Integrand size = 12, antiderivative size = 79 \[ \int \left (4-3 \sin ^2(x)\right )^{7/2} \, dx=\frac {224 E\left (x\left |\frac {3}{4}\right .\right )}{3}-\frac {200 \operatorname {EllipticF}\left (x,\frac {3}{4}\right )}{21}+\frac {100}{7} \cos (x) \sin (x) \sqrt {4-3 \sin ^2(x)}+\frac {18}{7} \cos (x) \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2}+\frac {3}{7} \cos (x) \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \] Output:
224/3*EllipticE(sin(x),1/2*3^(1/2))-200/21*InverseJacobiAM(x,1/2*3^(1/2))+ 100/7*cos(x)*sin(x)*(4-3*sin(x)^2)^(1/2)+18/7*cos(x)*sin(x)*(4-3*sin(x)^2) ^(3/2)+3/7*cos(x)*sin(x)*(4-3*sin(x)^2)^(5/2)
Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.68 \[ \int \left (4-3 \sin ^2(x)\right )^{7/2} \, dx=\frac {224 E\left (x\left |\frac {3}{4}\right .\right )}{3}-\frac {200 \operatorname {EllipticF}\left (x,\frac {3}{4}\right )}{21}+\frac {1}{448} \sqrt {10+6 \cos (2 x)} (2647 \sin (2 x)+396 \sin (4 x)+27 \sin (6 x)) \] Input:
Integrate[(4 - 3*Sin[x]^2)^(7/2),x]
Output:
(224*EllipticE[x, 3/4])/3 - (200*EllipticF[x, 3/4])/21 + (Sqrt[10 + 6*Cos[ 2*x]]*(2647*Sin[2*x] + 396*Sin[4*x] + 27*Sin[6*x]))/448
Time = 0.63 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {3042, 3659, 27, 3042, 3649, 27, 3042, 3649, 27, 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.
| \(\displaystyle \int \left (4-3 \sin ^2(x)\right )^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (4-3 \sin (x)^2\right )^{7/2}dx\) |
| \(\Big \downarrow \) 3659 |
| \(\displaystyle \frac {1}{7} \int 10 \left (10-9 \sin ^2(x)\right ) \left (4-3 \sin ^2(x)\right )^{3/2}dx+\frac {3}{7} \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{7} \int \left (10-9 \sin ^2(x)\right ) \left (4-3 \sin ^2(x)\right )^{3/2}dx+\frac {3}{7} \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {10}{7} \int \left (10-9 \sin (x)^2\right ) \left (4-3 \sin (x)^2\right )^{3/2}dx+\frac {3}{7} \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3649 |
| \(\displaystyle \frac {10}{7} \left (\frac {1}{5} \int 2 \left (82-75 \sin ^2(x)\right ) \sqrt {4-3 \sin ^2(x)}dx+\frac {9}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {3}{7} \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{7} \left (\frac {2}{5} \int \left (82-75 \sin ^2(x)\right ) \sqrt {4-3 \sin ^2(x)}dx+\frac {9}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {3}{7} \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {10}{7} \left (\frac {2}{5} \int \left (82-75 \sin (x)^2\right ) \sqrt {4-3 \sin (x)^2}dx+\frac {9}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {3}{7} \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3649 |
| \(\displaystyle \frac {10}{7} \left (\frac {2}{5} \left (\frac {1}{3} \int \frac {12 \left (57-49 \sin ^2(x)\right )}{\sqrt {4-3 \sin ^2(x)}}dx+25 \sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\right )+\frac {9}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {3}{7} \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{7} \left (\frac {2}{5} \left (4 \int \frac {57-49 \sin ^2(x)}{\sqrt {4-3 \sin ^2(x)}}dx+25 \sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\right )+\frac {9}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {3}{7} \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {10}{7} \left (\frac {2}{5} \left (4 \int \frac {57-49 \sin (x)^2}{\sqrt {4-3 \sin (x)^2}}dx+25 \sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\right )+\frac {9}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {3}{7} \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3651 |
| \(\displaystyle \frac {10}{7} \left (\frac {2}{5} \left (4 \left (\frac {49}{3} \int \sqrt {4-3 \sin ^2(x)}dx-\frac {25}{3} \int \frac {1}{\sqrt {4-3 \sin ^2(x)}}dx\right )+25 \sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\right )+\frac {9}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {3}{7} \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {10}{7} \left (\frac {2}{5} \left (4 \left (\frac {49}{3} \int \sqrt {4-3 \sin (x)^2}dx-\frac {25}{3} \int \frac {1}{\sqrt {4-3 \sin (x)^2}}dx\right )+25 \sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\right )+\frac {9}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {3}{7} \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3656 |
| \(\displaystyle \frac {10}{7} \left (\frac {2}{5} \left (4 \left (\frac {98 E\left (x\left |\frac {3}{4}\right .\right )}{3}-\frac {25}{3} \int \frac {1}{\sqrt {4-3 \sin (x)^2}}dx\right )+25 \sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\right )+\frac {9}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {3}{7} \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3661 |
