Integrand size = 12, antiderivative size = 132 \[ \int \frac {1}{\sqrt [3]{b \coth (c+d x)}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {b^{2/3}+2 (b \coth (c+d x))^{2/3}}{\sqrt {3} b^{2/3}}\right )}{2 \sqrt [3]{b} d}-\frac {\log \left (b^{2/3}-(b \coth (c+d x))^{2/3}\right )}{2 \sqrt [3]{b} d}+\frac {\log \left (b^{4/3}+b^{2/3} (b \coth (c+d x))^{2/3}+(b \coth (c+d x))^{4/3}\right )}{4 \sqrt [3]{b} d} \]
-1/2*ln(b^(2/3)-(b*coth(d*x+c))^(2/3))/b^(1/3)/d+1/4*ln(b^(4/3)+b^(2/3)*(b *coth(d*x+c))^(2/3)+(b*coth(d*x+c))^(4/3))/b^(1/3)/d+1/2*arctan(1/3*(b^(2/ 3)+2*(b*coth(d*x+c))^(2/3))/b^(2/3)*3^(1/2))*3^(1/2)/b^(1/3)/d
Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt [3]{b \coth (c+d x)}} \, dx=\frac {\sqrt [3]{\coth (c+d x)} \left (2 \sqrt {3} \arctan \left (\frac {1+2 \coth ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )-2 \log \left (1-\coth ^{\frac {2}{3}}(c+d x)\right )+\log \left (1+\coth ^{\frac {2}{3}}(c+d x)+\coth ^{\frac {4}{3}}(c+d x)\right )\right )}{4 d \sqrt [3]{b \coth (c+d x)}} \]
(Coth[c + d*x]^(1/3)*(2*Sqrt[3]*ArcTan[(1 + 2*Coth[c + d*x]^(2/3))/Sqrt[3] ] - 2*Log[1 - Coth[c + d*x]^(2/3)] + Log[1 + Coth[c + d*x]^(2/3) + Coth[c + d*x]^(4/3)]))/(4*d*(b*Coth[c + d*x])^(1/3))
Time = 0.32 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.84, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {3042, 3957, 25, 266, 807, 750, 16, 1142, 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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{b \coth (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt [3]{-i b \tan \left (i c+i d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle -\frac {b \int -\frac {1}{\sqrt [3]{b \coth (c+d x)} \left (b^2-b^2 \coth ^2(c+d x)\right )}d(b \coth (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \int \frac {1}{\sqrt [3]{b \coth (c+d x)} \left (b^2-b^2 \coth ^2(c+d x)\right )}d(b \coth (c+d x))}{d}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 b \int \frac {\sqrt [3]{b \coth (c+d x)}}{b^2-b^6 \coth ^6(c+d x)}d\sqrt [3]{b \coth (c+d x)}}{d}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {3 b \int \frac {1}{b^2-b^3 \coth ^3(c+d x)}d\left (b^2 \coth ^2(c+d x)\right )}{2 d}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {3 b \left (\frac {\int \frac {1}{b^{2/3}-b^2 \coth ^2(c+d x)}d\left (b^2 \coth ^2(c+d x)\right )}{3 b^{4/3}}+\frac {\int \frac {b^2 \coth ^2(c+d x)+2 b^{2/3}}{b^2 \coth ^2(c+d x)+b^{5/3} \coth (c+d x)+b^{4/3}}d\left (b^2 \coth ^2(c+d x)\right )}{3 b^{4/3}}\right )}{2 d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3 b \left (\frac {\int \frac {b^2 \coth ^2(c+d x)+2 b^{2/3}}{b^2 \coth ^2(c+d x)+b^{5/3} \coth (c+d x)+b^{4/3}}d\left (b^2 \coth ^2(c+d x)\right )}{3 b^{4/3}}-\frac {\log \left (b^{2/3}-b^2 \coth ^2(c+d x)\right )}{3 b^{4/3}}\right )}{2 d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3 b \left (\frac {\frac {3}{2} b^{2/3} \int \frac {1}{b^2 \coth ^2(c+d x)+b^{5/3} \coth (c+d x)+b^{4/3}}d\left (b^2 \coth ^2(c+d x)\right )+\frac {1}{2} \int \frac {2 b^2 \coth ^2(c+d x)+b^{2/3}}{b^2 \coth ^2(c+d x)+b^{5/3} \coth (c+d x)+b^{4/3}}d\left (b^2 \coth ^2(c+d x)\right )}{3 b^{4/3}}-\frac {\log \left (b^{2/3}-b^2 \coth ^2(c+d x)\right )}{3 b^{4/3}}\right )}{2 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 