3.121 \(\int \frac{\cot ^2(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx\)

Optimal. Leaf size=80 \[ \frac{6 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{5/3} F_1\left (\frac{7}{6};-\frac{1}{2},2;\frac{13}{6};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )}{7 a^2 f} \]

[Out]

(6*Sqrt[2]*AppellF1[7/6, -1/2, 2, 13/6, (1 + Sin[e + f*x])/2, 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e +
f*x]]*(a + a*Sin[e + f*x])^(5/3))/(7*a^2*f)

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Rubi [A]  time = 0.0949979, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2719, 137, 136} \[ \frac{6 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{5/3} F_1\left (\frac{7}{6};-\frac{1}{2},2;\frac{13}{6};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )}{7 a^2 f} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^2/(a + a*Sin[e + f*x])^(1/3),x]

[Out]

(6*Sqrt[2]*AppellF1[7/6, -1/2, 2, 13/6, (1 + Sin[e + f*x])/2, 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e +
f*x]]*(a + a*Sin[e + f*x])^(5/3))/(7*a^2*f)

Rule 2719

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_), x_Symbol] :> Dist[(Sqrt[a + b*Si
n[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])/(b*f*Cos[e + f*x]), Subst[Int[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p
+ 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m] && Inte
gerQ[p/2]

Rule 137

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^
FracPart[n]/((b/(b*c - a*d))^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*((b*c)/(b*c
- a*d) + (b*d*x)/(b*c - a*d))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&
!IntegerQ[n] && IntegerQ[p] &&  !GtQ[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x]

Rule 136

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*e - a*
f)^p*(a + b*x)^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f
))])/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rubi steps

\begin{align*} \int \frac{\cot ^2(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx &=\frac{\left (\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a-x} \sqrt [6]{a+x}}{x^2} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{\left (\sqrt{2} \sec (e+f x) (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [6]{a+x} \sqrt{\frac{1}{2}-\frac{x}{2 a}}}{x^2} \, dx,x,a \sin (e+f x)\right )}{a f \sqrt{\frac{a-a \sin (e+f x)}{a}}}\\ &=\frac{6 \sqrt{2} F_1\left (\frac{7}{6};-\frac{1}{2},2;\frac{13}{6};\frac{1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sec (e+f x) \sqrt{1-\sin (e+f x)} (a+a \sin (e+f x))^{5/3}}{7 a^2 f}\\ \end{align*}

Mathematica [F]  time = 12.7663, size = 0, normalized size = 0. \[ \int \frac{\cot ^2(e+f x)}{\sqrt [3]{a+a \sin (e+f x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cot[e + f*x]^2/(a + a*Sin[e + f*x])^(1/3),x]

[Out]

Integrate[Cot[e + f*x]^2/(a + a*Sin[e + f*x])^(1/3), x]

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cot \left ( fx+e \right ) \right ) ^{2}{\frac{1}{\sqrt [3]{a+a\sin \left ( fx+e \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^2/(a+a*sin(f*x+e))^(1/3),x)

[Out]

int(cot(f*x+e)^2/(a+a*sin(f*x+e))^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2/(a+a*sin(f*x+e))^(1/3),x, algorithm="maxima")

[Out]

integrate(cot(f*x + e)^2/(a*sin(f*x + e) + a)^(1/3), x)

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Fricas [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(cot(f*x+e)^2/(a+a*sin(f*x+e))^(1/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{2}{\left (e + f x \right )}}{\sqrt [3]{a \left (\sin{\left (e + f x \right )} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**2/(a+a*sin(f*x+e))**(1/3),x)

[Out]

Integral(cot(e + f*x)**2/(a*(sin(e + f*x) + 1))**(1/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2/(a+a*sin(f*x+e))^(1/3),x, algorithm="giac")

[Out]

integrate(cot(f*x + e)^2/(a*sin(f*x + e) + a)^(1/3), x)