3.1.18 \(\int (b \coth ^2(c+d x))^{3/2} \, dx\) [18]

Optimal. Leaf size=61 \[ -\frac {b \coth (c+d x) \sqrt {b \coth ^2(c+d x)}}{2 d}+\frac {b \sqrt {b \coth ^2(c+d x)} \log (\sinh (c+d x)) \tanh (c+d x)}{d} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 61, 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 {b \tanh (c+d x) \sqrt {b \coth ^2(c+d x)} \log (\sinh (c+d x))}{d}-\frac {b \coth (c+d x) \sqrt {b \coth ^2(c+d x)}}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*Coth[c + d*x]^2)^(3/2),x]

[Out]

-1/2*(b*Coth[c + d*x]*Sqrt[b*Coth[c + d*x]^2])/d + (b*Sqrt[b*Coth[c + d*x]^2]*Log[Sinh[c + d*x]]*Tanh[c + d*x]
)/d

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

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Mathematica [A]
time = 0.09, size = 56, normalized size = 0.92 \begin {gather*} -\frac {\left (b \coth ^2(c+d x)\right )^{3/2} \left (\coth ^2(c+d x)-2 \log (\cosh (c+d x))-2 \log (\tanh (c+d x))\right ) \tanh ^3(c+d x)}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((b*Coth[c + d*x]^2)^(3/2)*(Coth[c + d*x]^2 - 2*Log[Cosh[c + d*x]] - 2*Log[Tanh[c + d*x]])*Tanh[c + d*x]^
3)/d

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Maple [A]
time = 1.41, size = 53, normalized size = 0.87

