3.2.88 \(\int \frac {\coth ^2(d (a+b \log (c x^n)))}{x} \, dx\) [188]

Optimal. Leaf size=28 \[ -\frac {\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n}+\log (x) \]

[Out]

-coth(a*d+b*d*ln(c*x^n))/b/d/n+ln(x)

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Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3554, 8} \begin {gather*} \log (x)-\frac {\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[d*(a + b*Log[c*x^n])]^2/x,x]

[Out]

-(Coth[a*d + b*d*Log[c*x^n]]/(b*d*n)) + Log[x]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.08, size = 49, normalized size = 1.75 \begin {gather*} -\frac {\coth \left (a d+b d \log \left (c x^n\right )\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2\left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(28)=56\).
time = 2.32, size = 63, normalized size = 2.25

method	result	size
derivativedivides	\(\frac {-\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-\frac {\ln \left (\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-1\right )}{2}+\frac {\ln \left (\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )+1\right )}{2}}{n b d}\)	\(63\)
default	\(\frac {-\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-\frac {\ln \left (\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-1\right )}{2}+\frac {\ln \left (\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )+1\right )}{2}}{n b d}\)	\(63\)
risch	\(\ln \left (x \right )-\frac {2}{d b n \left ({\mathrm e}^{d \left (-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )+i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )-i b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )+2 b \ln \left (c \right )+2 b \ln \left (x^{n}\right )+2 a \right )}-1\right )}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 37, normalized size = 1.32 \begin {gather*} -\frac {2}{b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - b d n} + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(b*c^(2*b*d)*d*n*e^(2*b*d*log(x^n) + 2*a*d) - b*d*n) + log(x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (28) = 56\).
time = 0.35, size = 72, normalized size = 2.57 \begin {gather*} \frac {{\left (b d n \log \left (x\right ) + 1\right )} \sinh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) - \cosh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}{b d n \sinh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(a*d + b*d*log(c*x**n))**2/x, x)

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Giac [A]
time = 0.48, size = 37, normalized size = 1.32 \begin {gather*} -\frac {2}{{\left (c^{2 \, b d} x^{2 \, b d n} e^{\left (2 \, a d\right )} - 1\right )} b d n} + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/((c^(2*b*d)*x^(2*b*d*n)*e^(2*a*d) - 1)*b*d*n) + log(x)

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Mupad [B]
time = 1.19, size = 34, normalized size = 1.21 \begin {gather*} \ln \left (x\right )-\frac {2}{b\,d\,n\,\left ({\mathrm {e}}^{2\,a\,d}\,{\left (c\,x^n\right )}^{2\,b\,d}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*(a + b*log(c*x^n)))^2/x,x)

[Out]

log(x) - 2/(b*d*n*(exp(2*a*d)*(c*x^n)^(2*b*d) - 1))

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