3.1.21 \(\int \frac {1}{(b \coth ^2(c+d x))^{3/2}} \, dx\) [21]

Optimal. Leaf size=65 \[ \frac {\coth (c+d x) \log (\cosh (c+d x))}{b d \sqrt {b \coth ^2(c+d x)}}-\frac {\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}} \]

[Out]

coth(d*x+c)*ln(cosh(d*x+c))/b/d/(b*coth(d*x+c)^2)^(1/2)-1/2*tanh(d*x+c)/b/d/(b*coth(d*x+c)^2)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3739, 3554, 3556} \begin {gather*} \frac {\coth (c+d x) \log (\cosh (c+d x))}{b d \sqrt {b \coth ^2(c+d x)}}-\frac {\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*Coth[c + d*x]^2)^(-3/2),x]

[Out]

(Coth[c + d*x]*Log[Cosh[c + d*x]])/(b*d*Sqrt[b*Coth[c + d*x]^2]) - Tanh[c + d*x]/(2*b*d*Sqrt[b*Coth[c + d*x]^2
])

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]

Rule 3739

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps

\begin {align*} \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx &=\frac {\coth (c+d x) \int \tanh ^3(c+d x) \, dx}{b \sqrt {b \coth ^2(c+d x)}}\\ &=-\frac {\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}}+\frac {\coth (c+d x) \int \tanh (c+d x) \, dx}{b \sqrt {b \coth ^2(c+d x)}}\\ &=\frac {\coth (c+d x) \log (\cosh (c+d x))}{b d \sqrt {b \coth ^2(c+d x)}}-\frac {\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 48, normalized size = 0.74 \begin {gather*} \frac {2 \coth (c+d x) \log (\cosh (c+d x))-\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*Coth[c + d*x]^2)^(-3/2),x]

[Out]

(2*Coth[c + d*x]*Log[Cosh[c + d*x]] - Tanh[c + d*x])/(2*b*d*Sqrt[b*Coth[c + d*x]^2])

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Maple [A]
time = 1.41, size = 79, normalized size = 1.22

method	result	size
derivativedivides	\(-\frac {\coth \left (d x +c \right ) \left (\ln \left (\coth \left (d x +c \right )+1\right ) \left (\coth ^{2}\left (d x +c \right )\right )+\ln \left (\coth \left (d x +c \right )-1\right ) \left (\coth ^{2}\left (d x +c \right )\right )-2 \ln \left (\coth \left (d x +c \right )\right ) \left (\coth ^{2}\left (d x +c \right )\right )+1\right )}{2 d \left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}}}\)	\(79\)
default	\(-\frac {\coth \left (d x +c \right ) \left (\ln \left (\coth \left (d x +c \right )+1\right ) \left (\coth ^{2}\left (d x +c \right )\right )+\ln \left (\coth \left (d x +c \right )-1\right ) \left (\coth ^{2}\left (d x +c \right )\right )-2 \ln \left (\coth \left (d x +c \right )\right ) \left (\coth ^{2}\left (d x +c \right )\right )+1\right )}{2 d \left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}}}\)	\(79\)
risch	\(\frac {\left (1+{\mathrm e}^{2 d x +2 c}\right ) x}{b \left ({\mathrm e}^{2 d x +2 c}-1\right ) \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}}-\frac {2 \left (1+{\mathrm e}^{2 d x +2 c}\right ) \left (d x +c \right )}{b \left ({\mathrm e}^{2 d x +2 c}-1\right ) \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, d}+\frac {2 \,{\mathrm e}^{2 d x +2 c}}{b \left (1+{\mathrm e}^{2 d x +2 c}\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right ) \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, d}+\frac {\left (1+{\mathrm e}^{2 d x +2 c}\right ) \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{b \left ({\mathrm e}^{2 d x +2 c}-1\right ) \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, d}\)	\(274\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 84, normalized size = 1.29 \begin {gather*} -\frac {2 \, \sqrt {b} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (2 \, b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + b^{2} e^{\left (-4 \, d x - 4 \, c\right )} + b^{2}\right )} d} - \frac {d x + c}{b^{\frac {3}{2}} d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{b^{\frac {3}{2}} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*coth(d*x+c)^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 817 vs. \(2 (59) = 118\).
time = 0.46, size = 817, normalized size = 12.57 \begin {gather*} \frac {{\left (d x \cosh \left (d x + c\right )^{4} - {\left (d x e^{\left (2 \, d x + 2 \, c\right )} - d x\right )} \sinh \left (d x + c\right )^{4} - 4 \, {\left (d x \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} - d x \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (d x - 1\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d x \cosh \left (d x + c\right )^{2} + d x - {\left (3 \, d x \cosh \left (d x + c\right )^{2} + d x - 1\right )} e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} \sinh \left (d x + c\right )^{2} + d x - {\left (d x \cosh \left (d x + c\right )^{4} + 2 \, {\left (d x - 1\right )} \cosh \left (d x + c\right )^{2} + d x\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left ({\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} \sinh \left (d x + c\right )^{4} - \cosh \left (d x + c\right )^{4} + 4 \, {\left (\cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} - \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 2 \, {\left (3 \, \cosh \left (d x + c\right )^{2} - {\left (3 \, \cosh \left (d x + c\right )^{2} + 1\right )} e^{\left (2 \, d x + 2 \, c\right )} + 1\right )} \sinh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{4} + 2 \, \cosh \left (d x + c\right )^{2} + 1\right )} e^{\left (2 \, d x + 2 \, c\right )} - 4 \, {\left (\cosh \left (d x + c\right )^{3} - {\left (\cosh \left (d x + c\right )^{3} + \cosh \left (d x + c\right )\right )} e^{\left (2 \, d x + 2 \, c\right )} + \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 1\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \, {\left (d x \cosh \left (d x + c\right )^{3} + {\left (d x - 1\right )} \cosh \left (d x + c\right ) - {\left (d x \cosh \left (d x + c\right )^{3} + {\left (d x - 1\right )} \cosh \left (d x + c\right )\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (4 \, d x + 4 \, c\right )} - 2 \, e^{\left (2 \, d x + 2 \, c\right )} + 1}}}{b^{2} d \cosh \left (d x + c\right )^{4} + 2 \, b^{2} d \cosh \left (d x + c\right )^{2} + {\left (b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (b^{2} d \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + b^{2} d + 2 \, {\left (3 \, b^{2} d \cosh \left (d x + c\right )^{2} + b^{2} d + {\left (3 \, b^{2} d \cosh \left (d x + c\right )^{2} + b^{2} d\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \sinh \left (d x + c\right )^{2} + {\left (b^{2} d \cosh \left (d x + c\right )^{4} + 2 \, b^{2} d \cosh \left (d x + c\right )^{2} + b^{2} d\right )} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, {\left (b^{2} d \cosh \left (d x + c\right )^{3} + b^{2} d \cosh \left (d x + c\right ) + {\left (b^{2} d \cosh \left (d x + c\right )^{3} + b^{2} d \cosh \left (d x + c\right )\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \sinh \left (d x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*coth(d*x+c)^2)^(3/2),x, algorithm="fricas")