| \(\displaystyle \frac {3}{7} \sin (x) \left (4-3 \sin ^2(x)\right )^{5/2} \cos (x)+\frac {10}{7} \left (\frac {9}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)+\frac {2}{5} \left (25 \sin (x) \sqrt {4-3 \sin ^2(x)} \cos (x)+4 \left (\frac {98 E\left (x\left |\frac {3}{4}\right .\right )}{3}-\frac {25 \operatorname {EllipticF}\left (x,\frac {3}{4}\right )}{6}\right )\right )\right )\) |
Input:
Int[(4 - 3*Sin[x]^2)^(7/2),x]
Output:
(3*Cos[x]*Sin[x]*(4 - 3*Sin[x]^2)^(5/2))/7 + (10*((9*Cos[x]*Sin[x]*(4 - 3* Sin[x]^2)^(3/2))/5 + (2*(4*((98*EllipticE[x, 3/4])/3 - (25*EllipticF[x, 3/ 4])/6) + 25*Cos[x]*Sin[x]*Sqrt[4 - 3*Sin[x]^2]))/5))/7
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)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-B)*Cos[e + f*x]*Sin[e + f*x]*((a + b* Sin[e + f*x]^2)^p/(2*f*(p + 1))), x] + Simp[1/(2*(p + 1)) Int[(a + b*Sin[ e + f*x]^2)^(p - 1)*Simp[a*B + 2*a*A*(p + 1) + (2*A*b*(p + 1) + B*(b + 2*a* p + 2*b*p))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && G tQ[p, 0]
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[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*f*p)), x] + Sim p[1/(2*p) Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(2*a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[ a + b, 0] && GtQ[p, 1]
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]
Time = 5.51 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.66
| method | result | size |
| default | \(-\frac {\sqrt {-\left (-4+3 \sin \left (x \right )^{2}\right ) \cos \left (x \right )^{2}}\, \left (-243 \cos \left (x \right )^{8} \sin \left (x \right )-729 \cos \left (x \right )^{6} \sin \left (x \right )-1305 \cos \left (x \right )^{4} \sin \left (x \right )+200 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {3 \cos \left (x \right )^{2}+1}\, \operatorname {EllipticF}\left (\sin \left (x \right ), \frac {\sqrt {3}}{2}\right )-1568 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {3 \cos \left (x \right )^{2}+1}\, \operatorname {EllipticE}\left (\sin \left (x \right ), \frac {\sqrt {3}}{2}\right )-363 \cos \left (x \right )^{2} \sin \left (x \right )\right )}{21 \sqrt {3 \cos \left (x \right )^{4}+\cos \left (x \right )^{2}}\, \cos \left (x \right ) \sqrt {4-3 \sin \left (x \right )^{2}}}\) | \(131\) |
Input:
int((4-3*sin(x)^2)^(7/2),x,method=_RETURNVERBOSE)
Output:
-1/21*(-(-4+3*sin(x)^2)*cos(x)^2)^(1/2)*(-243*cos(x)^8*sin(x)-729*cos(x)^6 *sin(x)-1305*cos(x)^4*sin(x)+200*(cos(x)^2)^(1/2)*(3*cos(x)^2+1)^(1/2)*Ell ipticF(sin(x),1/2*3^(1/2))-1568*(cos(x)^2)^(1/2)*(3*cos(x)^2+1)^(1/2)*Elli pticE(sin(x),1/2*3^(1/2))-363*cos(x)^2*sin(x))/(3*cos(x)^4+cos(x)^2)^(1/2) /cos(x)/(4-3*sin(x)^2)^(1/2)
\[ \int \left (4-3 \sin ^2(x)\right )^{7/2} \, dx=\int { {\left (-3 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {7}{2}} \,d x } \] Input:
integrate((4-3*sin(x)^2)^(7/2),x, algorithm="fricas")
Output:
integral((27*cos(x)^6 + 27*cos(x)^4 + 9*cos(x)^2 + 1)*sqrt(3*cos(x)^2 + 1) , x)
Timed out. \[ \int \left (4-3 \sin ^2(x)\right )^{7/2} \, dx=\text {Timed out} \] Input:
integrate((4-3*sin(x)**2)**(7/2),x)
Output:
Timed out
\[ \int \left (4-3 \sin ^2(x)\right )^{7/2} \, dx=\int { {\left (-3 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {7}{2}} \,d x } \] Input:
integrate((4-3*sin(x)^2)^(7/2),x, algorithm="maxima")
Output:
integrate((-3*sin(x)^2 + 4)^(7/2), x)
\[ \int \left (4-3 \sin ^2(x)\right )^{7/2} \, dx=\int { {\left (-3 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {7}{2}} \,d x } \] Input:
integrate((4-3*sin(x)^2)^(7/2),x, algorithm="giac")
Output:
integrate((-3*sin(x)^2 + 4)^(7/2), x)
Timed out. \[ \int \left (4-3 \sin ^2(x)\right )^{7/2} \, dx=\int {\left (4-3\,{\sin \left (x\right )}^2\right )}^{7/2} \,d x \] Input:
int((4 - 3*sin(x)^2)^(7/2),x)
Output:
int((4 - 3*sin(x)^2)^(7/2), x)
\[ \int \left (4-3 \sin ^2(x)\right )^{7/2} \, dx=64 \left (\int \sqrt {-3 \sin \left (x \right )^{2}+4}d x \right )-27 \left (\int \sqrt {-3 \sin \left (x \right )^{2}+4}\, \sin \left (x \right )^{6}d x \right )+108 \left (\int \sqrt {-3 \sin \left (x \right )^{2}+4}\, \sin \left (x \right )^{4}d x \right )-144 \left (\int \sqrt {-3 \sin \left (x \right )^{2}+4}\, \sin \left (x \right )^{2}d x \right ) \] Input:
int((4-3*sin(x)^2)^(7/2),x)
Output:
64*int(sqrt( - 3*sin(x)**2 + 4),x) - 27*int(sqrt( - 3*sin(x)**2 + 4)*sin(x )**6,x) + 108*int(sqrt( - 3*sin(x)**2 + 4)*sin(x)**4,x) - 144*int(sqrt( - 3*sin(x)**2 + 4)*sin(x)**2,x)