Integrand size = 12, antiderivative size = 79 \[ \int \left (4-5 \sin ^2(x)\right )^{7/2} \, dx=\frac {1696 E\left (x\left |\frac {5}{4}\right .\right )}{35}+\frac {632 \operatorname {EllipticF}\left (x,\frac {5}{4}\right )}{105}+\frac {316}{21} \cos (x) \sin (x) \sqrt {4-5 \sin ^2(x)}+\frac {18}{7} \cos (x) \sin (x) \left (4-5 \sin ^2(x)\right )^{3/2}+\frac {5}{7} \cos (x) \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \] Output:
1696/35*EllipticE(sin(x),1/2*5^(1/2))+632/105*InverseJacobiAM(x,1/2*5^(1/2 ))+316/21*cos(x)*sin(x)*(4-5*sin(x)^2)^(1/2)+18/7*cos(x)*sin(x)*(4-5*sin(x )^2)^(3/2)+5/7*cos(x)*sin(x)*(4-5*sin(x)^2)^(5/2)
Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.66 \[ \int \left (4-5 \sin ^2(x)\right )^{7/2} \, dx=\frac {325632 E\left (x\left |\frac {5}{4}\right .\right )+40448 \operatorname {EllipticF}\left (x,\frac {5}{4}\right )+5 \sqrt {6+10 \cos (2 x)} (7267 \sin (2 x)+1980 \sin (4 x)+375 \sin (6 x))}{6720} \] Input:
Integrate[(4 - 5*Sin[x]^2)^(7/2),x]
Output:
(325632*EllipticE[x, 5/4] + 40448*EllipticF[x, 5/4] + 5*Sqrt[6 + 10*Cos[2* x]]*(7267*Sin[2*x] + 1980*Sin[4*x] + 375*Sin[6*x]))/6720
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.
| \(\displaystyle \int \left (4-5 \sin ^2(x)\right )^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (4-5 \sin (x)^2\right )^{7/2}dx\) |
| \(\Big \downarrow \) 3659 |
| \(\displaystyle \frac {1}{7} \int 2 \left (46-45 \sin ^2(x)\right ) \left (4-5 \sin ^2(x)\right )^{3/2}dx+\frac {5}{7} \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{7} \int \left (46-45 \sin ^2(x)\right ) \left (4-5 \sin ^2(x)\right )^{3/2}dx+\frac {5}{7} \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} \int \left (46-45 \sin (x)^2\right ) \left (4-5 \sin (x)^2\right )^{3/2}dx+\frac {5}{7} \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3649 |
| \(\displaystyle \frac {2}{7} \left (\frac {1}{5} \int 10 \left (74-79 \sin ^2(x)\right ) \sqrt {4-5 \sin ^2(x)}dx+9 \sin (x) \left (4-5 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {5}{7} \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{7} \left (2 \int \left (74-79 \sin ^2(x)\right ) \sqrt {4-5 \sin ^2(x)}dx+9 \sin (x) \left (4-5 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {5}{7} \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} \left (2 \int \left (74-79 \sin (x)^2\right ) \sqrt {4-5 \sin (x)^2}dx+9 \sin (x) \left (4-5 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {5}{7} \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3649 |
| \(\displaystyle \frac {2}{7} \left (2 \left (\frac {1}{3} \int \frac {4 \left (143-159 \sin ^2(x)\right )}{\sqrt {4-5 \sin ^2(x)}}dx+\frac {79}{3} \sqrt {4-5 \sin ^2(x)} \sin (x) \cos (x)\right )+9 \sin (x) \left (4-5 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {5}{7} \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{7} \left (2 \left (\frac {4}{3} \int \frac {143-159 \sin ^2(x)}{\sqrt {4-5 \sin ^2(x)}}dx+\frac {79}{3} \sqrt {4-5 \sin ^2(x)} \sin (x) \cos (x)\right )+9 \sin (x) \left (4-5 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {5}{7} \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} \left (2 \left (\frac {4}{3} \int \frac {143-159 \sin (x)^2}{\sqrt {4-5 \sin (x)^2}}dx+\frac {79}{3} \sqrt {4-5 \sin ^2(x)} \sin (x) \cos (x)\right )+9 \sin (x) \left (4-5 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {5}{7} \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3651 |
| \(\displaystyle \frac {2}{7} \left (2 \left (\frac {4}{3} \left (\frac {79}{5} \int \frac {1}{\sqrt {4-5 \sin ^2(x)}}dx+\frac {159}{5} \int \sqrt {4-5 \sin ^2(x)}dx\right )+\frac {79}{3} \sqrt {4-5 \sin ^2(x)} \sin (x) \cos (x)\right )+9 \sin (x) \left (4-5 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {5}{7} \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} \left (2 \left (\frac {4}{3} \left (\frac {79}{5} \int \frac {1}{\sqrt {4-5 \sin (x)^2}}dx+\frac {159}{5} \int \sqrt {4-5 \sin (x)^2}dx\right )+\frac {79}{3} \sqrt {4-5 \sin ^2(x)} \sin (x) \cos (x)\right )+9 \sin (x) \left (4-5 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {5}{7} \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3656 |
