3.115 \(\int \frac{\csc ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{\cos (c+d x) F_1\left (\frac{1}{2};2,\frac{11}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]

[Out]

-((AppellF1[1/2, 2, 11/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x])/(2^(5/6)*a*d*(1 + Sin[c +
d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))

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Rubi [A]  time = 0.137083, antiderivative size = 80, 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} \[ -\frac{\cos (c+d x) F_1\left (\frac{1}{2};2,\frac{11}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(4/3),x]

[Out]

-((AppellF1[1/2, 2, 11/6, 3/2, 1 - Sin[c + d*x], (1 - Sin[c + d*x])/2]*Cos[c + d*x])/(2^(5/6)*a*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)}{(a+a \sin (c+d x))^{4/3}} \, dx &=\frac{\sqrt [3]{1+\sin (c+d x)} \int \frac{\csc ^2(c+d x)}{(1+\sin (c+d x))^{4/3}} \, dx}{a \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac{\cos (c+d x) \operatorname{Subst}\left (\int \frac{1}{(1-x)^2 (2-x)^{11/6} \sqrt{x}} \, dx,x,1-\sin (c+d x)\right )}{a 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 )^{11/6}} \, dx,x,\sqrt{1-\sin (c+d x)}\right )}{a d \sqrt{1-\sin (c+d x)} \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac{F_1\left (\frac{1}{2};2,\frac{11}{6};\frac{3}{2};1-\sin (c+d x),\frac{1}{2} (1-\sin (c+d x))\right ) \cos (c+d x)}{2^{5/6} a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 14.0285, size = 230, normalized size = 2.88 \[ \frac{8\ 2^{2/3} \cos ^{\frac{8}{3}}\left (\frac{1}{4} (2 c+2 d x-\pi )\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x))) \left (14 i \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-i e^{-i (c+d x)}\right ) (\sin (c+d x)+i \cos (c+d x)+1)^{2/3} (\sin (2 (c+d x))+2 \cos (c+d x))+35 \sin (c+d x)-14 \cos (2 (c+d x))+6\right )}{55 d \left (-1+i e^{i (c+d x)}\right )^3 \left (e^{i (c+d x)}-i\right ) \left (-(-1)^{3/4} e^{-\frac{1}{2} i (c+d x)} \left (e^{i (c+d x)}+i\right )\right )^{2/3} (a (\sin (c+d x)+1))^{4/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(4/3),x]

[Out]

(8*2^(2/3)*Cos[(2*c - Pi + 2*d*x)/4]^(8/3)*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])*(6 - 14*Cos[2*(c + d*x)] +
35*Sin[c + d*x] + (14*I)*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)*(2*Cos[c + d*x] + Sin[2*(c + d*x)])))/(55*d*(-1 + I*E^(I*(c + d*x)))^3*(-I + E^(I*(c + d*x)))*(-((
(-1)^(3/4)*(I + E^(I*(c + d*x))))/E^((I/2)*(c + d*x))))^(2/3)*(a*(1 + Sin[c + d*x]))^(4/3))

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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} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{-{\frac{4}{3}}}}\, 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))^(4/3),x)

[Out]

int(csc(d*x+c)^2/(a+a*sin(d*x+c))^(4/3),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{4}{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))^(4/3),x, algorithm="maxima")

[Out]

integrate(csc(d*x + c)^2/(a*sin(d*x + c) + a)^(4/3), 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))^(4/3),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)**2/(a+a*sin(d*x+c))**(4/3),x)

[Out]

Timed out

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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{4}{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))^(4/3),x, algorithm="giac")

[Out]

integrate(csc(d*x + c)^2/(a*sin(d*x + c) + a)^(4/3), x)