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

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

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Rubi [A]  time = 0.20, 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 = {6719, 270} \[ -\frac {4 \sin (x) \cos ^5(x)}{9 \sqrt [4]{\sin ^{13}(x) \cos ^{11}(x)}}-\frac {8 \sin ^3(x) \cos ^3(x)}{\sqrt [4]{\sin ^{13}(x) \cos ^{11}(x)}}+\frac {4 \sin ^5(x) \cos (x)}{7 \sqrt [4]{\sin ^{13}(x) \cos ^{11}(x)}} \]

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 270

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 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[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 &=\operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.06, size = 35, normalized size = 0.50 \[ -\frac {4 \sin (x) \cos (x) (8 \cos (2 x)-16 \cos (4 x)+15)}{63 \sqrt [4]{\sin ^{13}(x) \cos ^{11}(x)}} \]

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

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 0.88, size = 101, normalized size = 1.44 \[ \frac {4 \, {\left (128 \, \cos \relax (x)^{4} - 144 \, \cos \relax (x)^{2} + 9\right )} \left ({\left (\cos \relax (x)^{23} - 6 \, \cos \relax (x)^{21} + 15 \, \cos \relax (x)^{19} - 20 \, \cos \relax (x)^{17} + 15 \, \cos \relax (x)^{15} - 6 \, \cos \relax (x)^{13} + \cos \relax (x)^{11}\right )} \sin \relax (x)\right )^{\frac {3}{4}}}{63 \, {\left (\cos \relax (x)^{22} - 6 \, \cos \relax (x)^{20} + 15 \, \cos \relax (x)^{18} - 20 \, \cos \relax (x)^{16} + 15 \, \cos \relax (x)^{14} - 6 \, \cos \relax (x)^{12} + \cos \relax (x)^{10}\right )}} \]

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

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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maple [F]  time = 0.44, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (\left (\cos ^{11}\relax (x )\right ) \left (\sin ^{13}\relax (x )\right )\right )^{\frac {1}{4}}}\, dx\]

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 = 1.01, size = 77, normalized size = 1.10 \[ \frac {4}{23} \, \tan \relax (x)^{\frac {23}{4}} + \frac {8}{15} \, \tan \relax (x)^{\frac {15}{4}} + \frac {4}{7} \, \tan \relax (x)^{\frac {7}{4}} - \frac {4 \, {\left (35 \, \tan \relax (x)^{7} + 161 \, \tan \relax (x)^{5} + 345 \, \tan \relax (x)^{3} - 805 \, \tan \relax (x)\right )}}{805 \, \tan \relax (x)^{\frac {5}{4}}} + \frac {4 \, {\left (21 \, \tan \relax (x)^{7} + 135 \, \tan \relax (x)^{5} - 945 \, \tan \relax (x)^{3} - 35 \, \tan \relax (x)\right )}}{315 \, \tan \relax (x)^{\frac {13}{4}}} \]

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

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

[Out]

Timed out

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