3.203 \(\int \frac{1}{x \coth ^{\frac{3}{2}}(a+b \log (c x^n))} \, dx\)

Optimal. Leaf size=71 \[ -\frac{2}{b n \sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}}-\frac{\tan ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]

[Out]

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

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Rubi [A]  time = 0.0517306, antiderivative size = 71, 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 = {3474, 3476, 329, 298, 203, 206} \[ -\frac{2}{b n \sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}}-\frac{\tan ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Coth[a + b*Log[c*x^n]]^(3/2)),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/(b*n*Sqrt[Coth
[a + b*Log[c*x^n]]])

Rule 3474

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 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 298

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

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])

Rubi steps

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

Mathematica [C]  time = 0.144225, size = 44, normalized size = 0.62 \[ -\frac{2 \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};\coth ^2\left (a+b \log \left (c x^n\right )\right )\right )}{b n \sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Coth[a + b*Log[c*x^n]]^(3/2)),x]

[Out]

(-2*Hypergeometric2F1[-1/4, 1, 3/4, Coth[a + b*Log[c*x^n]]^2])/(b*n*Sqrt[Coth[a + b*Log[c*x^n]]])

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Maple [A]  time = 0.015, size = 93, normalized size = 1.3 \begin{align*}{\frac{1}{2\,bn}\ln \left ( \sqrt{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )}+1 \right ) }-2\,{\frac{1}{bn\sqrt{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \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(1/x/coth(a+b*ln(c*x^n))^(3/2),x)

[Out]

1/2/b/n*ln(coth(a+b*ln(c*x^n))^(1/2)+1)-2/b/n/coth(a+b*ln(c*x^n))^(1/2)-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{1}{x \coth \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/coth(a+b*log(c*x^n))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/(x*coth(b*log(c*x^n) + a)^(3/2)), x)

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Fricas [B]  time = 2.84225, size = 2087, normalized size = 29.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/coth(a+b*log(c*x^n))^(3/2),x, algorithm="fricas")

[Out]

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)*sqrt(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) + s
inh(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)*sqr
t(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*cos
h(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(cos
h(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*log(x) + b*log(c) + a) + b*n*sinh(b*n*log(x) + b*log(c) + a)^2 +
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(1/x/coth(a+b*ln(c*x**n))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \coth \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/coth(a+b*log(c*x^n))^(3/2),x, algorithm="giac")

[Out]

integrate(1/(x*coth(b*log(c*x^n) + a)^(3/2)), x)