method	result	size
derivativedivides	\(-\frac {\left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}} \left (\coth ^{2}\left (d x +c \right )+\ln \left (\coth \left (d x +c \right )-1\right )+\ln \left (\coth \left (d x +c \right )+1\right )\right )}{2 d \coth \left (d x +c \right )^{3}}\)	\(53\)
default	\(-\frac {\left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}} \left (\coth ^{2}\left (d x +c \right )+\ln \left (\coth \left (d x +c \right )-1\right )+\ln \left (\coth \left (d x +c \right )+1\right )\right )}{2 d \coth \left (d x +c \right )^{3}}\)	\(53\)
risch	\(\frac {b \left ({\mathrm e}^{2 d x +2 c}-1\right ) \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, x}{1+{\mathrm e}^{2 d x +2 c}}-\frac {2 b \left ({\mathrm e}^{2 d x +2 c}-1\right ) \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, \left (d x +c \right )}{\left (1+{\mathrm e}^{2 d x +2 c}\right ) d}-\frac {2 b \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, {\mathrm e}^{2 d x +2 c}}{\left (1+{\mathrm e}^{2 d x +2 c}\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right ) d}+\frac {b \left ({\mathrm e}^{2 d x +2 c}-1\right ) \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{\left (1+{\mathrm e}^{2 d x +2 c}\right ) d}\)	\(266\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 97, normalized size = 1.59 \begin {gather*} -\frac {{\left (d x + c\right )} b^{\frac {3}{2}}}{d} - \frac {b^{\frac {3}{2}} \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {b^{\frac {3}{2}} \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {2 \, b^{\frac {3}{2}} e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (55) = 110\).
time = 0.51, size = 823, normalized size = 13.49 \begin {gather*} \frac {{\left (b d x \cosh \left (d x + c\right )^{4} - {\left (b d x e^{\left (2 \, d x + 2 \, c\right )} - b d x\right )} \sinh \left (d x + c\right )^{4} - 4 \, {\left (b d x \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} - b d x \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + b d x - 2 \, {\left (b d x - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b d x \cosh \left (d x + c\right )^{2} - b d x - {\left (3 \, b d x \cosh \left (d x + c\right )^{2} - b d x + b\right )} e^{\left (2 \, d x + 2 \, c\right )} + b\right )} \sinh \left (d x + c\right )^{2} - {\left (b d x \cosh \left (d x + c\right )^{4} + b d x - 2 \, {\left (b d x - b\right )} \cosh \left (d x + c\right )^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (b \cosh \left (d x + c\right )^{4} - {\left (b e^{\left (2 \, d x + 2 \, c\right )} - b\right )} \sinh \left (d x + c\right )^{4} - 4 \, {\left (b \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} - b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 2 \, b \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} - {\left (3 \, b \cosh \left (d x + c\right )^{2} - b\right )} e^{\left (2 \, d x + 2 \, c\right )} - b\right )} \sinh \left (d x + c\right )^{2} - {\left (b \cosh \left (d x + c\right )^{4} - 2 \, b \cosh \left (d x + c\right )^{2} + b\right )} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} - b \cosh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right )^{3} - b \cosh \left (d x + c\right )\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \sinh \left (d x + c\right ) + b\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \, {\left (b d x \cosh \left (d x + c\right )^{3} - {\left (b d x - b\right )} \cosh \left (d x + c\right ) - {\left (b d x \cosh \left (d x + c\right )^{3} - {\left (b d x - b\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}}}{d \cosh \left (d x + c\right )^{4} + {\left (d e^{\left (2 \, d x + 2 \, c\right )} + d\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (d \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} + {\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sinh \left (d x + c\right )^{2} + {\left (d \cosh \left (d x + c\right )^{4} - 2 \, d \cosh \left (d x + c\right )^{2} + d\right )} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right ) + {\left (d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \sinh \left (d x + c\right ) + d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*d*x*cosh(d*x + c)^4 - (b*d*x*e^(2*d*x + 2*c) - b*d*x)*sinh(d*x + c)^4 - 4*(b*d*x*cosh(d*x + c)*e^(2*d*x + 2
*c) - b*d*x*cosh(d*x + c))*sinh(d*x + c)^3 + b*d*x - 2*(b*d*x - b)*cosh(d*x + c)^2 + 2*(3*b*d*x*cosh(d*x + c)^
2 - b*d*x - (3*b*d*x*cosh(d*x + c)^2 - b*d*x + b)*e^(2*d*x + 2*c) + b)*sinh(d*x + c)^2 - (b*d*x*cosh(d*x + c)^
4 + b*d*x - 2*(b*d*x - b)*cosh(d*x + c)^2)*e^(2*d*x + 2*c) - (b*cosh(d*x + c)^4 - (b*e^(2*d*x + 2*c) - b)*sinh
(d*x + c)^4 - 4*(b*cosh(d*x + c)*e^(2*d*x + 2*c) - b*cosh(d*x + c))*sinh(d*x + c)^3 - 2*b*cosh(d*x + c)^2 + 2*
(3*b*cosh(d*x + c)^2 - (3*b*cosh(d*x + c)^2 - b)*e^(2*d*x + 2*c) - b)*sinh(d*x + c)^2 - (b*cosh(d*x + c)^4 - 2
*b*cosh(d*x + c)^2 + b)*e^(2*d*x + 2*c) + 4*(b*cosh(d*x + c)^3 - b*cosh(d*x + c) - (b*cosh(d*x + c)^3 - b*cosh
(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x + c) + b)*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 4*(b*d*x
*cosh(d*x + c)^3 - (b*d*x - b)*cosh(d*x + c) - (b*d*x*cosh(d*x + c)^3 - (b*d*x - b)*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))/(d*cosh(d*x + c)^4 + (d*e^(2*d*x + 2*c) + d)*sinh(d*x + c)^4 + 4*(d*cosh(d*x + c)*e^(2*d*x + 2*c) + d*cos
h(d*x + c))*sinh(d*x + c)^3 - 2*d*cosh(d*x + c)^2 + 2*(3*d*cosh(d*x + c)^2 + (3*d*cosh(d*x + c)^2 - d)*e^(2*d*
x + 2*c) - d)*sinh(d*x + c)^2 + (d*cosh(d*x + c)^4 - 2*d*cosh(d*x + c)^2 + d)*e^(2*d*x + 2*c) + 4*(d*cosh(d*x
+ c)^3 - d*cosh(d*x + c) + (d*cosh(d*x + c)^3 - d*cosh(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x + c) + d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \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((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 = 90, normalized size = 1.48 \begin {gather*} -\frac {{\left ({\left (d x + c\right )} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) - \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) + \frac {2 \, e^{\left (2 \, d x + 2 \, c\right )} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}\right )} b^{\frac {3}{2}}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

int((b*coth(c + d*x)^2)^(3/2), x)

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