3.200 \(\int \frac{\coth ^{\frac{3}{2}}(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=70 \[ -\frac{2 \sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac{\tan ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]

[Out]

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

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Rubi [A]  time = 0.0502198, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {3473, 3476, 329, 212, 206, 203} \[ -\frac{2 \sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac{\tan ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*Log[c*x^n]]^(3/2)/x,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*Sqrt[Coth[a + b*
Log[c*x^n]]])/(b*n)

Rule 3473

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

Rule 3476

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]

Rule 329

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 212

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] &&
!GtQ[a/b, 0]

Rule 206

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{\coth ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \coth ^{\frac{3}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{2 \sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\coth (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{2 \sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac{\operatorname{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 \sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac{2 \operatorname{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 \sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac{\operatorname{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{\operatorname{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{\tan ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac{2 \sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}\\ \end{align*}

Mathematica [A]  time = 0.146253, size = 57, normalized size = 0.81 \[ \frac{-2 \sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}+\tan ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )+\tanh ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + b*Log[c*x^n]]^(3/2)/x,x]

[Out]

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

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Maple [A]  time = 0.015, size = 92, normalized size = 1.3 \begin{align*} -2\,{\frac{\sqrt{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )}}{bn}}-{\frac{1}{2\,bn}\ln \left ( \sqrt{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )}-1 \right ) }+{\frac{1}{2\,bn}\ln \left ( \sqrt{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )}+1 \right ) }+{\frac{1}{bn}\arctan \left ( \sqrt{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a+b*ln(c*x^n))^(3/2)/x,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+b*log(c*x^n))^(3/2)/x,x, algorithm="maxima")

[Out]

integrate(coth(b*log(c*x^n) + a)^(3/2)/x, x)

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Fricas [B]  time = 2.51166, size = 1100, normalized size = 15.71 \begin{align*} -\frac{4 \, \sqrt{\frac{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}} + 2 \, \arctan \left (-\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} +{\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sqrt{\frac{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}\right ) + \log \left (-\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} +{\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sqrt{\frac{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}\right )}{2 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+b*log(c*x^n))^(3/2)/x,x, algorithm="fricas")

[Out]

-1/2*(4*sqrt(cosh(b*n*log(x) + b*log(c) + a)/sinh(b*n*log(x) + b*log(c) + a)) + 2*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)*sqrt(cosh(b*n*log(x) + b*log(c) + a)/sinh(b*n*log(x) + b*log(c) +
a))) + 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))))/(b*n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+b*ln(c*x**n))**(3/2)/x,x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+b*log(c*x^n))^(3/2)/x,x, algorithm="giac")

[Out]

Timed out