3.5.6 \(\int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx\) [406]

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

[Out]

-1/16*arcsin(cos(x)-sin(x))+1/16*ln(cos(x)+sin(x)+sin(2*x)^(1/2))+1/5*sin(x)^5/sin(2*x)^(5/2)-1/4*sin(x)/sin(2
*x)^(1/2)

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Rubi [A]
time = 0.06, 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 = {4379, 4393, 4390} \begin {gather*} -\frac {1}{16} \text {ArcSin}(\cos (x)-\sin (x))+\frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {\sin (x)}{4 \sqrt {\sin (2 x)}}+\frac {1}{16} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]/16 + Sin[x]^5/(5*Sin[2*x]^(5/2)) - Sin[x
]/(4*Sqrt[Sin[2*x]])

Rule 4379

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[(-e^2)*(e*Sin
[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))), In
t[(e*Sin[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 4390

Int[cos[(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 4393

Int[((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_)/sin[(a_.) + (b_.)*(x_)], x_Symbol] :> Dist[2*g, Int[Cos[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 {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx &=\frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {1}{4} \int \frac {\sin ^3(x)}{\sin ^{\frac {3}{2}}(2 x)} \, dx\\ &=\frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {\sin (x)}{4 \sqrt {\sin (2 x)}}+\frac {1}{16} \int \csc (x) \sqrt {\sin (2 x)} \, dx\\ &=\frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {\sin (x)}{4 \sqrt {\sin (2 x)}}+\frac {1}{8} \int \frac {\cos (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 {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {\sin (x)}{4 \sqrt {\sin (2 x)}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 50, normalized size = 0.82 \begin {gather*} \frac {1}{80} \left (5 \left (-\sin ^{-1}(\cos (x)-\sin (x))+\log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )\right )+2 \sec (x) \left (-6+\sec ^2(x)\right ) \sqrt {\sin (2 x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(-ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]) + 2*Sec[x]*(-6 + Sec[x]^2)*Sqrt[Sin[2*x]
])/80

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.14, size = 510, normalized size = 8.36

method	result	size
default	\(\frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \left (5 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{14}\left (\frac {x}{2}\right )\right )+35 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{12}\left (\frac {x}{2}\right )\right )+10 \left (\tan ^{15}\left (\frac {x}{2}\right )\right )+105 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{10}\left (\frac {x}{2}\right )\right )+66 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )+175 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{8}\left (\frac {x}{2}\right )\right )-1014 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )+175 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+2002 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )+105 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{4}\left (\frac {x}{2}\right )\right )-2002 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )+35 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+1014 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )+5 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right )-66 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )-10 \tan \left (\frac {x}{2}\right )\right )}{2688 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{7} \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}}\)	\(510\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2688*(-tan(1/2*x)/(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^2-1)*(5*(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(
-tan(1/2*x))^(1/2)*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*tan(1/2*x)^14+35*(1+tan(1/2*x))^(1/2)*(-2*tan(1
/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*tan(1/2*x)^12+10*tan(1/2*x)^15+
105*(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2
))*tan(1/2*x)^10+66*tan(1/2*x)^13+175*(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*Ellipti
cF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*tan(1/2*x)^8-1014*tan(1/2*x)^11+175*(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2
)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*tan(1/2*x)^6+2002*tan(1/2*x)^9+105*(1+
tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*tan(
1/2*x)^4-2002*tan(1/2*x)^7+35*(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((1+ta
n(1/2*x))^(1/2),1/2*2^(1/2))*tan(1/2*x)^2+1014*tan(1/2*x)^5+5*(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-t
an(1/2*x))^(1/2)*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))-66*tan(1/2*x)^3-10*tan(1/2*x))/(tan(1/2*x)*(tan(1
/2*x)^2-1))^(1/2)/(1+tan(1/2*x)^2)^7/(tan(1/2*x)^3-tan(1/2*x))^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(x)^7/sin(2*x)^(7/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (47) = 94\).
time = 1.23, size = 181, normalized size = 2.97 \begin {gather*} \frac {10 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} + \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) - 1}\right ) \cos \left (x\right )^{3} - 10 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} - \cos \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right ) - \sin \left (x\right )}\right ) \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{3} \log \left (-32 \, \cos \left (x\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (x\right )^{3} - {\left (4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )} + 32 \, \cos \left (x\right )^{2} + 16 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) - 48 \, \cos \left (x\right )^{3} - 8 \, \sqrt {2} {\left (6 \, \cos \left (x\right )^{2} - 1\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )}}{320 \, \cos \left (x\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/320*(10*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))*cos(x)^3 - 10*arctan(-(2*sqrt(2)*sqrt(cos(x)*sin(x)) - cos(x) - sin(x))/(cos(x) - sin(x)))*cos(x)^3 - 5*
cos(x)^3*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) - 48*cos(x)^3 - 8*sqrt(2)*(6*cos(x)^2 - 1)*sqrt(cos(x)*sin(x)))/cos(x)^3

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(x)^7/sin(2*x)^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\sin \left (x\right )}^7}{{\sin \left (2\,x\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sin(x)^7/sin(2*x)^(7/2), x)

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