b \left (\frac {\frac {1}{2} \int \frac {2 b^2 \coth ^2(c+d x)+b^{2/3}}{b^2 \coth ^2(c+d x)+b^{5/3} \coth (c+d x)+b^{4/3}}d\left (b^2 \coth ^2(c+d x)\right )-3 \int \frac {1}{-2 \sqrt [3]{b} \coth (c+d x)-4}d\left (2 \sqrt [3]{b} \coth (c+d x)+1\right )}{3 b^{4/3}}-\frac {\log \left (b^{2/3}-b^2 \coth ^2(c+d x)\right )}{3 b^{4/3}}\right )}{2 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 b \left (\frac {\frac {1}{2} \int \frac {2 b^2 \coth ^2(c+d x)+b^{2/3}}{b^2 \coth ^2(c+d x)+b^{5/3} \coth (c+d x)+b^{4/3}}d\left (b^2 \coth ^2(c+d x)\right )+\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{b} \coth (c+d x)+1}{\sqrt {3}}\right )}{3 b^{4/3}}-\frac {\log \left (b^{2/3}-b^2 \coth ^2(c+d x)\right )}{3 b^{4/3}}\right )}{2 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 b \left (\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{b} \coth (c+d x)+1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (b^{5/3} \coth (c+d x)+b^{4/3}+b^2 \coth ^2(c+d x)\right )}{3 b^{4/3}}-\frac {\log \left (b^{2/3}-b^2 \coth ^2(c+d x)\right )}{3 b^{4/3}}\right )}{2 d}\) |
(3*b*(-1/3*Log[b^(2/3) - b^2*Coth[c + d*x]^2]/b^(4/3) + (Sqrt[3]*ArcTan[(1 + 2*b^(1/3)*Coth[c + d*x])/Sqrt[3]] + Log[b^(4/3) + b^(5/3)*Coth[c + d*x] + b^2*Coth[c + d*x]^2]/2)/(3*b^(4/3))))/(2*d)
3.1.12.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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])
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]
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*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]
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]
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]
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]
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]
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]
Time = 0.07 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(-\frac {3 b \left (\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}-\left (b^{2}\right )^{\frac {1}{3}}\right )}{6 \left (b^{2}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {4}{3}}+\left (b^{2}\right )^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {2}{3}}\right )}{12 \left (b^{2}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}}{\left (b^{2}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{6 \left (b^{2}\right )^{\frac {2}{3}}}\right )}{d}\) | \(109\) |
| default | \(-\frac {3 b \left (\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}-\left (b^{2}\right )^{\frac {1}{3}}\right )}{6 \left (b^{2}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {4}{3}}+\left (b^{2}\right )^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}+\left (b^{2}\right )^{\frac {2}{3}}\right )}{12 \left (b^{2}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}}{\left (b^{2}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{6 \left (b^{2}\right )^{\frac {2}{3}}}\right )}{d}\) | \(109\) |
-3/d*b*(1/6/(b^2)^(2/3)*ln((b*coth(d*x+c))^(2/3)-(b^2)^(1/3))-1/12/(b^2)^( 2/3)*ln((b*coth(d*x+c))^(4/3)+(b^2)^(1/3)*(b*coth(d*x+c))^(2/3)+(b^2)^(2/3 ))-1/6/(b^2)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(b^2)^(1/3)*(b*coth(d*x+c ))^(2/3)+1)))
Leaf count of result is larger than twice the leaf count of optimal. 742 vs. \(2 (99) = 198\).