| \(\displaystyle \frac {2}{7} \left (2 \left (\frac {4}{3} \left (\frac {79}{5} \int \frac {1}{\sqrt {4-5 \sin (x)^2}}dx+\frac {318 E\left (x\left |\frac {5}{4}\right .\right )}{5}\right )+\frac {79}{3} \sqrt {4-5 \sin ^2(x)} \sin (x) \cos (x)\right )+9 \sin (x) \left (4-5 \sin ^2(x)\right )^{3/2} \cos (x)\right )+\frac {5}{7} \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \cos (x)\) |
| \(\Big \downarrow \) 3661 |
| \(\displaystyle \frac {5}{7} \sin (x) \left (4-5 \sin ^2(x)\right )^{5/2} \cos (x)+\frac {2}{7} \left (9 \sin (x) \left (4-5 \sin ^2(x)\right )^{3/2} \cos (x)+2 \left (\frac {79}{3} \sin (x) \sqrt {4-5 \sin ^2(x)} \cos (x)+\frac {4}{3} \left (\frac {79 \operatorname {EllipticF}\left (x,\frac {5}{4}\right )}{10}+\frac {318 E\left (x\left |\frac {5}{4}\right .\right )}{5}\right )\right )\right )\) |
Input:
Int[(4 - 5*Sin[x]^2)^(7/2),x]
Output:
(5*Cos[x]*Sin[x]*(4 - 5*Sin[x]^2)^(5/2))/7 + (2*(9*Cos[x]*Sin[x]*(4 - 5*Si n[x]^2)^(3/2) + 2*((4*((318*EllipticE[x, 5/4])/5 + (79*EllipticF[x, 5/4])/ 10))/3 + (79*Cos[x]*Sin[x]*Sqrt[4 - 5*Sin[x]^2])/3)))/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 = 5.52 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.68
| method | result | size |
| default | \(\frac {\sqrt {-\left (-4+5 \sin \left (x \right )^{2}\right ) \cos \left (x \right )^{2}}\, \left (9375 \cos \left (x \right )^{8} \sin \left (x \right )+1125 \cos \left (x \right )^{6} \sin \left (x \right )+6325 \cos \left (x \right )^{4} \sin \left (x \right )+632 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {5 \cos \left (x \right )^{2}-1}\, \operatorname {EllipticF}\left (\sin \left (x \right ), \frac {\sqrt {5}}{2}\right )+5088 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {5 \cos \left (x \right )^{2}-1}\, \operatorname {EllipticE}\left (\sin \left (x \right ), \frac {\sqrt {5}}{2}\right )-1385 \cos \left (x \right )^{2} \sin \left (x \right )\right )}{105 \sqrt {5 \cos \left (x \right )^{4}-\cos \left (x \right )^{2}}\, \cos \left (x \right ) \sqrt {4-5 \sin \left (x \right )^{2}}}\) | \(133\) |
Input:
int((4-5*sin(x)^2)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/105*(-(-4+5*sin(x)^2)*cos(x)^2)^(1/2)*(9375*cos(x)^8*sin(x)+1125*cos(x)^ 6*sin(x)+6325*cos(x)^4*sin(x)+632*(cos(x)^2)^(1/2)*(5*cos(x)^2-1)^(1/2)*El lipticF(sin(x),1/2*5^(1/2))+5088*(cos(x)^2)^(1/2)*(5*cos(x)^2-1)^(1/2)*Ell ipticE(sin(x),1/2*5^(1/2))-1385*cos(x)^2*sin(x))/(5*cos(x)^4-cos(x)^2)^(1/ 2)/cos(x)/(4-5*sin(x)^2)^(1/2)
\[ \int \left (4-5 \sin ^2(x)\right )^{7/2} \, dx=\int { {\left (-5 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {7}{2}} \,d x } \] Input:
integrate((4-5*sin(x)^2)^(7/2),x, algorithm="fricas")
Output:
integral((125*cos(x)^6 - 75*cos(x)^4 + 15*cos(x)^2 - 1)*sqrt(5*cos(x)^2 - 1), x)
Timed out. \[ \int \left (4-5 \sin ^2(x)\right )^{7/2} \, dx=\text {Timed out} \] Input:
integrate((4-5*sin(x)**2)**(7/2),x)
Output:
Timed out
\[ \int \left (4-5 \sin ^2(x)\right )^{7/2} \, dx=\int { {\left (-5 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {7}{2}} \,d x } \] Input:
integrate((4-5*sin(x)^2)^(7/2),x, algorithm="maxima")
Output:
integrate((-5*sin(x)^2 + 4)^(7/2), x)
\[ \int \left (4-5 \sin ^2(x)\right )^{7/2} \, dx=\int { {\left (-5 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {7}{2}} \,d x } \] Input:
integrate((4-5*sin(x)^2)^(7/2),x, algorithm="giac")
Output:
integrate((-5*sin(x)^2 + 4)^(7/2), x)
Timed out. \[ \int \left (4-5 \sin ^2(x)\right )^{7/2} \, dx=\int {\left (4-5\,{\sin \left (x\right )}^2\right )}^{7/2} \,d x \] Input:
int((4 - 5*sin(x)^2)^(7/2),x)
Output:
int((4 - 5*sin(x)^2)^(7/2), x)
\[ \int \left (4-5 \sin ^2(x)\right )^{7/2} \, dx=64 \left (\int \sqrt {-5 \sin \left (x \right )^{2}+4}d x \right )-125 \left (\int \sqrt {-5 \sin \left (x \right )^{2}+4}\, \sin \left (x \right )^{6}d x \right )+300 \left (\int \sqrt {-5 \sin \left (x \right )^{2}+4}\, \sin \left (x \right )^{4}d x \right )-240 \left (\int \sqrt {-5 \sin \left (x \right )^{2}+4}\, \sin \left (x \right )^{2}d x \right ) \] Input:
int((4-5*sin(x)^2)^(7/2),x)
Output:
64*int(sqrt( - 5*sin(x)**2 + 4),x) - 125*int(sqrt( - 5*sin(x)**2 + 4)*sin( x)**6,x) + 300*int(sqrt( - 5*sin(x)**2 + 4)*sin(x)**4,x) - 240*int(sqrt( - 5*sin(x)**2 + 4)*sin(x)**2,x)