3.109 \(\int \frac{\csc ^2(c+d x)}{\sqrt [3]{a+a \sin (c+d x)}} \, dx\)

Optimal. Leaf size=77 \[ -\frac{\sqrt [6]{2} \cos (c+d x) F_1\left (\frac{1}{2};2,\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right )}{d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]

[Out]

-((2^(1/6)*AppellF1[1/2, 2, 5/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x])/(d*(1 + Sin[c + d*
x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))

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Rubi [A]  time = 0.119338, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2787, 2785, 130, 429} \[ -\frac{\sqrt [6]{2} \cos (c+d x) F_1\left (\frac{1}{2};2,\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right )}{d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(1/3),x]

[Out]

-((2^(1/6)*AppellF1[1/2, 2, 5/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x])/(d*(1 + Sin[c + d*
x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))

Rule 2787

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[(a^In
tPart[m]*(a + b*Sin[e + f*x])^FracPart[m])/(1 + (b*Sin[e + f*x])/a)^FracPart[m], Int[(1 + (b*Sin[e + f*x])/a)^
m*(d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m] &&  !GtQ
[a, 0]

Rule 2785

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Dist[(b*(d
/b)^n*Cos[e + f*x])/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]]), Subst[Int[((a - x)^n*(2*a - x)^(m -
1/2))/Sqrt[x], x], x, a - b*Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !In
tegerQ[m] && GtQ[a, 0] && GtQ[d/b, 0]

Rule 130

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + (b*x^k)/e)^m*(c + (d*x^k)/e)^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{\csc ^2(c+d x)}{\sqrt [3]{a+a \sin (c+d x)}} \, dx &=\frac{\sqrt [3]{1+\sin (c+d x)} \int \frac{\csc ^2(c+d x)}{\sqrt [3]{1+\sin (c+d x)}} \, dx}{\sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac{\cos (c+d x) \operatorname{Subst}\left (\int \frac{1}{(1-x)^2 (2-x)^{5/6} \sqrt{x}} \, dx,x,1-\sin (c+d x)\right )}{d \sqrt{1-\sin (c+d x)} \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac{(2 \cos (c+d x)) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2 \left (2-x^2\right )^{5/6}} \, dx,x,\sqrt{1-\sin (c+d x)}\right )}{d \sqrt{1-\sin (c+d x)} \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac{\sqrt [6]{2} F_1\left (\frac{1}{2};2,\frac{5}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right ) \cos (c+d x)}{d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 8.7488, size = 184, normalized size = 2.39 \[ \frac{2\ 2^{2/3} \cos ^{\frac{2}{3}}\left (\frac{1}{4} (2 c+2 d x-\pi )\right ) (\cos (c+d x)+i \sin (c+d x)) \left (4 i \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-i e^{-i (c+d x)}\right ) \cos (c+d x) (\sin (c+d x)+i \cos (c+d x)+1)^{2/3}+4 \sin (c+d x)+1\right )}{5 d \left (-(-1)^{3/4} e^{-\frac{1}{2} i (c+d x)} \left (e^{i (c+d x)}+i\right )\right )^{2/3} \left (1+e^{2 i (c+d x)}\right ) \sqrt [3]{a (\sin (c+d x)+1)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(1/3),x]

[Out]

(2*2^(2/3)*Cos[(2*c - Pi + 2*d*x)/4]^(2/3)*(Cos[c + d*x] + I*Sin[c + d*x])*(1 + 4*Sin[c + d*x] + (4*I)*Cos[c +
d*x]*Hypergeometric2F1[1/3, 2/3, 4/3, (-I)/E^(I*(c + d*x))]*(1 + I*Cos[c + d*x] + Sin[c + d*x])^(2/3)))/(5*d*
(-(((-1)^(3/4)*(I + E^(I*(c + d*x))))/E^((I/2)*(c + d*x))))^(2/3)*(1 + E^((2*I)*(c + d*x)))*(a*(1 + Sin[c + d*
x]))^(1/3))

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Maple [F]  time = 0.105, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \csc \left ( dx+c \right ) \right ) ^{2}{\frac{1}{\sqrt [3]{a+a\sin \left ( dx+c \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^2/(a+a*sin(d*x+c))^(1/3),x)

[Out]

int(csc(d*x+c)^2/(a+a*sin(d*x+c))^(1/3),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2/(a+a*sin(d*x+c))^(1/3),x, algorithm="maxima")

[Out]

integrate(csc(d*x + c)^2/(a*sin(d*x + c) + 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(csc(d*x+c)^2/(a+a*sin(d*x+c))^(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{\csc ^{2}{\left (c + d x \right )}}{\sqrt [3]{a \left (\sin{\left (c + d x \right )} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**2/(a+a*sin(d*x+c))**(1/3),x)

[Out]

Integral(csc(c + d*x)**2/(a*(sin(c + d*x) + 1))**(1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2/(a+a*sin(d*x+c))^(1/3),x, algorithm="giac")

[Out]

integrate(csc(d*x + c)^2/(a*sin(d*x + c) + a)^(1/3), x)