Time = 0.29 (sec) , antiderivative size = 1598, normalized size of antiderivative = 12.11 \[ \int \frac {1}{\sqrt [3]{b \coth (c+d x)}} \, dx=\text {Too large to display} \]
[1/4*(sqrt(3)*b*sqrt((-b)^(1/3)/b)*log((3*b*cosh(d*x + c)^4 + 12*b*cosh(d* x + c)*sinh(d*x + c)^3 + 3*b*sinh(d*x + c)^4 + 2*b*cosh(d*x + c)^2 + 2*(9* b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 3*(cosh(d*x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2 - 2*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*(-b)^(1/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(2/3) - sqrt(3)*((co sh(d*x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*c osh(d*x + c)^2 - 1)*sinh(d*x + c)^2 - 2*cosh(d*x + c)^2 + 4*(cosh(d*x + c) ^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*(-b)^(2/3)*(b*cosh(d*x + c)/sinh(d* x + c))^(2/3) - (b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b *sinh(d*x + c)^4 - 2*b*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - b)*sinh( d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - b*cosh(d*x + c))*sinh(d*x + c) + b)*(- b)^(1/3) - 2*(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)^3*sinh(d*x + c) + 6*b* cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*si nh(d*x + c)^4 - b)*(b*cosh(d*x + c)/sinh(d*x + c))^(1/3))*sqrt((-b)^(1/3)/ b) + 4*(3*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c) + 3*b)/(cosh( d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 2*(-b)^(2 /3)*log(-(-b)^(2/3) + (b*cosh(d*x + c)/sinh(d*x + c))^(2/3)) + (-b)^(2/3)* log(((cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*(-b)^(2/3)*(b*cosh(d*x + c)/sinh(d*x + c))^(2/3) - (b*cosh(d*x + c)^...
\[ \int \frac {1}{\sqrt [3]{b \coth (c+d x)}} \, dx=\int \frac {1}{\sqrt [3]{b \coth {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {1}{\sqrt [3]{b \coth (c+d x)}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (99) = 198\).
Time = 0.39 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.64 \[ \int \frac {1}{\sqrt [3]{b \coth (c+d x)}} \, dx=\frac {b {\left (\frac {2 \, \sqrt {3} {\left | b \right |}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}}{3 \, {\left | b \right |}^{\frac {2}{3}}}\right )}{b^{2}} + \frac {{\left | b \right |}^{\frac {2}{3}} \log \left (\left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}} {\left | b \right |}^{\frac {2}{3}} + {\left | b \right |}^{\frac {4}{3}} + \frac {{\left (b e^{\left (2 \, d x + 2 \, c\right )} + b\right )} \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {1}{3}}}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )}{b^{2}} - \frac {2 \, {\left | b \right |}^{\frac {2}{3}} \log \left ({\left | \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}} - {\left | b \right |}^{\frac {2}{3}} \right |}\right )}{b^{2}}\right )}}{4 \, d} \]
1/4*b*(2*sqrt(3)*abs(b)^(2/3)*arctan(1/3*sqrt(3)*(2*((b*e^(2*d*x + 2*c) + b)/(e^(2*d*x + 2*c) - 1))^(2/3) + abs(b)^(2/3))/abs(b)^(2/3))/b^2 + abs(b) ^(2/3)*log(((b*e^(2*d*x + 2*c) + b)/(e^(2*d*x + 2*c) - 1))^(2/3)*abs(b)^(2 /3) + abs(b)^(4/3) + (b*e^(2*d*x + 2*c) + b)*((b*e^(2*d*x + 2*c) + b)/(e^( 2*d*x + 2*c) - 1))^(1/3)/(e^(2*d*x + 2*c) - 1))/b^2 - 2*abs(b)^(2/3)*log(a bs(((b*e^(2*d*x + 2*c) + b)/(e^(2*d*x + 2*c) - 1))^(2/3) - abs(b)^(2/3)))/ b^2)/d
Time = 2.31 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt [3]{b \coth (c+d x)}} \, dx=\frac {\ln \left (162\,{\left (-b\right )}^{11/3}+162\,b^3\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{2/3}\right )}{2\,{\left (-b\right )}^{1/3}\,d}+\frac {\ln \left (\frac {81\,{\left (-b\right )}^{11/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{d^3}+\frac {162\,b^3\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{2/3}}{d^3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,{\left (-b\right )}^{1/3}\,d}-\frac {\ln \left (\frac {81\,{\left (-b\right )}^{11/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{d^3}-\frac {162\,b^3\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{2/3}}{d^3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,{\left (-b\right )}^{1/3}\,d} \]
log(162*(-b)^(11/3) + 162*b^3*(b*coth(c + d*x))^(2/3))/(2*(-b)^(1/3)*d) + (log((81*(-b)^(11/3)*(3^(1/2)*1i - 1))/d^3 + (162*b^3*(b*coth(c + d*x))^(2 /3))/d^3)*(3^(1/2)*1i - 1))/(4*(-b)^(1/3)*d) - (log((81*(-b)^(11/3)*(3^(1/ 2)*1i + 1))/d^3 - (162*b^3*(b*coth(c + d*x))^(2/3))/d^3)*(3^(1/2)*1i + 1)) /(4*(-b)^(1/3)*d)