3.407 \(\int \frac {\cos ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx\)

Optimal. Leaf size=61 \[ -\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {1}{16} \sin ^{-1}(\cos (x)-\sin (x))+\frac {\cos (x)}{4 \sqrt {\sin (2 x)}}-\frac {1}{16} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4293, 4307, 4306} \[ -\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {1}{16} \sin ^{-1}(\cos (x)-\sin (x))+\frac {\cos (x)}{4 \sqrt {\sin (2 x)}}-\frac {1}{16} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^7/Sin[2*x]^(7/2),x]

[Out]

-ArcSin[Cos[x] - Sin[x]]/16 - Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]/16 - Cos[x]^5/(5*Sin[2*x]^(5/2)) + Cos[x]/
(4*Sqrt[Sin[2*x]])

Rule 4293

Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[(e^2*(e*Cos[a
+ b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(p + 1)), x] + Dist[(e^4*(m + p - 1))/(4*g^2*(p + 1)), Int[(
e*Cos[a + b*x])^(m - 4)*(g*Sin[c + d*x])^(p + 2), x], x] /; FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0]
&& EqQ[d/b, 2] &&  !IntegerQ[p] && GtQ[m, 2] && LtQ[p, -1] && (GtQ[m, 3] || EqQ[p, -3/2]) && IntegersQ[2*m, 2*
p]

Rule 4306

Int[sin[(a_.) + (b_.)*(x_)]/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> -Simp[ArcSin[Cos[a + b*x] - Sin[a + b*
x]]/d, x] - Simp[Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[c + d*x]]]/d, x] /; FreeQ[{a, b, c, d}, x] && EqQ[
b*c - a*d, 0] && EqQ[d/b, 2]

Rule 4307

Int[((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_)/cos[(a_.) + (b_.)*(x_)], x_Symbol] :> Dist[2*g, Int[Sin[a + b*x]*(g*S
in[c + d*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] &&  !IntegerQ
[p] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {\cos ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx &=-\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {1}{4} \int \frac {\cos ^3(x)}{\sin ^{\frac {3}{2}}(2 x)} \, dx\\ &=-\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}+\frac {\cos (x)}{4 \sqrt {\sin (2 x)}}+\frac {1}{16} \int \sec (x) \sqrt {\sin (2 x)} \, dx\\ &=-\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}+\frac {\cos (x)}{4 \sqrt {\sin (2 x)}}+\frac {1}{8} \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx\\ &=-\frac {1}{16} \sin ^{-1}(\cos (x)-\sin (x))-\frac {1}{16} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )-\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}+\frac {\cos (x)}{4 \sqrt {\sin (2 x)}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 56, normalized size = 0.92 \[ \sqrt {\sin (2 x)} \left (\frac {3 \csc (x)}{20}-\frac {\csc ^3(x)}{40}\right )+\frac {1}{16} \left (-\sin ^{-1}(\cos (x)-\sin (x))-\log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^7/Sin[2*x]^(7/2),x]

[Out]

(-ArcSin[Cos[x] - Sin[x]] - Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]])/16 + ((3*Csc[x])/20 - Csc[x]^3/40)*Sqrt[Sin
[2*x]]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cos[x]^7/Sin[2*x]^(7/2),x]

[Out]

Could not integrate

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fricas [B]  time = 1.02, size = 205, normalized size = 3.36 \[ \frac {10 \, {\left (\cos \relax (x)^{2} - 1\right )} \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} {\left (\cos \relax (x) - \sin \relax (x)\right )} + \cos \relax (x) \sin \relax (x)}{\cos \relax (x)^{2} + 2 \, \cos \relax (x) \sin \relax (x) - 1}\right ) \sin \relax (x) - 10 \, {\left (\cos \relax (x)^{2} - 1\right )} \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} - \cos \relax (x) - \sin \relax (x)}{\cos \relax (x) - \sin \relax (x)}\right ) \sin \relax (x) + 5 \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (-32 \, \cos \relax (x)^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \relax (x)^{3} - {\left (4 \, \cos \relax (x)^{2} + 1\right )} \sin \relax (x) - 5 \, \cos \relax (x)\right )} \sqrt {\cos \relax (x) \sin \relax (x)} + 32 \, \cos \relax (x)^{2} + 16 \, \cos \relax (x) \sin \relax (x) + 1\right ) \sin \relax (x) + 8 \, \sqrt {2} {\left (6 \, \cos \relax (x)^{2} - 5\right )} \sqrt {\cos \relax (x) \sin \relax (x)} + 48 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)}{320 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^7/sin(2*x)^(7/2),x, algorithm="fricas")

[Out]

