3.96 \(\int \csc (c+d x) (a+a \sin (c+d x))^{2/3} \, dx\)

Optimal. Leaf size=77 \[ -\frac{2 \sqrt [6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} F_1\left (\frac{1}{2};1,-\frac{1}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right )}{d (\sin (c+d x)+1)^{7/6}} \]

[Out]

(-2*2^(1/6)*AppellF1[1/2, 1, -1/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x]*(a + a*Sin[c + d*
x])^(2/3))/(d*(1 + Sin[c + d*x])^(7/6))

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Rubi [A]  time = 0.110812, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2787, 2785, 130, 429} \[ -\frac{2 \sqrt [6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} F_1\left (\frac{1}{2};1,-\frac{1}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right )}{d (\sin (c+d x)+1)^{7/6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*2^(1/6)*AppellF1[1/2, 1, -1/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x]*(a + a*Sin[c + d*
x])^(2/3))/(d*(1 + Sin[c + d*x])^(7/6))

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 \csc (c+d x) (a+a \sin (c+d x))^{2/3} \, dx &=\frac{(a+a \sin (c+d x))^{2/3} \int \csc (c+d x) (1+\sin (c+d x))^{2/3} \, dx}{(1+\sin (c+d x))^{2/3}}\\ &=-\frac{\left (\cos (c+d x) (a+a \sin (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [6]{2-x}}{(1-x) \sqrt{x}} \, dx,x,1-\sin (c+d x)\right )}{d \sqrt{1-\sin (c+d x)} (1+\sin (c+d x))^{7/6}}\\ &=-\frac{\left (2 \cos (c+d x) (a+a \sin (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [6]{2-x^2}}{1-x^2} \, dx,x,\sqrt{1-\sin (c+d x)}\right )}{d \sqrt{1-\sin (c+d x)} (1+\sin (c+d x))^{7/6}}\\ &=-\frac{2 \sqrt [6]{2} F_1\left (\frac{1}{2};1,-\frac{1}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right ) \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{d (1+\sin (c+d x))^{7/6}}\\ \end{align*}

Mathematica [F]  time = 2.72419, size = 0, normalized size = 0. \[ \int \csc (c+d x) (a+a \sin (c+d x))^{2/3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \begin{align*} \int \csc \left ( dx+c \right ) \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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