3.5.15 \(\int \frac {1}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \, dx\) [415]

Optimal. Leaf size=70 \[ -\frac {4 \cos ^5(x) \sin (x)}{9 \sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}}-\frac {8 \cos ^3(x) \sin ^3(x)}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}}+\frac {4 \cos (x) \sin ^5(x)}{7 \sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \]

[Out]

-4/9*cos(x)^5*sin(x)/(cos(x)^11*sin(x)^13)^(1/4)-8*cos(x)^3*sin(x)^3/(cos(x)^11*sin(x)^13)^(1/4)+4/7*cos(x)*si
n(x)^5/(cos(x)^11*sin(x)^13)^(1/4)

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Rubi [A]
time = 0.14, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6851, 276} \begin {gather*} \frac {4 \sin ^5(x) \cos (x)}{7 \sqrt [4]{\sin ^{13}(x) \cos ^{11}(x)}}-\frac {4 \sin (x) \cos ^5(x)}{9 \sqrt [4]{\sin ^{13}(x) \cos ^{11}(x)}}-\frac {8 \sin ^3(x) \cos ^3(x)}{\sqrt [4]{\sin ^{13}(x) \cos ^{11}(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Cos[x]^11*Sin[x]^13)^(-1/4),x]

[Out]

(-4*Cos[x]^5*Sin[x])/(9*(Cos[x]^11*Sin[x]^13)^(1/4)) - (8*Cos[x]^3*Sin[x]^3)/(Cos[x]^11*Sin[x]^13)^(1/4) + (4*
Cos[x]*Sin[x]^5)/(7*(Cos[x]^11*Sin[x]^13)^(1/4))

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[p])*w^(n*FracPart[p]))), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \, dx &=\text {Subst}\left (\int \frac {1}{\sqrt [4]{\frac {x^{13}}{\left (1+x^2\right )^{12}}} \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=\frac {\left (\cos ^6(x) \tan ^{\frac {13}{4}}(x)\right ) \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^{13/4}} \, dx,x,\tan (x)\right )}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}}\\ &=\frac {\left (\cos ^6(x) \tan ^{\frac {13}{4}}(x)\right ) \text {Subst}\left (\int \left (\frac {1}{x^{13/4}}+\frac {2}{x^{5/4}}+x^{3/4}\right ) \, dx,x,\tan (x)\right )}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}}\\ &=-\frac {4 \cos ^5(x) \sin (x)}{9 \sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}}-\frac {8 \cos ^3(x) \sin ^3(x)}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}}+\frac {4 \cos (x) \sin ^5(x)}{7 \sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 35, normalized size = 0.50 \begin {gather*} -\frac {4 \cos (x) (15+8 \cos (2 x)-16 \cos (4 x)) \sin (x)}{63 \sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Cos[x]^11*Sin[x]^13)^(-1/4),x]

[Out]

(-4*Cos[x]*(15 + 8*Cos[2*x] - 16*Cos[4*x])*Sin[x])/(63*(Cos[x]^11*Sin[x]^13)^(1/4))

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Maple [F]
time = 0.28, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (\left (\cos ^{11}\left (x \right )\right ) \left (\sin ^{13}\left (x \right )\right )\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^11*sin(x)^13)^(1/4),x)

[Out]

int(1/(cos(x)^11*sin(x)^13)^(1/4),x)

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Maxima [A]
time = 1.59, size = 77, normalized size = 1.10 \begin {gather*} \frac {4}{23} \, \tan \left (x\right )^{\frac {23}{4}} + \frac {8}{15} \, \tan \left (x\right )^{\frac {15}{4}} + \frac {4}{7} \, \tan \left (x\right )^{\frac {7}{4}} - \frac {4 \, {\left (35 \, \tan \left (x\right )^{7} + 161 \, \tan \left (x\right )^{5} + 345 \, \tan \left (x\right )^{3} - 805 \, \tan \left (x\right )\right )}}{805 \, \tan \left (x\right )^{\frac {5}{4}}} + \frac {4 \, {\left (21 \, \tan \left (x\right )^{7} + 135 \, \tan \left (x\right )^{5} - 945 \, \tan \left (x\right )^{3} - 35 \, \tan \left (x\right )\right )}}{315 \, \tan \left (x\right )^{\frac {13}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)^11*sin(x)^13)^(1/4),x, algorithm="maxima")

