3.43 \(\int \frac{1}{(b \coth ^4(c+d x))^{3/2}} \, dx\)
Optimal. Leaf size=118 \[ \frac{x \coth ^2(c+d x)}{b \sqrt{b \coth ^4(c+d x)}}-\frac{\coth (c+d x)}{b d \sqrt{b \coth ^4(c+d x)}}-\frac{\tanh ^3(c+d x)}{5 b d \sqrt{b \coth ^4(c+d x)}}-\frac{\tanh (c+d x)}{3 b d \sqrt{b \coth ^4(c+d x)}} \]
[Out]
-(Coth[c + d*x]/(b*d*Sqrt[b*Coth[c + d*x]^4])) + (x*Coth[c + d*x]^2)/(b*Sqrt[b*Coth[c + d*x]^4]) - Tanh[c + d*
x]/(3*b*d*Sqrt[b*Coth[c + d*x]^4]) - Tanh[c + d*x]^3/(5*b*d*Sqrt[b*Coth[c + d*x]^4])
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Rubi [A] time = 0.0459973, antiderivative size = 118, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used =
{3658, 3473, 8} \[ \frac{x \coth ^2(c+d x)}{b \sqrt{b \coth ^4(c+d x)}}-\frac{\coth (c+d x)}{b d \sqrt{b \coth ^4(c+d x)}}-\frac{\tanh ^3(c+d x)}{5 b d \sqrt{b \coth ^4(c+d x)}}-\frac{\tanh (c+d x)}{3 b d \sqrt{b \coth ^4(c+d x)}} \]
Antiderivative was successfully verified.
[In]
Int[(b*Coth[c + d*x]^4)^(-3/2),x]
[Out]
-(Coth[c + d*x]/(b*d*Sqrt[b*Coth[c + d*x]^4])) + (x*Coth[c + d*x]^2)/(b*Sqrt[b*Coth[c + d*x]^4]) - Tanh[c + d*
x]/(3*b*d*Sqrt[b*Coth[c + d*x]^4]) - Tanh[c + d*x]^3/(5*b*d*Sqrt[b*Coth[c + d*x]^4])
Rule 3658
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
]])
Rule 3473
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 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rubi steps
\begin{align*} \int \frac{1}{\left (b \coth ^4(c+d x)\right )^{3/2}} \, dx &=\frac{\coth ^2(c+d x) \int \tanh ^6(c+d x) \, dx}{b \sqrt{b \coth ^4(c+d x)}}\\ &=-\frac{\tanh ^3(c+d x)}{5 b d \sqrt{b \coth ^4(c+d x)}}+\frac{\coth ^2(c+d x) \int \tanh ^4(c+d x) \, dx}{b \sqrt{b \coth ^4(c+d x)}}\\ &=-\frac{\tanh (c+d x)}{3 b d \sqrt{b \coth ^4(c+d x)}}-\frac{\tanh ^3(c+d x)}{5 b d \sqrt{b \coth ^4(c+d x)}}+\frac{\coth ^2(c+d x) \int \tanh ^2(c+d x) \, dx}{b \sqrt{b \coth ^4(c+d x)}}\\ &=-\frac{\coth (c+d x)}{b d \sqrt{b \coth ^4(c+d x)}}-\frac{\tanh (c+d x)}{3 b d \sqrt{b \coth ^4(c+d x)}}-\frac{\tanh ^3(c+d x)}{5 b d \sqrt{b \coth ^4(c+d x)}}+\frac{\coth ^2(c+d x) \int 1 \, dx}{b \sqrt{b \coth ^4(c+d x)}}\\ &=-\frac{\coth (c+d x)}{b d \sqrt{b \coth ^4(c+d x)}}+\frac{x \coth ^2(c+d x)}{b \sqrt{b \coth ^4(c+d x)}}-\frac{\tanh (c+d x)}{3 b d \sqrt{b \coth ^4(c+d x)}}-\frac{\tanh ^3(c+d x)}{5 b d \sqrt{b \coth ^4(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.232707, size = 68, normalized size = 0.58 \[ \frac{-3 \tanh ^3(c+d x)-5 \tanh (c+d x)-15 \coth (c+d x)+15 \tanh ^{-1}(\tanh (c+d x)) \coth ^2(c+d x)}{15 b d \sqrt{b \coth ^4(c+d x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[(b*Coth[c + d*x]^4)^(-3/2),x]
