*(18*sqrt(3*cos(x)^2 + 1)*cos(x)*sin(x) - (3*cos(x)^2 + 1)*elliptic_e (arcsin(1/3*I*sqrt(3)*cos(x) - 1/3*sqrt(3)*sin(x)), 9) - (3*cos(x)^2 + 1)* elliptic_e(arcsin(-1/3*I*sqrt(3)*cos(x) - 1/3*sqrt(3)*sin(x)), 9) + 16*(3* cos(x)^2 + 1)*elliptic_f(arcsin(1/3*I*sqrt(3)*cos(x) - 1/3*sqrt(3)*sin(x)) , 9) + 16*(3*cos(x)^2 + 1)*elliptic_f(arcsin(-1/3*I*sqrt(3)*cos(x) - 1/3*s qrt(3)*sin(x)), 9))/(3*cos(x)^2 + 1)
\[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (4 - 3 \sin ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(4-3*sin(x)**2)**(3/2),x)
Output:
Integral((4 - 3*sin(x)**2)**(-3/2), x)
\[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-3 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(4-3*sin(x)^2)^(3/2),x, algorithm="maxima")
Output:
integrate((-3*sin(x)^2 + 4)^(-3/2), x)
\[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-3 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(4-3*sin(x)^2)^(3/2),x, algorithm="giac")
Output:
integrate((-3*sin(x)^2 + 4)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (4-3\,{\sin \left (x\right )}^2\right )}^{3/2}} \,d x \] Input:
int(1/(4 - 3*sin(x)^2)^(3/2),x)
Output:
int(1/(4 - 3*sin(x)^2)^(3/2), x)
\[ \int \frac {1}{\left (4-3 \sin ^2(x)\right )^{3/2}} \, dx=\int \frac {\sqrt {-3 \sin \left (x \right )^{2}+4}}{9 \sin \left (x \right )^{4}-24 \sin \left (x \right )^{2}+16}d x \] Input:
int(1/(4-3*sin(x)^2)^(3/2),x)
Output:
int(sqrt( - 3*sin(x)**2 + 4)/(9*sin(x)**4 - 24*sin(x)**2 + 16),x)