3.2.92 \(\int \frac {\coth ^4(a+b \log (c x^n))}{x} \, dx\) [192]

Optimal. Leaf size=45 \[ -\frac {\coth \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\log (x) \]

[Out]

-coth(a+b*ln(c*x^n))/b/n-1/3*coth(a+b*ln(c*x^n))^3/b/n+ln(x)

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Rubi [A]
time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3554, 8} \begin {gather*} -\frac {\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right )}{b n}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*Log[c*x^n]]^4/x,x]

[Out]

-(Coth[a + b*Log[c*x^n]]/(b*n)) - Coth[a + b*Log[c*x^n]]^3/(3*b*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 ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \coth ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\text {Subst}\left (\int \coth ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\coth \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\text {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\coth \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b 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 = 44, normalized size = 0.98 \begin {gather*} -\frac {\coth ^3\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\tanh ^2\left (a+b \log \left (c x^n\right )\right )\right )}{3 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + b*Log[c*x^n]]^4/x,x]

[Out]

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

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Maple [A]
time = 2.36, size = 69, normalized size = 1.53

method	result	size
derivativedivides	\(\frac {-\frac {\left (\coth ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3}-\coth \left (a +b \ln \left (c \,x^{n}\right )\right )-\frac {\ln \left (\coth \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}+\frac {\ln \left (\coth \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\)	\(69\)
default	\(\frac {-\frac {\left (\coth ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3}-\coth \left (a +b \ln \left (c \,x^{n}\right )\right )-\frac {\ln \left (\coth \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}+\frac {\ln \left (\coth \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\)	\(69\)
risch	\(\ln \left (x \right )-\frac {4 \left (3 \left (x^{n}\right )^{4 b} c^{4 b} {\mathrm e}^{4 a} {\mathrm e}^{-2 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{2 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{2 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{-2 i b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )}-3 \left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{-i b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )}+2\right )}{3 b n \left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{-i b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )}-1\right )^{3}}\)	\(329\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a+b*ln(c*x^n))^4/x,x,method=_RETURNVERBOSE)

[Out]

1/n/b*(-1/3*coth(a+b*ln(c*x^n))^3-coth(a+b*ln(c*x^n))-1/2*ln(coth(a+b*ln(c*x^n))-1)+1/2*ln(coth(a+b*ln(c*x^n))
+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 499 vs. \(2 (43) = 86\).
time = 0.35, size = 499, normalized size = 11.09 \begin {gather*} -\frac {18 \, c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 27 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 11}{12 \, {\left (b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n\right )}} - \frac {6 \, c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 15 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 11}{12 \, {\left (b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n\right )}} - \frac {2 \, {\left (3 \, c^{4 \, b} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 3 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + 1\right )}}{3 \, {\left (b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n\right )}} - \frac {3 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 1}{2 \, {\left (b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n\right )}} - \frac {2}{3 \, {\left (b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n\right )}} + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+b*log(c*x^n))^4/x,x, algorithm="maxima")

[Out]

-1/12*(18*c^(4*b)*e^(4*b*log(x^n) + 4*a) - 27*c^(2*b)*e^(2*b*log(x^n) + 2*a) + 11)/(b*c^(6*b)*n*e^(6*b*log(x^n
) + 6*a) - 3*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 3*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) - b*n) - 1/12*(6*c^(4*b
)*e^(4*b*log(x^n) + 4*a) - 15*c^(2*b)*e^(2*b*log(x^n) + 2*a) + 11)/(b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) - 3*b*c
^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 3*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) - b*n) - 2/3*(3*c^(4*b)*e^(4*b*log(x^n)
+ 4*a) - 3*c^(2*b)*e^(2*b*log(x^n) + 2*a) + 1)/(b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) - 3*b*c^(4*b)*n*e^(4*b*log
(x^n) + 4*a) + 3*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) - b*n) - 1/2*(3*c^(2*b)*e^(2*b*log(x^n) + 2*a) - 1)/(b*c^(
6*b)*n*e^(6*b*log(x^n) + 6*a) - 3*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 3*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) -
b*n) - 2/3/(b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) - 3*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 3*b*c^(2*b)*n*e^(2*b*l
og(x^n) + 2*a) - b*n) + log(x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (43) = 86\).
time = 0.36, size = 171, normalized size = 3.80 \begin {gather*} \frac {{\left (3 \, b n \log \left (x\right ) + 4\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 4 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 12 \, \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} + 3 \, {\left ({\left (3 \, b n \log \left (x\right ) + 4\right )} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 3 \, b n \log \left (x\right ) - 4\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{3 \, {\left (b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - b n\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+b*log(c*x^n))^4/x,x, algorithm="fricas")

[Out]

1/3*((3*b*n*log(x) + 4)*sinh(b*n*log(x) + b*log(c) + a)^3 - 4*cosh(b*n*log(x) + b*log(c) + a)^3 - 12*cosh(b*n*
log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^2 + 3*((3*b*n*log(x) + 4)*cosh(b*n*log(x) + b*log(c) +
a)^2 - 3*b*n*log(x) - 4)*sinh(b*n*log(x) + b*log(c) + a))/(b*n*sinh(b*n*log(x) + b*log(c) + a)^3 + 3*(b*n*cosh
(b*n*log(x) + b*log(c) + a)^2 - b*n)*sinh(b*n*log(x) + b*log(c) + a))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+b*ln(c*x**n))**4/x,x)

[Out]

Exception raised: TypeError >> Invalid NaN comparison

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Giac [A]
time = 0.50, size = 67, normalized size = 1.49 \begin {gather*} -\frac {4 \, {\left (3 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 3 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 2\right )}}{3 \, {\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{3} b n} + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+b*log(c*x^n))^4/x,x, algorithm="giac")

[Out]

-4/3*(3*c^(4*b)*x^(4*b*n)*e^(4*a) - 3*c^(2*b)*x^(2*b*n)*e^(2*a) + 2)/((c^(2*b)*x^(2*b*n)*e^(2*a) - 1)^3*b*n) +
log(x)

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Mupad [B]
time = 1.21, size = 163, normalized size = 3.62 \begin {gather*} \ln \left (x\right )-\frac {\frac {4}{3\,b\,n}+\frac {4\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}{3\,b\,n}}{3\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-3\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}+{\mathrm {e}}^{6\,a}\,{\left (c\,x^n\right )}^{6\,b}-1}-\frac {4}{3\,b\,n\,\left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-1\right )}-\frac {4\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}{3\,b\,n\,\left ({\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}-2\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + b*log(c*x^n))^4/x,x)

[Out]

log(x) - (4/(3*b*n) + (4*exp(4*a)*(c*x^n)^(4*b))/(3*b*n))/(3*exp(2*a)*(c*x^n)^(2*b) - 3*exp(4*a)*(c*x^n)^(4*b)
+ exp(6*a)*(c*x^n)^(6*b) - 1) - 4/(3*b*n*(exp(2*a)*(c*x^n)^(2*b) - 1)) - (4*exp(2*a)*(c*x^n)^(2*b))/(3*b*n*(e
xp(4*a)*(c*x^n)^(4*b) - 2*exp(2*a)*(c*x^n)^(2*b) + 1))

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