3.3.3 \(\int \frac {1}{x \coth ^{\frac {3}{2}}(a+b \log (c x^n))} \, dx\) [203]

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

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

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Rubi [A]
time = 0.04, antiderivative size = 71, 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 = {3555, 3557, 335, 304, 209, 212} \begin {gather*} -\frac {\text {ArcTan}\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 )}}+\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[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 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 304

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.11, size = 44, normalized size = 0.62 \begin {gather*} -\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 )}} \end {gather*}

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 = 4.71, size = 76, normalized size = 1.07

method	result	size
derivativedivides	\(\frac {-\arctan \left (\sqrt {\coth }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )-\frac {2}{\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}}-\frac {\ln \left (\sqrt {\coth }\left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}+\frac {\ln \left (\sqrt {\coth }\left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\)	\(76\)
default	\(\frac {-\arctan \left (\sqrt {\coth }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )-\frac {2}{\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}}-\frac {\ln \left (\sqrt {\coth }\left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}+\frac {\ln \left (\sqrt {\coth }\left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/coth(a+b*ln(c*x^n))^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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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(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] Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (65) = 130\).
time = 0.36, size = 625, normalized size = 8.80 \begin {gather*} \frac {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 )} \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 ) - 4 \, \cosh \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 )} \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 ) - 8 \, \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 ) - 4 \, \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 4 \, {\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 )}} - 4}{2 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, b n \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 ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n\right )}} \end {gather*}

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

[Out]

Timed out

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Mupad [B]
time = 1.74, size = 65, normalized size = 0.92 \begin {gather*} \frac {\mathrm {atanh}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {\mathrm {atan}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {2}{b\,n\,\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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