3.24 \(\int \csc ^{\frac{5}{3}}(c+d x) \sqrt{a+a \csc (c+d x)} \, dx\)

Optimal. Leaf size=514 \[ \frac{8 \sqrt{2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{7 d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{12 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{6 a \cos (c+d x) \csc ^{\frac{5}{3}}(c+d x)}{7 d \sqrt{a \csc (c+d x)+a}}+\frac{24 a \cot (c+d x)}{7 d \left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right ) \sqrt{a \csc (c+d x)+a}} \]

[Out]

(24*a*Cot[c + d*x])/(7*d*(1 + Sqrt[3] - Csc[c + d*x]^(1/3))*Sqrt[a + a*Csc[c + d*x]]) - (6*a*Cos[c + d*x]*Csc[
c + d*x]^(5/3))/(7*d*Sqrt[a + a*Csc[c + d*x]]) - (12*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d
*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticE[
ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(7*d*Sqrt[(1 -
Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]]) + (8
*Sqrt[2]*3^(3/4)*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/
(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c
+ d*x]^(1/3))], -7 - 4*Sqrt[3]])/(7*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a
- a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])

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Rubi [A]  time = 0.298501, antiderivative size = 514, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3806, 50, 63, 303, 218, 1877} \[ \frac{8 \sqrt{2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{12 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{6 a \cos (c+d x) \csc ^{\frac{5}{3}}(c+d x)}{7 d \sqrt{a \csc (c+d x)+a}}+\frac{24 a \cot (c+d x)}{7 d \left (-\sqrt [3]{\csc (c+d x)}+\sqrt{3}+1\right ) \sqrt{a \csc (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^(5/3)*Sqrt[a + a*Csc[c + d*x]],x]

[Out]

(24*a*Cot[c + d*x])/(7*d*(1 + Sqrt[3] - Csc[c + d*x]^(1/3))*Sqrt[a + a*Csc[c + d*x]]) - (6*a*Cos[c + d*x]*Csc[
c + d*x]^(5/3))/(7*d*Sqrt[a + a*Csc[c + d*x]]) - (12*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d
*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticE[
ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(7*d*Sqrt[(1 -
Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]]) + (8
*Sqrt[2]*3^(3/4)*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/
(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c
+ d*x]^(1/3))], -7 - 4*Sqrt[3]])/(7*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a
- a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])

Rule 3806

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(a^2*d*
Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]]), Subst[Int[(d*x)^(n - 1)/Sqrt[a - b*x], x]
, x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 303

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(Sq
rt[2]*s)/(Sqrt[2 + Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a +
b*x^3], x], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 1877

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 - Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 - Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 + Sqrt[3])*s + r*x)), x
] - Simp[(3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin{align*} \int \csc ^{\frac{5}{3}}(c+d x) \sqrt{a+a \csc (c+d x)} \, dx &=\frac{\left (a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{x^{2/3}}{\sqrt{a-a x}} \, dx,x,\csc (c+d x)\right )}{d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}\\ &=-\frac{6 a \cos (c+d x) \csc ^{\frac{5}{3}}(c+d x)}{7 d \sqrt{a+a \csc (c+d x)}}+\frac{\left (4 a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{x} \sqrt{a-a x}} \, dx,x,\csc (c+d x)\right )}{7 d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}\\ &=-\frac{6 a \cos (c+d x) \csc ^{\frac{5}{3}}(c+d x)}{7 d \sqrt{a+a \csc (c+d x)}}+\frac{\left (12 a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{7 d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}\\ &=-\frac{6 a \cos (c+d x) \csc ^{\frac{5}{3}}(c+d x)}{7 d \sqrt{a+a \csc (c+d x)}}-\frac{\left (12 a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1-\sqrt{3}-x}{\sqrt{a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{7 d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}-\frac{\left (12 \sqrt{2 \left (2-\sqrt{3}\right )} a^2 \cot (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{7 d \sqrt{a-a \csc (c+d x)} \sqrt{a+a \csc (c+d x)}}\\ &=\frac{24 a \cot (c+d x)}{7 d \left (1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{a+a \csc (c+d x)}}-\frac{6 a \cos (c+d x) \csc ^{\frac{5}{3}}(c+d x)}{7 d \sqrt{a+a \csc (c+d x)}}-\frac{12 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac{2}{3}}(c+d x)}{\left (1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}\right )^2}} E\left (\sin ^{-1}\left (\frac{1-\sqrt{3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{7 d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt{a+a \csc (c+d x)}}+\frac{8 \sqrt{2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt{\frac{1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac{2}{3}}(c+d x)}{\left (1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}\right )^2}} F\left (\sin ^{-1}\left (\frac{1-\sqrt{3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{7 d \sqrt{\frac{1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt{3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt{a+a \csc (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 1.18692, size = 120, normalized size = 0.23 \[ -\frac{2 \sqrt{a (\csc (c+d x)+1)} \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (3 (\csc (c+d x)+4)-8 \sqrt [3]{\csc (c+d x)} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{4}{3},\frac{3}{2},1-\csc (c+d x)\right )\right )}{7 d \sqrt [3]{\csc (c+d x)} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^(5/3)*Sqrt[a + a*Csc[c + d*x]],x]

[Out]

(-2*Sqrt[a*(1 + Csc[c + d*x])]*(3*(4 + Csc[c + d*x]) - 8*Csc[c + d*x]^(1/3)*Hypergeometric2F1[1/2, 4/3, 3/2, 1
- Csc[c + d*x]])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(7*d*Csc[c + d*x]^(1/3)*(Cos[(c + d*x)/2] + Sin[(c +
d*x)/2]))

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Maple [F]  time = 0.335, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( dx+c \right ) \right ) ^{{\frac{5}{3}}}\sqrt{a+a\csc \left ( dx+c \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^(5/3)*(a+a*csc(d*x+c))^(1/2),x)

[Out]

int(csc(d*x+c)^(5/3)*(a+a*csc(d*x+c))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac{5}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^(5/3)*(a+a*csc(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^(5/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac{5}{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^(5/3)*(a+a*csc(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^(5/3), x)

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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)**(5/3)*(a+a*csc(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac{5}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^(5/3)*(a+a*csc(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^(5/3), x)