Integrand size = 13, antiderivative size = 72 \[ \int \left (a-a \sin ^2(x)\right )^{7/2} \, dx=\frac {16}{35} a^3 \sqrt {a \cos ^2(x)} \tan (x)+\frac {8}{35} a^2 \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {6}{35} a \left (a \cos ^2(x)\right )^{5/2} \tan (x)+\frac {1}{7} \left (a \cos ^2(x)\right )^{7/2} \tan (x) \] Output:
16/35*a^3*(a*cos(x)^2)^(1/2)*tan(x)+8/35*a^2*(a*cos(x)^2)^(3/2)*tan(x)+6/3 5*a*(a*cos(x)^2)^(5/2)*tan(x)+1/7*(a*cos(x)^2)^(7/2)*tan(x)
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.54 \[ \int \left (a-a \sin ^2(x)\right )^{7/2} \, dx=-\frac {1}{35} a^3 \sqrt {a \cos ^2(x)} \left (-35+35 \sin ^2(x)-21 \sin ^4(x)+5 \sin ^6(x)\right ) \tan (x) \] Input:
Integrate[(a - a*Sin[x]^2)^(7/2),x]
Output:
-1/35*(a^3*Sqrt[a*Cos[x]^2]*(-35 + 35*Sin[x]^2 - 21*Sin[x]^4 + 5*Sin[x]^6) *Tan[x])
Time = 0.50 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {3042, 3655, 3042, 3682, 3042, 3682, 3042, 3682, 3042, 3686, 3042, 3117}
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.
| \(\displaystyle \int \left (a-a \sin ^2(x)\right )^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-a \sin (x)^2\right )^{7/2}dx\) |
| \(\Big \downarrow \) 3655 |
| \(\displaystyle \int \left (a \cos ^2(x)\right )^{7/2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (x+\frac {\pi }{2}\right )^2\right )^{7/2}dx\) |
| \(\Big \downarrow \) 3682 |
| \(\displaystyle \frac {6}{7} a \int \left (a \cos ^2(x)\right )^{5/2}dx+\frac {1}{7} \tan (x) \left (a \cos ^2(x)\right )^{7/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6}{7} a \int \left (a \sin \left (x+\frac {\pi }{2}\right )^2\right )^{5/2}dx+\frac {1}{7} \tan (x) \left (a \cos ^2(x)\right )^{7/2}\) |
| \(\Big \downarrow \) 3682 |
| \(\displaystyle \frac {6}{7} a \left (\frac {4}{5} a \int \left (a \cos ^2(x)\right )^{3/2}dx+\frac {1}{5} \tan (x) \left (a \cos ^2(x)\right )^{5/2}\right )+\frac {1}{7} \tan (x) \left (a \cos ^2(x)\right )^{7/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6}{7} a \left (\frac {4}{5} a \int \left (a \sin \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}dx+\frac {1}{5} \tan (x) \left (a \cos ^2(x)\right )^{5/2}\right )+\frac {1}{7} \tan (x) \left (a \cos ^2(x)\right )^{7/2}\) |
| \(\Big \downarrow \) 3682 |
| \(\displaystyle \frac {6}{7} a \left (\frac {4}{5} a \left (\frac {2}{3} a \int \sqrt {a \cos ^2(x)}dx+\frac {1}{3} \tan (x) \left (a \cos ^2(x)\right )^{3/2}\right )+\frac {1}{5} \tan (x) \left (a \cos ^2(x)\right )^{5/2}\right )+\frac {1}{7} \tan (x) \left (a \cos ^2(x)\right )^{7/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6}{7} a \left (\frac {4}{5} a \left (\frac {2}{3} a \int \sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2}dx+\frac {1}{3} \tan (x) \left (a \cos ^2(x)\right )^{3/2}\right )+\frac {1}{5} \tan (x) \left (a \cos ^2(x)\right )^{5/2}\right )+\frac {1}{7} \tan (x) \left (a \cos ^2(x)\right )^{7/2}\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle \frac {6}{7} a \left (\frac {4}{5} a \left (\frac {2}{3} a \sec (x) \sqrt {a \cos ^2(x)} \int \cos (x)dx+\frac {1}{3} \tan (x) \left (a \cos ^2(x)\right )^{3/2}\right )+\frac {1}{5} \tan (x) \left (a \cos ^2(x)\right )^{5/2}\right )+\frac {1}{7} \tan (x) \left (a \cos ^2(x)\right )^{7/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6}{7} a \left (\frac {4}{5} a \left (\frac {2}{3} a \sec (x) \sqrt {a \cos ^2(x)} \int \sin \left (x+\frac {\pi }{2}\right )dx+\frac {1}{3} \tan (x) \left (a \cos ^2(x)\right )^{3/2}\right )+\frac {1}{5} \tan (x) \left (a \cos ^2(x)\right )^{5/2}\right )+\frac {1}{7} \tan (x) \left (a \cos ^2(x)\right )^{7/2}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {1}{7} \tan (x) \left (a \cos ^2(x)\right )^{7/2}+\frac {6}{7} a \left (\frac {1}{5} \tan (x) \left (a \cos ^2(x)\right )^{5/2}+\frac {4}{5} a \left (\frac {1}{3} \tan (x) \left (a \cos ^2(x)\right )^{3/2}+\frac {2}{3} a \tan (x) \sqrt {a \cos ^2(x)}\right )\right )\) |
Input:
Int[(a - a*Sin[x]^2)^(7/2),x]
Output:
((a*Cos[x]^2)^(7/2)*Tan[x])/7 + (6*a*(((a*Cos[x]^2)^(5/2)*Tan[x])/5 + (4*a *((2*a*Sqrt[a*Cos[x]^2]*Tan[x])/3 + ((a*Cos[x]^2)^(3/2)*Tan[x])/3))/5))/7
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(a*cos[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a + b, 0]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-Cot[e + f*x ])*((b*Sin[e + f*x]^2)^p/(2*f*p)), x] + Simp[b*((2*p - 1)/(2*p)) Int[(b*S in[e + f*x]^2)^(p - 1), x], x] /; FreeQ[{b, e, f}, x] && !IntegerQ[p] && G tQ[p, 1]
Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Si n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 0.70 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.53
| method | result | size |
| default | \(-\frac {\cos \left (x \right ) a^{4} \sin \left (x \right ) \left (-5 \cos \left (x \right )^{6}-6 \cos \left (x \right )^{4}-8 \cos \left (x \right )^{2}-16\right )}{35 \sqrt {a \cos \left (x \right )^{2}}}\) | \(38\) |
| risch | \(-\frac {i a^{3} {\mathrm e}^{8 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{896 \left ({\mathrm e}^{2 i x}+1\right )}-\frac {35 i a^{3} {\mathrm e}^{2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{128 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {35 i a^{3} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{128 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {7 i a^{3} {\mathrm e}^{-2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{128 \left ({\mathrm e}^{2 i x}+1\right )}-\frac {11 i a^{3} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}\, \cos \left (6 x \right )}{1120 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {27 a^{3} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}\, \sin \left (6 x \right )}{2240 \left ({\mathrm e}^{2 i x}+1\right )}-\frac {7 i a^{3} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}\, \cos \left (4 x \right )}{160 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {21 a^{3} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}\, \sin \left (4 x \right )}{320 \left ({\mathrm e}^{2 i x}+1\right )}\) | \(295\) |
Input:
int((a-a*sin(x)^2)^(7/2),x,method=_RETURNVERBOSE)
Output:
-1/35*cos(x)*a^4*sin(x)*(-5*cos(x)^6-6*cos(x)^4-8*cos(x)^2-16)/(a*cos(x)^2 )^(1/2)
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.68 \[ \int \left (a-a \sin ^2(x)\right )^{7/2} \, dx=\frac {{\left (5 \, a^{3} \cos \left (x\right )^{6} + 6 \, a^{3} \cos \left (x\right )^{4} + 8 \, a^{3} \cos \left (x\right )^{2} + 16 \, a^{3}\right )} \sqrt {a \cos \left (x\right )^{2}} \sin \left (x\right )}{35 \, \cos \left (x\right )} \] Input:
integrate((a-a*sin(x)^2)^(7/2),x, algorithm="fricas")
Output:
1/35*(5*a^3*cos(x)^6 + 6*a^3*cos(x)^4 + 8*a^3*cos(x)^2 + 16*a^3)*sqrt(a*co s(x)^2)*sin(x)/cos(x)
Timed out. \[ \int \left (a-a \sin ^2(x)\right )^{7/2} \, dx=\text {Timed out} \] Input:
integrate((a-a*sin(x)**2)**(7/2),x)
Output:
Timed out
Time = 0.15 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.56 \[ \int \left (a-a \sin ^2(x)\right )^{7/2} \, dx=\frac {1}{2240} \, {\left (5 \, a^{3} \sin \left (7 \, x\right ) + 49 \, a^{3} \sin \left (5 \, x\right ) + 245 \, a^{3} \sin \left (3 \, x\right ) + 1225 \, a^{3} \sin \left (x\right )\right )} \sqrt {a} \] Input:
integrate((a-a*sin(x)^2)^(7/2),x, algorithm="maxima")
Output:
1/2240*(5*a^3*sin(7*x) + 49*a^3*sin(5*x) + 245*a^3*sin(3*x) + 1225*a^3*sin (x))*sqrt(a)
Time = 0.52 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.54 \[ \int \left (a-a \sin ^2(x)\right )^{7/2} \, dx=-\frac {2 \, {\left (35 \, a^{\frac {7}{2}} {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{6} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right ) - 140 \, a^{\frac {7}{2}} {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{4} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right ) + 336 \, a^{\frac {7}{2}} {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right ) - 320 \, a^{\frac {7}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right )\right )}}{35 \, {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{7}} \] Input:
integrate((a-a*sin(x)^2)^(7/2),x, algorithm="giac")
Output:
-2/35*(35*a^(7/2)*(1/tan(1/2*x) + tan(1/2*x))^6*sgn(tan(1/2*x)^4 - 1) - 14 0*a^(7/2)*(1/tan(1/2*x) + tan(1/2*x))^4*sgn(tan(1/2*x)^4 - 1) + 336*a^(7/2 )*(1/tan(1/2*x) + tan(1/2*x))^2*sgn(tan(1/2*x)^4 - 1) - 320*a^(7/2)*sgn(ta n(1/2*x)^4 - 1))/(1/tan(1/2*x) + tan(1/2*x))^7
Timed out. \[ \int \left (a-a \sin ^2(x)\right )^{7/2} \, dx=\int {\left (a-a\,{\sin \left (x\right )}^2\right )}^{7/2} \,d x \] Input:
int((a - a*sin(x)^2)^(7/2),x)
Output:
int((a - a*sin(x)^2)^(7/2), x)
\[ \int \left (a-a \sin ^2(x)\right )^{7/2} \, dx=\sqrt {a}\, a^{3} \left (\int \sqrt {-\sin \left (x \right )^{2}+1}d x -\left (\int \sqrt {-\sin \left (x \right )^{2}+1}\, \sin \left (x \right )^{6}d x \right )+3 \left (\int \sqrt {-\sin \left (x \right )^{2}+1}\, \sin \left (x \right )^{4}d x \right )-3 \left (\int \sqrt {-\sin \left (x \right )^{2}+1}\, \sin \left (x \right )^{2}d x \right )\right ) \] Input:
int((a-a*sin(x)^2)^(7/2),x)
Output:
sqrt(a)*a**3*(int(sqrt( - sin(x)**2 + 1),x) - int(sqrt( - sin(x)**2 + 1)*s in(x)**6,x) + 3*int(sqrt( - sin(x)**2 + 1)*sin(x)**4,x) - 3*int(sqrt( - si n(x)**2 + 1)*sin(x)**2,x))