[Out]

(d*x*cosh(d*x + c)^4 - (d*x*e^(2*d*x + 2*c) - d*x)*sinh(d*x + c)^4 - 4*(d*x*cosh(d*x + c)*e^(2*d*x + 2*c) - d*
x*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(d*x - 1)*cosh(d*x + c)^2 + 2*(3*d*x*cosh(d*x + c)^2 + d*x - (3*d*x*cosh(
d*x + c)^2 + d*x - 1)*e^(2*d*x + 2*c) - 1)*sinh(d*x + c)^2 + d*x - (d*x*cosh(d*x + c)^4 + 2*(d*x - 1)*cosh(d*x
+ c)^2 + d*x)*e^(2*d*x + 2*c) + ((e^(2*d*x + 2*c) - 1)*sinh(d*x + c)^4 - cosh(d*x + c)^4 + 4*(cosh(d*x + c)*e
^(2*d*x + 2*c) - cosh(d*x + c))*sinh(d*x + c)^3 - 2*(3*cosh(d*x + c)^2 - (3*cosh(d*x + c)^2 + 1)*e^(2*d*x + 2*
c) + 1)*sinh(d*x + c)^2 - 2*cosh(d*x + c)^2 + (cosh(d*x + c)^4 + 2*cosh(d*x + c)^2 + 1)*e^(2*d*x + 2*c) - 4*(c
osh(d*x + c)^3 - (cosh(d*x + c)^3 + cosh(d*x + c))*e^(2*d*x + 2*c) + cosh(d*x + c))*sinh(d*x + c) - 1)*log(2*c
osh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 4*(d*x*cosh(d*x + c)^3 + (d*x - 1)*cosh(d*x + c) - (d*x*cosh(d
*x + c)^3 + (d*x - 1)*cosh(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x + c))*sqrt((b*e^(4*d*x + 4*c) + 2*b*e^(2*d*x +
2*c) + b)/(e^(4*d*x + 4*c) - 2*e^(2*d*x + 2*c) + 1))/(b^2*d*cosh(d*x + c)^4 + 2*b^2*d*cosh(d*x + c)^2 + (b^2*d
*e^(2*d*x + 2*c) + b^2*d)*sinh(d*x + c)^4 + 4*(b^2*d*cosh(d*x + c)*e^(2*d*x + 2*c) + b^2*d*cosh(d*x + c))*sinh
(d*x + c)^3 + b^2*d + 2*(3*b^2*d*cosh(d*x + c)^2 + b^2*d + (3*b^2*d*cosh(d*x + c)^2 + b^2*d)*e^(2*d*x + 2*c))*
sinh(d*x + c)^2 + (b^2*d*cosh(d*x + c)^4 + 2*b^2*d*cosh(d*x + c)^2 + b^2*d)*e^(2*d*x + 2*c) + 4*(b^2*d*cosh(d*
x + c)^3 + b^2*d*cosh(d*x + c) + (b^2*d*cosh(d*x + c)^3 + b^2*d*cosh(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x + c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \coth ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*coth(d*x+c)**2)**(3/2),x)

[Out]

Integral((b*coth(c + d*x)**2)**(-3/2), x)

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Giac [A]
time = 0.42, size = 104, normalized size = 1.60 \begin {gather*} -\frac {\frac {d x + c}{\sqrt {b} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )} - \frac {\log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{\sqrt {b} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )} - \frac {2 \, e^{\left (2 \, d x + 2 \, c\right )}}{\sqrt {b} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}}{b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*coth(d*x+c)^2)^(3/2),x, algorithm="giac")

[Out]

-((d*x + c)/(sqrt(b)*sgn(e^(4*d*x + 4*c) - 1)) - log(e^(2*d*x + 2*c) + 1)/(sqrt(b)*sgn(e^(4*d*x + 4*c) - 1)) -
2*e^(2*d*x + 2*c)/(sqrt(b)*(e^(2*d*x + 2*c) + 1)^2*sgn(e^(4*d*x + 4*c) - 1)))/(b*d)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*coth(c + d*x)^2)^(3/2),x)

[Out]

int(1/(b*coth(c + d*x)^2)^(3/2), x)

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