3.1.19 \(\int \sqrt [3]{c \cot (a+b x)} \, dx\) [19]

Optimal. Leaf size=131 \[ \frac {\sqrt {3} \sqrt [3]{c} \text {ArcTan}\left (\frac {c^{2/3}-2 (c \cot (a+b x))^{2/3}}{\sqrt {3} c^{2/3}}\right )}{2 b}+\frac {\sqrt [3]{c} \log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b}-\frac {\sqrt [3]{c} \log \left (c^{4/3}-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}\right )}{4 b} \]

[Out]

1/2*c^(1/3)*ln(c^(2/3)+(c*cot(b*x+a))^(2/3))/b-1/4*c^(1/3)*ln(c^(4/3)-c^(2/3)*(c*cot(b*x+a))^(2/3)+(c*cot(b*x+
a))^(4/3))/b+1/2*c^(1/3)*arctan(1/3*(c^(2/3)-2*(c*cot(b*x+a))^(2/3))/c^(2/3)*3^(1/2))*3^(1/2)/b

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Rubi [A]
time = 0.07, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3557, 335, 281, 298, 31, 648, 631, 210, 642} \begin {gather*} \frac {\sqrt {3} \sqrt [3]{c} \text {ArcTan}\left (\frac {c^{2/3}-2 (c \cot (a+b x))^{2/3}}{\sqrt {3} c^{2/3}}\right )}{2 b}+\frac {\sqrt [3]{c} \log \left ((c \cot (a+b x))^{2/3}+c^{2/3}\right )}{2 b}-\frac {\sqrt [3]{c} \log \left (-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}+c^{4/3}\right )}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Cot[a + b*x])^(1/3),x]

[Out]

(Sqrt[3]*c^(1/3)*ArcTan[(c^(2/3) - 2*(c*Cot[a + b*x])^(2/3))/(Sqrt[3]*c^(2/3))])/(2*b) + (c^(1/3)*Log[c^(2/3)
+ (c*Cot[a + b*x])^(2/3)])/(2*b) - (c^(1/3)*Log[c^(4/3) - c^(2/3)*(c*Cot[a + b*x])^(2/3) + (c*Cot[a + b*x])^(4
/3)])/(4*b)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rubi steps

\begin {align*} \int \sqrt [3]{c \cot (a+b x)} \, dx &=-\frac {c \text {Subst}\left (\int \frac {\sqrt [3]{x}}{c^2+x^2} \, dx,x,c \cot (a+b x)\right )}{b}\\ &=-\frac {(3 c) \text {Subst}\left (\int \frac {x^3}{c^2+x^6} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b}\\ &=-\frac {(3 c) \text {Subst}\left (\int \frac {x}{c^2+x^3} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{2 b}\\ &=\frac {\sqrt [3]{c} \text {Subst}\left (\int \frac {1}{c^{2/3}+x} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{2 b}-\frac {\sqrt [3]{c} \text {Subst}\left (\int \frac {c^{2/3}+x}{c^{4/3}-c^{2/3} x+x^2} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{2 b}\\ &=\frac {\sqrt [3]{c} \log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b}-\frac {\sqrt [3]{c} \text {Subst}\left (\int \frac {-c^{2/3}+2 x}{c^{4/3}-c^{2/3} x+x^2} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{4 b}-\frac {(3 c) \text {Subst}\left (\int \frac {1}{c^{4/3}-c^{2/3} x+x^2} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{4 b}\\ &=\frac {\sqrt [3]{c} \log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b}-\frac {\sqrt [3]{c} \log \left (c^{4/3}-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}\right )}{4 b}-\frac {\left (3 \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 (c \cot (a+b x))^{2/3}}{c^{2/3}}\right )}{2 b}\\ &=\frac {\sqrt {3} \sqrt [3]{c} \tan ^{-1}\left (\frac {1-\frac {2 (c \cot (a+b x))^{2/3}}{c^{2/3}}}{\sqrt {3}}\right )}{2 b}+\frac {\sqrt [3]{c} \log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b}-\frac {\sqrt [3]{c} \log \left (c^{4/3}-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}\right )}{4 b}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.04, size = 40, normalized size = 0.31 \begin {gather*} -\frac {3 (c \cot (a+b x))^{4/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\cot ^2(a+b x)\right )}{4 b c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*Cot[a + b*x])^(1/3),x]

[Out]

(-3*(c*Cot[a + b*x])^(4/3)*Hypergeometric2F1[2/3, 1, 5/3, -Cot[a + b*x]^2])/(4*b*c)

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Maple [A]
time = 0.22, size = 108, normalized size = 0.82

method	result	size
derivativedivides	\(-\frac {3 c \left (-\frac {\ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{6 \left (c^{2}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {4}{3}}-\left (c^{2}\right )^{\frac {1}{3}} \left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\left (c^{2}\right )^{\frac {2}{3}}\right )}{12 \left (c^{2}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}}{\left (c^{2}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 \left (c^{2}\right )^{\frac {1}{3}}}\right )}{b}\)	\(108\)
default	\(-\frac {3 c \left (-\frac {\ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{6 \left (c^{2}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {4}{3}}-\left (c^{2}\right )^{\frac {1}{3}} \left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\left (c^{2}\right )^{\frac {2}{3}}\right )}{12 \left (c^{2}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}}{\left (c^{2}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 \left (c^{2}\right )^{\frac {1}{3}}}\right )}{b}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*cot(b*x+a))^(1/3),x,method=_RETURNVERBOSE)