[Out]
(-15*Coth[c + d*x] + 15*ArcTanh[Tanh[c + d*x]]*Coth[c + d*x]^2 - 5*Tanh[c + d*x] - 3*Tanh[c + d*x]^3)/(15*b*d*
Sqrt[b*Coth[c + d*x]^4])
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Maple [A] time = 0.022, size = 84, normalized size = 0.7 \begin{align*}{\frac{{\rm coth} \left (dx+c\right ) \left ( 15\,\ln \left ({\rm coth} \left (dx+c\right )+1 \right ) \left ({\rm coth} \left (dx+c\right ) \right ) ^{5}-15\,\ln \left ({\rm coth} \left (dx+c\right )-1 \right ) \left ({\rm coth} \left (dx+c\right ) \right ) ^{5}-30\, \left ({\rm coth} \left (dx+c\right ) \right ) ^{4}-10\, \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}-6 \right ) }{30\,d} \left ( b \left ({\rm coth} \left (dx+c\right ) \right ) ^{4} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*coth(d*x+c)^4)^(3/2),x)
[Out]
1/30/d*coth(d*x+c)*(15*ln(coth(d*x+c)+1)*coth(d*x+c)^5-15*ln(coth(d*x+c)-1)*coth(d*x+c)^5-30*coth(d*x+c)^4-10*
coth(d*x+c)^2-6)/(b*coth(d*x+c)^4)^(3/2)
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Maxima [A] time = 1.80108, size = 209, normalized size = 1.77 \begin{align*} -\frac{2 \,{\left (70 \, \sqrt{b} e^{\left (-2 \, d x - 2 \, c\right )} + 140 \, \sqrt{b} e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, \sqrt{b} e^{\left (-6 \, d x - 6 \, c\right )} + 45 \, \sqrt{b} e^{\left (-8 \, d x - 8 \, c\right )} + 23 \, \sqrt{b}\right )}}{15 \,{\left (5 \, b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, b^{2} e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, b^{2} e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, b^{2} e^{\left (-8 \, d x - 8 \, c\right )} + b^{2} e^{\left (-10 \, d x - 10 \, c\right )} + b^{2}\right )} d} + \frac{d x + c}{b^{\frac{3}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*coth(d*x+c)^4)^(3/2),x, algorithm="maxima")
[Out]
-2/15*(70*sqrt(b)*e^(-2*d*x - 2*c) + 140*sqrt(b)*e^(-4*d*x - 4*c) + 90*sqrt(b)*e^(-6*d*x - 6*c) + 45*sqrt(b)*e
^(-8*d*x - 8*c) + 23*sqrt(b))/((5*b^2*e^(-2*d*x - 2*c) + 10*b^2*e^(-4*d*x - 4*c) + 10*b^2*e^(-6*d*x - 6*c) + 5
*b^2*e^(-8*d*x - 8*c) + b^2*e^(-10*d*x - 10*c) + b^2)*d) + (d*x + c)/(b^(3/2)*d)
