3.2.100 \(\int \frac {\coth ^{\frac {3}{2}}(a+b \log (c x^n))}{x} \, dx\) [200]

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

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

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Rubi [A]
time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, 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 = {3554, 3557, 335, 218, 212, 209} \begin {gather*} \frac {\text {ArcTan}\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}+\frac {\tanh ^{-1}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \end {gather*}

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 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 3554

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

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Mathematica [A]
time = 0.11, size = 57, normalized size = 0.81 \begin {gather*} \frac {\text {ArcTan}\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 )-2 \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n} \end {gather*}

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 = 4.65, size = 74, normalized size = 1.06

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (64) = 128\).
time = 0.37, size = 334, normalized size = 4.77 \begin {gather*} -\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 {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Mupad [B]
time = 1.86, size = 51, normalized size = 0.73 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )+\mathrm {atanh}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )-2\,\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}}{b\,n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + b*log(c*x^n))^(3/2)/x,x)

[Out]

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

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