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

3.1.19.1 Optimal result

Integrand size = 12, antiderivative size = 131 \[ \int \sqrt [3]{c \cot (a+b x)} \, dx=\frac {\sqrt {3} \sqrt [3]{c} \arctan \left (\frac {c^{2/3}-2 (c \cot (a+b x))^{2/3}}{\sqrt {3} c^{2/3}}\right )}{2 b}+\frac {\sqrt [3]{c} \log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b}-\frac {\sqrt [3]{c} \log \left (c^{4/3}-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}\right )}{4 b} \]

output

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
 

3.1.19.2 Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.81 \[ \int \sqrt [3]{c \cot (a+b x)} \, dx=\frac {(c \cot (a+b x))^{4/3} \left (\log \left (1+\sqrt [3]{\cot ^2(a+b x)}\right )-\sqrt [3]{-1} \log \left (1-\sqrt [3]{-1} \sqrt [3]{\cot ^2(a+b x)}\right )+(-1)^{2/3} \log \left (1+(-1)^{2/3} \sqrt [3]{\cot ^2(a+b x)}\right )\right )}{2 b c \cot ^2(a+b x)^{2/3}} \]

input

Integrate[(c*Cot[a + b*x])^(1/3),x]
 

output

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

3.1.19.3 Rubi [A] (warning: unable to verify)

Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {3042, 3957, 266, 807, 821, 16, 1142, 25, 1082, 217, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt [3]{-c \tan \left (a+b x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 3957

\(\displaystyle -\frac {c \int \frac {\sqrt [3]{c \cot (a+b x)}}{\cot ^2(a+b x) c^2+c^2}d(c \cot (a+b x))}{b}\)

\(\Big \downarrow \) 266

\(\displaystyle -\frac {3 c \int \frac {c^3 \cot ^3(a+b x)}{c^6 \cot ^6(a+b x)+c^2}d\sqrt [3]{c \cot (a+b x)}}{b}\)

\(\Big \downarrow \) 807

\(\displaystyle -\frac {3 c \int \frac {c^2 \cot ^2(a+b x)}{c^3 \cot ^3(a+b x)+c^2}d\left (c^2 \cot ^2(a+b x)\right )}{2 b}\)

\(\Big \downarrow \) 821

\(\displaystyle -\frac {3 c \left (\frac {\int \frac {c^2 \cot ^2(a+b x)+c^{2/3}}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )}{3 c^{2/3}}-\frac {\int \frac {1}{c^2 \cot ^2(a+b x)+c^{2/3}}d\left (c^2 \cot ^2(a+b x)\right )}{3 c^{2/3}}\right )}{2 b}\)

\(\Big \downarrow \) 16

\(\displaystyle -\frac {3 c \left (\frac {\int \frac {c^2 \cot ^2(a+b x)+c^{2/3}}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )}{3 c^{2/3}}-\frac {\log \left (c^2 \cot ^2(a+b x)+c^{2/3}\right )}{3 c^{2/3}}\right )}{2 b}\)

\(\Big \downarrow \) 1142

\(\displaystyle -\frac {3 c \left (\frac {\frac {3}{2} c^{2/3} \int \frac {1}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )+\frac {1}{2} \int -\frac {c^{2/3}-2 c^2 \cot ^2(a+b x)}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )}{3 c^{2/3}}-\frac {\log \left (c^2 \cot ^2(a+b x)+c^{2/3}\right )}{3 c^{2/3}}\right )}{2 b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3 c \left (\frac {\frac {3}{2} c^{2/3} \int \frac {1}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )-\frac {1}{2} \int \frac {c^{2/3}-2 c^2 \cot ^2(a+b x)}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )}{3 c^{2/3}}-\frac {\log \left (c^2 \cot ^2(a+b x)+c^{2/3}\right )}{3 c^{2/3}}\right )}{2 b}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {3 c \left (\frac {3 \int \frac {1}{2 \sqrt [3]{c} \cot (a+b x)-4}d\left (1-2 \sqrt [3]{c} \cot (a+b x)\right )-\frac {1}{2} \int \frac {c^{2/3}-2 c^2 \cot ^2(a+b x)}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )}{3 c^{2/3}}-\frac {\log \left (c^2 \cot ^2(a+b x)+c^{2/3}\right )}{3 c^{2/3}}\right )}{2 b}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {3 c \left (\frac {-\frac {1}{2} \int \frac {c^{2/3}-2 c^2 \cot ^2(a+b x)}{c^2 \cot ^2(a+b x)-c^{5/3} \cot (a+b x)+c^{4/3}}d\left (c^2 \cot ^2(a+b x)\right )-\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} \cot (a+b x)}{\sqrt {3}}\right )}{3 c^{2/3}}-\frac {\log \left (c^2 \cot ^2(a+b x)+c^{2/3}\right )}{3 c^{2/3}}\right )}{2 b}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {3 c \left (\frac {\frac {1}{2} \log \left (-c^{5/3} \cot (a+b x)+c^2 \cot ^2(a+b x)+c^{4/3}\right )-\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} \cot (a+b x)}{\sqrt {3}}\right )}{3 c^{2/3}}-\frac {\log \left (c^2 \cot ^2(a+b x)+c^{2/3}\right )}{3 c^{2/3}}\right )}{2 b}\)

