3.26 \(\int \frac {\sqrt {a+a \csc (c+d x)}}{\sqrt [3]{\csc (c+d x)}} \, dx\)

Optimal. Leaf size=508 \[ -\frac {\sqrt {2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt {a \csc (c+d x)+a}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{2 d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt {a \csc (c+d x)+a}}-\frac {3 a \cos (c+d x) \csc ^{\frac {2}{3}}(c+d x)}{d \sqrt {a \csc (c+d x)+a}}-\frac {3 a \cot (c+d x)}{d \left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right ) \sqrt {a \csc (c+d x)+a}} \]

[Out]

-3*a*cos(d*x+c)*csc(d*x+c)^(2/3)/d/(a+a*csc(d*x+c))^(1/2)-3*a*cot(d*x+c)/d/(1-csc(d*x+c)^(1/3)+3^(1/2))/(a+a*c
sc(d*x+c))^(1/2)-3^(3/4)*a^2*cot(d*x+c)*(1-csc(d*x+c)^(1/3))*EllipticF((1-csc(d*x+c)^(1/3)-3^(1/2))/(1-csc(d*x
+c)^(1/3)+3^(1/2)),I*3^(1/2)+2*I)*2^(1/2)*((1+csc(d*x+c)^(1/3)+csc(d*x+c)^(2/3))/(1-csc(d*x+c)^(1/3)+3^(1/2))^
2)^(1/2)/d/(a-a*csc(d*x+c))/(a+a*csc(d*x+c))^(1/2)/((1-csc(d*x+c)^(1/3))/(1-csc(d*x+c)^(1/3)+3^(1/2))^2)^(1/2)
+3/2*3^(1/4)*a^2*cot(d*x+c)*(1-csc(d*x+c)^(1/3))*EllipticE((1-csc(d*x+c)^(1/3)-3^(1/2))/(1-csc(d*x+c)^(1/3)+3^
(1/2)),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2))*((1+csc(d*x+c)^(1/3)+csc(d*x+c)^(2/3))/(1-csc(d*x+c)^(1/3)+3^(
1/2))^2)^(1/2)/d/(a-a*csc(d*x+c))/(a+a*csc(d*x+c))^(1/2)/((1-csc(d*x+c)^(1/3))/(1-csc(d*x+c)^(1/3)+3^(1/2))^2)
^(1/2)

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Rubi [A]  time = 0.27, antiderivative size = 508, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3806, 51, 63, 303, 218, 1877} \[ -\frac {\sqrt {2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt {a \csc (c+d x)+a}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{2 d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt {a \csc (c+d x)+a}}-\frac {3 a \cos (c+d x) \csc ^{\frac {2}{3}}(c+d x)}{d \sqrt {a \csc (c+d x)+a}}-\frac {3 a \cot (c+d x)}{d \left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right ) \sqrt {a \csc (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*Cot[c + d*x])/(d*(1 + Sqrt[3] - Csc[c + d*x]^(1/3))*Sqrt[a + a*Csc[c + d*x]]) - (3*a*Cos[c + d*x]*Csc[c
+ d*x]^(2/3))/(d*Sqrt[a + a*Csc[c + d*x]]) + (3*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[ArcSi
n[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(2*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]]) - (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 + Sq
rt[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]])/(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 51

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

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+a \csc (c+d x)}}{\sqrt [3]{\csc (c+d x)}} \, dx &=\frac {\left (a^2 \cot (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{x^{4/3} \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 {3 a \cos (c+d x) \csc ^{\frac {2}{3}}(c+d x)}{d \sqrt {a+a \csc (c+d x)}}-\frac {\left (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 )}{2 d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}}\\ &=-\frac {3 a \cos (c+d x) \csc ^{\frac {2}{3}}(c+d x)}{d \sqrt {a+a \csc (c+d x)}}-\frac {\left (3 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 )}{2 d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}}\\ &=-\frac {3 a \cos (c+d x) \csc ^{\frac {2}{3}}(c+d x)}{d \sqrt {a+a \csc (c+d x)}}+\frac {\left (3 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 )}{2 d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}}+\frac {\left (3 \sqrt {\frac {1}{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 )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}}\\ &=-\frac {3 a \cot (c+d x)}{d \left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {a+a \csc (c+d x)}}-\frac {3 a \cos (c+d x) \csc ^{\frac {2}{3}}(c+d x)}{d \sqrt {a+a \csc (c+d x)}}+\frac {3 \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 )}{2 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 {\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 )}{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*}

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Mathematica [C]  time = 0.76, size = 46, normalized size = 0.09 \[ -\frac {2 a \cot (c+d x) \, _2F_1\left (\frac {1}{2},\frac {4}{3};\frac {3}{2};1-\csc (c+d x)\right )}{d \sqrt {a (\csc (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*Cot[c + d*x]*Hypergeometric2F1[1/2, 4/3, 3/2, 1 - Csc[c + d*x]])/(d*Sqrt[a*(1 + Csc[c + d*x])])

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fricas [F]  time = 1.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \csc \left (d x + c\right ) + a}}{\csc \left (d x + c\right )^{\frac {1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*csc(d*x + c) + a)/csc(d*x + c)^(1/3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \csc \left (d x + c\right ) + a}}{\csc \left (d x + c\right )^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 2.68, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a +a \csc \left (d x +c \right )}}{\csc \left (d x +c \right )^{\frac {1}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \csc \left (d x + c\right ) + a}}{\csc \left (d x + c\right )^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {a+\frac {a}{\sin \left (c+d\,x\right )}}}{{\left (\frac {1}{\sin \left (c+d\,x\right )}\right )}^{1/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \left (\csc {\left (c + d x \right )} + 1\right )}}{\sqrt [3]{\csc {\left (c + d x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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