3.5.63 \(\int \coth ^3(a+b x) \, dx\) [463]

Optimal. Leaf size=27 \[ -\frac {\coth ^2(a+b x)}{2 b}+\frac {\log (\sinh (a+b x))}{b} \]

[Out]

-1/2*coth(b*x+a)^2/b+ln(sinh(b*x+a))/b

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Rubi [A]
time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 3556} \begin {gather*} \frac {\log (\sinh (a+b x))}{b}-\frac {\coth ^2(a+b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*x]^3,x]

[Out]

-1/2*Coth[a + b*x]^2/b + Log[Sinh[a + b*x]]/b

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 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \coth ^3(a+b x) \, dx &=-\frac {\coth ^2(a+b x)}{2 b}+\int \coth (a+b x) \, dx\\ &=-\frac {\coth ^2(a+b x)}{2 b}+\frac {\log (\sinh (a+b x))}{b}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 34, normalized size = 1.26 \begin {gather*} -\frac {\coth ^2(a+b x)-2 \log (\cosh (a+b x))-2 \log (\tanh (a+b x))}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + b*x]^3,x]

[Out]

-1/2*(Coth[a + b*x]^2 - 2*Log[Cosh[a + b*x]] - 2*Log[Tanh[a + b*x]])/b

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Maple [A]
time = 1.01, size = 23, normalized size = 0.85

method	result	size
derivativedivides	\(\frac {\ln \left (\sinh \left (b x +a \right )\right )-\frac {\left (\coth ^{2}\left (b x +a \right )\right )}{2}}{b}\)	\(23\)
default	\(\frac {\ln \left (\sinh \left (b x +a \right )\right )-\frac {\left (\coth ^{2}\left (b x +a \right )\right )}{2}}{b}\)	\(23\)
risch	\(-x -\frac {2 a}{b}-\frac {2 \,{\mathrm e}^{2 b x +2 a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^3*csch(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/b*(ln(sinh(b*x+a))-1/2*coth(b*x+a)^2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (25) = 50\).
time = 0.25, size = 79, normalized size = 2.93 \begin {gather*} x + \frac {a}{b} + \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2 \, e^{\left (-2 \, b x - 2 \, a\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*csch(b*x+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (25) = 50\).
time = 0.46, size = 346, normalized size = 12.81 \begin {gather*} -\frac {b x \cosh \left (b x + a\right )^{4} + 4 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b x \sinh \left (b x + a\right )^{4} - 2 \, {\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x + 1\right )} \sinh \left (b x + a\right )^{2} + b x - {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 4 \, {\left (b x \cosh \left (b x + a\right )^{3} - {\left (b x - 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*csch(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cosh ^{3}{\left (a + b x \right )} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)**3*csch(b*x+a)**3,x)

[Out]

Integral(cosh(a + b*x)**3*csch(a + b*x)**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (25) = 50\).
time = 0.43, size = 66, normalized size = 2.44 \begin {gather*} -\frac {2 \, b x + 2 \, a + \frac {3 \, e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 3}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} - 2 \, \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^3*csch(b*x+a)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.50, size = 25, normalized size = 0.93 \begin {gather*} \frac {\ln \left (\mathrm {sinh}\left (a+b\,x\right )\right )}{b}-\frac {1}{2\,b\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^3/sinh(a + b*x)^3,x)

[Out]

log(sinh(a + b*x))/b - 1/(2*b*sinh(a + b*x)^2)

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