[Out]

-3/b*c*(-1/6/(c^2)^(1/3)*ln((c*cot(b*x+a))^(2/3)+(c^2)^(1/3))+1/12/(c^2)^(1/3)*ln((c*cot(b*x+a))^(4/3)-(c^2)^(
1/3)*(c*cot(b*x+a))^(2/3)+(c^2)^(2/3))+1/6*3^(1/2)/(c^2)^(1/3)*arctan(1/3*3^(1/2)*(2/(c^2)^(1/3)*(c*cot(b*x+a)
)^(2/3)-1)))

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Maxima [A]
time = 0.54, size = 102, normalized size = 0.78 \begin {gather*} -\frac {c {\left (\frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (c^{\frac {2}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {2}{3}}} + \frac {\log \left (c^{\frac {4}{3}} - c^{\frac {2}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {4}{3}}\right )}{c^{\frac {2}{3}}} - \frac {2 \, \log \left (c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {2}{3}}}\right )}}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cot(b*x+a))^(1/3),x, algorithm="maxima")

[Out]

-1/4*c*(2*sqrt(3)*arctan(-1/3*sqrt(3)*(c^(2/3) - 2*(c/tan(b*x + a))^(2/3))/c^(2/3))/c^(2/3) + log(c^(4/3) - c^
(2/3)*(c/tan(b*x + a))^(2/3) + (c/tan(b*x + a))^(4/3))/c^(2/3) - 2*log(c^(2/3) + (c/tan(b*x + a))^(2/3))/c^(2/
3))/b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (100) = 200\).
time = 2.04, size = 211, normalized size = 1.61 \begin {gather*} -\frac {2 \, \sqrt {3} c^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} c - 2 \, \sqrt {3} c^{\frac {1}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}}}{3 \, c}\right ) - 2 \, c^{\frac {1}{3}} \log \left (c^{\frac {2}{3}} + \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}}\right ) + c^{\frac {1}{3}} \log \left (\frac {c^{\frac {4}{3}} \sin \left (2 \, b x + 2 \, a\right ) - c^{\frac {2}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}} \sin \left (2 \, b x + 2 \, a\right ) + {\left (c \cos \left (2 \, b x + 2 \, a\right ) + c\right )} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}}{\sin \left (2 \, b x + 2 \, a\right )}\right )}{4 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cot(b*x+a))^(1/3),x, algorithm="fricas")

[Out]

-1/4*(2*sqrt(3)*c^(1/3)*arctan(-1/3*(sqrt(3)*c - 2*sqrt(3)*c^(1/3)*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))
^(2/3))/c) - 2*c^(1/3)*log(c^(2/3) + ((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(2/3)) + c^(1/3)*log((c^(4/3)
*sin(2*b*x + 2*a) - c^(2/3)*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(2/3)*sin(2*b*x + 2*a) + (c*cos(2*b*x
+ 2*a) + c)*((c*cos(2*b*x + 2*a) + c)/sin(2*b*x + 2*a))^(1/3))/sin(2*b*x + 2*a)))/b

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{c \cot {\left (a + b x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cot(b*x+a))**(1/3),x)

[Out]

Integral((c*cot(a + b*x))**(1/3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cot(b*x+a))^(1/3),x, algorithm="giac")

[Out]

integrate((c*cot(b*x + a))^(1/3), x)

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Mupad [B]
time = 0.52, size = 134, normalized size = 1.02 \begin {gather*} \frac {c^{1/3}\,\ln \left (81\,c^{16/3}\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}+81\,c^6\right )}{2\,b}-\frac {c^{1/3}\,\ln \left (\frac {81\,c^6}{b^4}-\frac {81\,c^{16/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}}{b^4}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}+\frac {c^{1/3}\,\ln \left (\frac {81\,c^6}{b^4}+\frac {162\,c^{16/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}}{b^4}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*cot(a + b*x))^(1/3),x)

[Out]

(c^(1/3)*log(81*c^(16/3)*(c*cot(a + b*x))^(2/3) + 81*c^6))/(2*b) - (c^(1/3)*log((81*c^6)/b^4 - (81*c^(16/3)*((
3^(1/2)*1i)/2 + 1/2)*(c*cot(a + b*x))^(2/3))/b^4)*((3^(1/2)*1i)/2 + 1/2))/(2*b) + (c^(1/3)*log((81*c^6)/b^4 +
(162*c^(16/3)*((3^(1/2)*1i)/4 - 1/4)*(c*cot(a + b*x))^(2/3))/b^4)*((3^(1/2)*1i)/4 - 1/4))/b

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