[Out]

4/23*tan(x)^(23/4) + 8/15*tan(x)^(15/4) + 4/7*tan(x)^(7/4) - 4/805*(35*tan(x)^7 + 161*tan(x)^5 + 345*tan(x)^3
- 805*tan(x))/tan(x)^(5/4) + 4/315*(21*tan(x)^7 + 135*tan(x)^5 - 945*tan(x)^3 - 35*tan(x))/tan(x)^(13/4)

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Fricas [A]
time = 1.50, size = 101, normalized size = 1.44 \begin {gather*} \frac {4 \, {\left (128 \, \cos \left (x\right )^{4} - 144 \, \cos \left (x\right )^{2} + 9\right )} \left ({\left (\cos \left (x\right )^{23} - 6 \, \cos \left (x\right )^{21} + 15 \, \cos \left (x\right )^{19} - 20 \, \cos \left (x\right )^{17} + 15 \, \cos \left (x\right )^{15} - 6 \, \cos \left (x\right )^{13} + \cos \left (x\right )^{11}\right )} \sin \left (x\right )\right )^{\frac {3}{4}}}{63 \, {\left (\cos \left (x\right )^{22} - 6 \, \cos \left (x\right )^{20} + 15 \, \cos \left (x\right )^{18} - 20 \, \cos \left (x\right )^{16} + 15 \, \cos \left (x\right )^{14} - 6 \, \cos \left (x\right )^{12} + \cos \left (x\right )^{10}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)^11*sin(x)^13)^(1/4),x, algorithm="fricas")

[Out]

4/63*(128*cos(x)^4 - 144*cos(x)^2 + 9)*((cos(x)^23 - 6*cos(x)^21 + 15*cos(x)^19 - 20*cos(x)^17 + 15*cos(x)^15
- 6*cos(x)^13 + cos(x)^11)*sin(x))^(3/4)/(cos(x)^22 - 6*cos(x)^20 + 15*cos(x)^18 - 20*cos(x)^16 + 15*cos(x)^14
- 6*cos(x)^12 + cos(x)^10)

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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(1/(cos(x)**11*sin(x)**13)**(1/4),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(1/(cos(x)^11*sin(x)^13)^(1/4),x, algorithm="giac")

[Out]

integrate((cos(x)^11*sin(x)^13)^(-1/4), x)

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Mupad [B]
time = 3.55, size = 110, normalized size = 1.57 \begin {gather*} -\frac {2^{3/4}\,\left (-32\,{\cos \left (2\,x\right )}^2+8\,\cos \left (2\,x\right )+31\right )\,{\left (924\,\sin \left (2\,x\right )-132\,\sin \left (4\,x\right )-660\,\sin \left (6\,x\right )+165\,\sin \left (8\,x\right )+330\,\sin \left (10\,x\right )-110\,\sin \left (12\,x\right )-110\,\sin \left (14\,x\right )+44\,\sin \left (16\,x\right )+22\,\sin \left (18\,x\right )-10\,\sin \left (20\,x\right )-2\,\sin \left (22\,x\right )+\sin \left (24\,x\right )\right )}^{3/4}}{2016\,{\left (\cos \left (2\,x\right )-1\right )}^6\,{\left (\cos \left (2\,x\right )+1\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^11*sin(x)^13)^(1/4),x)

[Out]

-(2^(3/4)*(8*cos(2*x) - 32*cos(2*x)^2 + 31)*(924*sin(2*x) - 132*sin(4*x) - 660*sin(6*x) + 165*sin(8*x) + 330*s
in(10*x) - 110*sin(12*x) - 110*sin(14*x) + 44*sin(16*x) + 22*sin(18*x) - 10*sin(20*x) - 2*sin(22*x) + sin(24*x
))^(3/4))/(2016*(cos(2*x) - 1)^6*(cos(2*x) + 1)^5)

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