1/320*(10*(cos(x)^2 - 1)*arctan(-(sqrt(2)*sqrt(cos(x)*sin(x))*(cos(x) - sin(x)) + cos(x)*sin(x))/(cos(x)^2 + 2
*cos(x)*sin(x) - 1))*sin(x) - 10*(cos(x)^2 - 1)*arctan(-(2*sqrt(2)*sqrt(cos(x)*sin(x)) - cos(x) - sin(x))/(cos
(x) - sin(x)))*sin(x) + 5*(cos(x)^2 - 1)*log(-32*cos(x)^4 + 4*sqrt(2)*(4*cos(x)^3 - (4*cos(x)^2 + 1)*sin(x) -
5*cos(x))*sqrt(cos(x)*sin(x)) + 32*cos(x)^2 + 16*cos(x)*sin(x) + 1)*sin(x) + 8*sqrt(2)*(6*cos(x)^2 - 5)*sqrt(c
os(x)*sin(x)) + 48*(cos(x)^2 - 1)*sin(x))/((cos(x)^2 - 1)*sin(x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \relax (x)^{7}}{\sin \left (2 \, x\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^7/sin(2*x)^(7/2),x, algorithm="giac")

[Out]

integrate(cos(x)^7/sin(2*x)^(7/2), x)

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maple [C]  time = 0.26, size = 1108, normalized size = 18.16

method	result	size
default	\(\frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (192 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticE \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{6}\left (\frac {x}{2}\right )\right )-96 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{6}\left (\frac {x}{2}\right )\right )-\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{10}\left (\frac {x}{2}\right )\right )-384 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticE \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+192 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+96 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \left (\tan ^{8}\left (\frac {x}{2}\right )\right )+3 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{8}\left (\frac {x}{2}\right )\right )+48 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{8}\left (\frac {x}{2}\right )\right )+192 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticE \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-96 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-192 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+14 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{6}\left (\frac {x}{2}\right )\right )-144 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+96 \left (\tan ^{4}\left (\frac {x}{2}\right )\right ) \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}+14 \left (\tan ^{4}\left (\frac {x}{2}\right )\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}+144 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+3 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-48 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\right )}{160 \tan \left (\frac {x}{2}\right )^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\, \left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\)	\(1108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^7/sin(2*x)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/160*(-tan(1/2*x)/(tan(1/2*x)^2-1))^(1/2)/tan(1/2*x)^3*(192*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)+1
)^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticE((tan(1/2*x)+1)^(1/2),1/2*2^(1/2))*(tan(1/2*x)*(t
an(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan(1/2*x)^6-96*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)+1)^(1/2)*(-
2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((tan(1/2*x)+1)^(1/2),1/2*2^(1/2))*(tan(1/2*x)*(tan(1/2*x)-
1)*(tan(1/2*x)+1))^(1/2)*tan(1/2*x)^6-(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*
x)+1))^(1/2)*tan(1/2*x)^10-384*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)+1)^(1/2)*(-2*tan(1/2*x)+2)^(1/2
)*(-tan(1/2*x))^(1/2)*EllipticE((tan(1/2*x)+1)^(1/2),1/2*2^(1/2))*(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(
1/2)*tan(1/2*x)^4+192*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)+1)^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1
/2*x))^(1/2)*EllipticF((tan(1/2*x)+1)^(1/2),1/2*2^(1/2))*(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan(
1/2*x)^4+96*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^3-tan(1/2*x))^(1/2)*tan(1/2*x)^8+3*(tan(1/2*x)*(ta
n(1/2*x)^2-1))^(1/2)*(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan(1/2*x)^8+48*(tan(1/2*x)^3-tan(1/2*x)
)^(1/2)*(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan(1/2*x)^8+192*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*
(tan(1/2*x)+1)^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticE((tan(1/2*x)+1)^(1/2),1/2*2^(1/2))*(
tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan(1/2*x)^2-96*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x
)+1)^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((tan(1/2*x)+1)^(1/2),1/2*2^(1/2))*(tan(1/2*x)
*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan(1/2*x)^2-192*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^3-tan(1
/2*x))^(1/2)*tan(1/2*x)^6+14*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1
/2)*tan(1/2*x)^6-144*(tan(1/2*x)^3-tan(1/2*x))^(1/2)*(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan(1/2*
x)^6+96*tan(1/2*x)^4*(tan(1/2*x)^3-tan(1/2*x))^(1/2)*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)+14*tan(1/2*x)^4*(tan(
1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)+144*(tan(1/2*x)^3-tan(1/2*x))^
(1/2)*(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan(1/2*x)^4+3*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan
(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan(1/2*x)^2-48*(tan(1/2*x)^3-tan(1/2*x))^(1/2)*(tan(1/2*x)*(tan(
1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*tan(1/2*x)^2-(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1))^(1/2)*(tan(1/2*x)*(tan
(1/2*x)^2-1))^(1/2))/(tan(1/2*x)^2-1)/(tan(1/2*x)^3-tan(1/2*x))^(1/2)/(tan(1/2*x)*(tan(1/2*x)-1)*(tan(1/2*x)+1
))^(1/2)/(tan(1/2*x)-1)/(tan(1/2*x)+1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \relax (x)^{7}}{\sin \left (2 \, x\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^7/sin(2*x)^(7/2),x, algorithm="maxima")

[Out]

integrate(cos(x)^7/sin(2*x)^(7/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\cos \relax (x)}^7}{{\sin \left (2\,x\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^7/sin(2*x)^(7/2),x)

[Out]

int(cos(x)^7/sin(2*x)^(7/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**7/sin(2*x)**(7/2),x)

[Out]

Timed out

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