Integrand size = 19, antiderivative size = 71 \[ \int \frac {1}{x \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2}{b n \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}} \]
-arctan(coth(a+b*ln(c*x^n))^(1/2))/b/n+arctanh(coth(a+b*ln(c*x^n))^(1/2))/ b/n-2/b/n/coth(a+b*ln(c*x^n))^(1/2)
Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {-2-\arctan \left (\sqrt [4]{\coth ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt [4]{\coth ^2\left (a+b \log \left (c x^n\right )\right )}+\text {arctanh}\left (\sqrt [4]{\coth ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt [4]{\coth ^2\left (a+b \log \left (c x^n\right )\right )}}{b n \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}} \]
(-2 - ArcTan[(Coth[a + b*Log[c*x^n]]^2)^(1/4)]*(Coth[a + b*Log[c*x^n]]^2)^ (1/4) + ArcTanh[(Coth[a + b*Log[c*x^n]]^2)^(1/4)]*(Coth[a + b*Log[c*x^n]]^ 2)^(1/4))/(b*n*Sqrt[Coth[a + b*Log[c*x^n]]])
Time = 0.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {3039, 3042, 3955, 3042, 3957, 25, 266, 827, 216, 219}
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.
| \(\displaystyle \int \frac {1}{x \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \frac {1}{\coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\left (-i \tan \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )\right )^{3/2}}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {\int \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )-\frac {2}{b \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2}{b \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}+\int \sqrt {-i \tan \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {-\frac {\int -\frac {\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{1-\coth ^2\left (a+b \log \left (c x^n\right )\right )}d\coth \left (a+b \log \left (c x^n\right )\right )}{b}-\frac {2}{b \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{1-\coth ^2\left (a+b \log \left (c x^n\right )\right )}d\coth \left (a+b \log \left (c x^n\right )\right )}{b}-\frac {2}{b \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {2 \int \frac {\coth \left (a+b \log \left (c x^n\right )\right )}{1-\coth ^2\left (a+b \log \left (c x^n\right )\right )}d\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b}-\frac {2}{b \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth \left (a+b \log \left (c x^n\right )\right )}d\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}-\frac {1}{2} \int \frac {1}{\coth \left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b}-\frac {2}{b \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth \left (a+b \log \left (c x^n\right )\right )}d\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}-\frac {1}{2} \arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{b}-\frac {2}{b \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )-\frac {1}{2} \arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{b}-\frac {2}{b \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}}{n}\) |
((2*(-1/2*ArcTan[Sqrt[Coth[a + b*Log[c*x^n]]]] + ArcTanh[Sqrt[Coth[a + b*L og[c*x^n]]]]/2))/b - 2/(b*Sqrt[Coth[a + b*Log[c*x^n]]]))/n
3.3.3.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x] )^(n + 1)/(b*d*(n + 1)), x] - Simp[1/b^2 Int[(b*Tan[c + d*x])^(n + 2), x] , x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
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 = 0.50 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.07
| method | result | size |
| derivativedivides | \(\frac {-\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-1\right )}{2}+\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}+1\right )}{2}-\frac {2}{\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}}-\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )}{n b}\) | \(76\) |
| default | \(\frac {-\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-1\right )}{2}+\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}+1\right )}{2}-\frac {2}{\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}}-\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )}{n b}\) | \(76\) |
1/n/b*(-1/2*ln(coth(a+b*ln(c*x^n))^(1/2)-1)+1/2*ln(coth(a+b*ln(c*x^n))^(1/ 2)+1)-2/coth(a+b*ln(c*x^n))^(1/2)-arctan(coth(a+b*ln(c*x^n))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (65) = 130\).
Time = 0.27 (sec) , antiderivative size = 625, normalized size of antiderivative = 8.80 \[ \int \frac {1}{x \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Too large to display} \]
1/2*(2*(cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)^2 + 1)*arctan(-cosh(b*n*log(x) + b*log(c) + a)^2 - 2*cosh(b*n*log(x) + b*log(c ) + a)*sinh(b*n*log(x) + b*log(c) + a) - sinh(b*n*log(x) + b*log(c) + a)^2 + (cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)* sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)^2 - 1)*s qrt(cosh(b*n*log(x) + b*log(c) + a)/sinh(b*n*log(x) + b*log(c) + a))) - 4* cosh(b*n*log(x) + b*log(c) + a)^2 - (cosh(b*n*log(x) + b*log(c) + a)^2 + 2 *cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b* n*log(x) + b*log(c) + a)^2 + 1)*log(-cosh(b*n*log(x) + b*log(c) + a)^2 - 2 *cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) - sinh(b* n*log(x) + b*log(c) + a)^2 + (cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b *n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x ) + b*log(c) + a)^2 - 1)*sqrt(cosh(b*n*log(x) + b*log(c) + a)/sinh(b*n*log (x) + b*log(c) + a))) - 8*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) - 4*sinh(b*n*log(x) + b*log(c) + a)^2 - 4*(cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b *log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)^2 - 1)*sqrt(cosh(b*n*log(x) + b*log(c) + a)/sinh(b*n*log(x) + b*log(c) + a)) - 4)/(b*n*cosh(b*n*log(x ) + b*log(c) + a)^2 + 2*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*lo...
\[ \int \frac {1}{x \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x \coth ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
\[ \int \frac {1}{x \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \coth \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{x \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]
Time = 2.43 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\mathrm {atanh}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {\mathrm {atan}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {2}{b\,n\,\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}} \]