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

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

[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))

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Rubi [A]  time = 0.195545, antiderivative size = 70, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0543274, size = 35, normalized size = 0.5 \[ -\frac{4 \sin (x) \cos (x) (8 \cos (2 x)-16 \cos (4 x)+15)}{63 \sqrt [4]{\sin ^{13}(x) \cos ^{11}(x)}} \]

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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Maple [F]  time = 0.29, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [4]{ \left ( \cos \left ( x \right ) \right ) ^{11} \left ( \sin \left ( x \right ) \right ) ^{13}}}}\, dx \end{align*}

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 = 1.51458, size = 104, normalized size = 1.49 \begin{align*} \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{align*}

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 = 3.10193, size = 336, normalized size = 4.8 \begin{align*} \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{align*}

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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\cos \left (x\right )^{11} \sin \left (x\right )^{13}\right )^{\frac{1}{4}}}\,{d x} \end{align*}

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)