3.197 \(\int \frac{1}{x \sqrt{\tanh (a+b \log (c x^n))}} \, dx\)

Optimal. Leaf size=47 \[ \frac{\tanh ^{-1}\left (\sqrt{\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tan ^{-1}\left (\sqrt{\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]

[Out]

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

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Rubi [A]  time = 0.0398733, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3476, 329, 212, 206, 203} \[ \frac{\tanh ^{-1}\left (\sqrt{\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tan ^{-1}\left (\sqrt{\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.108694, size = 47, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\sqrt{\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tan ^{-1}\left (\sqrt{\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[Tanh[a + b*Log[c*x^n]]]),x]

[Out]

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

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Maple [A]  time = 0.017, size = 44, normalized size = 0.9 \begin{align*}{\frac{1}{bn}\arctan \left ( \sqrt{\tanh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) } \right ) }+{\frac{1}{bn}{\it Artanh} \left ( \sqrt{\tanh \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/tanh(a+b*ln(c*x^n))^(1/2),x)

[Out]

arctan(tanh(a+b*ln(c*x^n))^(1/2))/b/n+arctanh(tanh(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 \sqrt{\tanh \left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x*sqrt(tanh(b*log(c*x^n) + a))), x)

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Fricas [B]  time = 2.06853, size = 999, normalized size = 21.26 \begin{align*} \frac{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{\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\cosh \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{\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\cosh \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(1/x/tanh(a+b*log(c*x^n))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.10488, size = 66, normalized size = 1.4 \begin{align*} - \frac{\log{\left (\sqrt{\tanh{\left (a + b \log{\left (c x^{n} \right )} \right )}} - 1 \right )}}{2 b n} + \frac{\log{\left (\sqrt{\tanh{\left (a + b \log{\left (c x^{n} \right )} \right )}} + 1 \right )}}{2 b n} + \frac{\operatorname{atan}{\left (\sqrt{\tanh{\left (a + b \log{\left (c x^{n} \right )} \right )}} \right )}}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sqrt(tanh(a + b*log(c*x**n))) - 1)/(2*b*n) + log(sqrt(tanh(a + b*log(c*x**n))) + 1)/(2*b*n) + atan(sqrt(t
anh(a + b*log(c*x**n))))/(b*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x*sqrt(tanh(b*log(c*x^n) + a))), x)