input

Int[(c*Cot[a + b*x])^(1/3),x]
 

output

(-3*c*(-1/3*Log[c^(2/3) + c^2*Cot[a + b*x]^2]/c^(2/3) + (-(Sqrt[3]*ArcTan[ 
(1 - 2*c^(1/3)*Cot[a + b*x])/Sqrt[3]]) + Log[c^(4/3) - c^(5/3)*Cot[a + b*x 
] + c^2*Cot[a + b*x]^2]/2)/(3*c^(2/3))))/(2*b)
 

3.1.19.3.1 Defintions of rubi rules used

rule 16

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 217

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 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 807

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m 
+ 1, n]}, Simp[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 821

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 
1)   Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[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 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[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])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[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 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[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]
 

3.1.19.4 Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.82

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

input

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

output

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

3.1.19.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (100) = 200\).

Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.61 \[ \int \sqrt [3]{c \cot (a+b x)} \, dx=-\frac {2 \, \sqrt {3} c^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} c - 2 \, \sqrt {3} c^{\frac {1}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}}}{3 \, c}\right ) - 2 \, c^{\frac {1}{3}} \log \left (c^{\frac {2}{3}} + \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}}\right ) + c^{\frac {1}{3}} \log \left (\frac {c^{\frac {4}{3}} \sin \left (2 \, b x + 2 \, a\right ) - c^{\frac {2}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}} \sin \left (2 \, b x + 2 \, a\right ) + {\left (c \cos \left (2 \, b x + 2 \, a\right ) + c\right )} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}}{\sin \left (2 \, b x + 2 \, a\right )}\right )}{4 \, b} \]

input

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

output

-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)*s 
in(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)/si 
n(2*b*x + 2*a))^(1/3))/sin(2*b*x + 2*a)))/b
 

3.1.19.6 Sympy [F]

\[ \int \sqrt [3]{c \cot (a+b x)} \, dx=\int \sqrt [3]{c \cot {\left (a + b x \right )}}\, dx \]

input

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

output

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

3.1.19.7 Maxima [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.78 \[ \int \sqrt [3]{c \cot (a+b x)} \, dx=-\frac {c {\left (\frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (c^{\frac {2}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {2}{3}}} + \frac {\log \left (c^{\frac {4}{3}} - c^{\frac {2}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {4}{3}}\right )}{c^{\frac {2}{3}}} - \frac {2 \, \log \left (c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {2}{3}}}\right )}}{4 \, b} \]

input

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

output

-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
 

3.1.19.8 Giac [F]

\[ \int \sqrt [3]{c \cot (a+b x)} \, dx=\int { \left (c \cot \left (b x + a\right )\right )^{\frac {1}{3}} \,d x } \]

input

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

output

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

3.1.19.9 Mupad [B] (verification not implemented)

Time = 12.66 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.02 \[ \int \sqrt [3]{c \cot (a+b x)} \, dx=\frac {c^{1/3}\,\ln \left (81\,c^{16/3}\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}+81\,c^6\right )}{2\,b}-\frac {c^{1/3}\,\ln \left (\frac {81\,c^6}{b^4}-\frac {81\,c^{16/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}}{b^4}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}+\frac {c^{1/3}\,\ln \left (\frac {81\,c^6}{b^4}+\frac {162\,c^{16/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}}{b^4}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b} \]

input

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

output

(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 + (1 
62*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