\(\int \frac {1}{x \coth ^{\frac {5}{2}}(a+b \log (c x^n))} \, dx\) [204]

Optimal result

Integrand size = 19, antiderivative size = 72 \[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2}{3 b n \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[Out]

arctan(coth(a+b*ln(c*x^n))^(1/2))/b/n+arctanh(coth(a+b*ln(c*x^n))^(1/2))/b/n-2/3/b/n/coth(a+b*ln(c*x^n))^(3/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3555, 3557, 335, 218, 212, 209} \[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2}{3 b n \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[In]

Int[1/(x*Coth[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

ArcTan[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n) + ArcTanh[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n) - 2/(3*b*n*Coth[a + b
*Log[c*x^n]]^(3/2))

Rule 209

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

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 3555

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)), x] - Dist[
1/b^2, Int[(b*Tan[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]

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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\coth ^{\frac {5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2}{3 b n \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\coth (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2}{3 b n \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,\coth \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \\ & = -\frac {2}{3 b n \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {2 \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \\ & = -\frac {2}{3 b n \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \\ & = \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}{3 b n \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {-2+3 \arctan \left (\sqrt [4]{\coth ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \coth ^2\left (a+b \log \left (c x^n\right )\right )^{3/4}+3 \text {arctanh}\left (\sqrt [4]{\coth ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \coth ^2\left (a+b \log \left (c x^n\right )\right )^{3/4}}{3 b n \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[In]

Integrate[1/(x*Coth[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

(-2 + 3*ArcTan[(Coth[a + b*Log[c*x^n]]^2)^(1/4)]*(Coth[a + b*Log[c*x^n]]^2)^(3/4) + 3*ArcTanh[(Coth[a + b*Log[
c*x^n]]^2)^(1/4)]*(Coth[a + b*Log[c*x^n]]^2)^(3/4))/(3*b*n*Coth[a + b*Log[c*x^n]]^(3/2))

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.03

method	result	size
derivativedivides	\(\frac {\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}+1\right )}{2}-\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-1\right )}{2}-\frac {2}{3 {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}}{n b}\)	\(74\)
default	\(\frac {\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}+1\right )}{2}-\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-1\right )}{2}-\frac {2}{3 {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}}{n b}\)	\(74\)

[In]

int(1/x/coth(a+b*ln(c*x^n))^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1104 vs. \(2 (64) = 128\).

Time = 0.28 (sec) , antiderivative size = 1104, normalized size of antiderivative = 15.33 \[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Too large to display} \]

[In]

integrate(1/x/coth(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")

[Out]

-1/6*(4*cosh(b*n*log(x) + b*log(c) + a)^4 + 16*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)
^3 + 4*sinh(b*n*log(x) + b*log(c) + a)^4 + 8*(3*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(x) + b*log
(c) + a)^2 + 6*(cosh(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(
c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(x) +
b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 + cosh(b*n*log(x)
+ b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 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*lo
g(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)^2 + 3*(cosh(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b
*log(c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(
x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 + cosh(b*n*l
og(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 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))) + 16*(cosh(b*n*log(x
) + b*log(c) + a)^3 + 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)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + sinh(b*n*log(x) + b*lo
g(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a)^2 - 2*cosh(b*n*log(x
) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 - cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x)
+ b*log(c) + a) + 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)^4 + 4*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + b*n*sinh(
b*n*log(x) + b*log(c) + a)^4 + 2*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + 2*(3*b*n*cosh(b*n*log(x) + b*log(c) +
a)^2 + b*n)*sinh(b*n*log(x) + b*log(c) + a)^2 + b*n + 4*(b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + b*n*cosh(b*n
*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a))

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/x/coth(a+b*ln(c*x**n))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \coth \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/x/coth(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/(x*coth(b*log(c*x^n) + a)^(5/2)), x)

Giac [F(-1)]

Timed out. \[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/x/coth(a+b*log(c*x^n))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 3.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\mathrm {atan}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}+\frac {\mathrm {atanh}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {2}{3\,b\,n\,{\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}} \]

[In]

int(1/(x*coth(a + b*log(c*x^n))^(5/2)),x)

[Out]

atan(coth(a + b*log(c*x^n))^(1/2))/(b*n) + atanh(coth(a + b*log(c*x^n))^(1/2))/(b*n) - 2/(3*b*n*coth(a + b*log
(c*x^n))^(3/2))