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Fricas [B] time = 2.67664, size = 9179, normalized size = 77.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*coth(d*x+c)^4)^(3/2),x, algorithm="fricas")
[Out]
1/15*(15*d*x*cosh(d*x + c)^10 + 15*(d*x*e^(4*d*x + 4*c) - 2*d*x*e^(2*d*x + 2*c) + d*x)*sinh(d*x + c)^10 + 150*
(d*x*cosh(d*x + c)*e^(4*d*x + 4*c) - 2*d*x*cosh(d*x + c)*e^(2*d*x + 2*c) + d*x*cosh(d*x + c))*sinh(d*x + c)^9
+ 15*(5*d*x + 6)*cosh(d*x + c)^8 + 15*(45*d*x*cosh(d*x + c)^2 + 5*d*x + (45*d*x*cosh(d*x + c)^2 + 5*d*x + 6)*e
^(4*d*x + 4*c) - 2*(45*d*x*cosh(d*x + c)^2 + 5*d*x + 6)*e^(2*d*x + 2*c) + 6)*sinh(d*x + c)^8 + 120*(15*d*x*cos
h(d*x + c)^3 + (5*d*x + 6)*cosh(d*x + c) + (15*d*x*cosh(d*x + c)^3 + (5*d*x + 6)*cosh(d*x + c))*e^(4*d*x + 4*c
) - 2*(15*d*x*cosh(d*x + c)^3 + (5*d*x + 6)*cosh(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x + c)^7 + 30*(5*d*x + 6)*c
osh(d*x + c)^6 + 30*(105*d*x*cosh(d*x + c)^4 + 14*(5*d*x + 6)*cosh(d*x + c)^2 + 5*d*x + (105*d*x*cosh(d*x + c)
^4 + 14*(5*d*x + 6)*cosh(d*x + c)^2 + 5*d*x + 6)*e^(4*d*x + 4*c) - 2*(105*d*x*cosh(d*x + c)^4 + 14*(5*d*x + 6)
*cosh(d*x + c)^2 + 5*d*x + 6)*e^(2*d*x + 2*c) + 6)*sinh(d*x + c)^6 + 60*(63*d*x*cosh(d*x + c)^5 + 14*(5*d*x +
6)*cosh(d*x + c)^3 + 3*(5*d*x + 6)*cosh(d*x + c) + (63*d*x*cosh(d*x + c)^5 + 14*(5*d*x + 6)*cosh(d*x + c)^3 +
3*(5*d*x + 6)*cosh(d*x + c))*e^(4*d*x + 4*c) - 2*(63*d*x*cosh(d*x + c)^5 + 14*(5*d*x + 6)*cosh(d*x + c)^3 + 3*
(5*d*x + 6)*cosh(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x + c)^5 + 10*(15*d*x + 28)*cosh(d*x + c)^4 + 10*(315*d*x*c
osh(d*x + c)^6 + 105*(5*d*x + 6)*cosh(d*x + c)^4 + 45*(5*d*x + 6)*cosh(d*x + c)^2 + 15*d*x + (315*d*x*cosh(d*x
+ c)^6 + 105*(5*d*x + 6)*cosh(d*x + c)^4 + 45*(5*d*x + 6)*cosh(d*x + c)^2 + 15*d*x + 28)*e^(4*d*x + 4*c) - 2*
(315*d*x*cosh(d*x + c)^6 + 105*(5*d*x + 6)*cosh(d*x + c)^4 + 45*(5*d*x + 6)*cosh(d*x + c)^2 + 15*d*x + 28)*e^(
2*d*x + 2*c) + 28)*sinh(d*x + c)^4 + 40*(45*d*x*cosh(d*x + c)^7 + 21*(5*d*x + 6)*cosh(d*x + c)^5 + 15*(5*d*x +
6)*cosh(d*x + c)^3 + (15*d*x + 28)*cosh(d*x + c) + (45*d*x*cosh(d*x + c)^7 + 21*(5*d*x + 6)*cosh(d*x + c)^5 +
15*(5*d*x + 6)*cosh(d*x + c)^3 + (15*d*x + 28)*cosh(d*x + c))*e^(4*d*x + 4*c) - 2*(45*d*x*cosh(d*x + c)^7 + 2
1*(5*d*x + 6)*cosh(d*x + c)^5 + 15*(5*d*x + 6)*cosh(d*x + c)^3 + (15*d*x + 28)*cosh(d*x + c))*e^(2*d*x + 2*c))
*sinh(d*x + c)^3 + 5*(15*d*x + 28)*cosh(d*x + c)^2 + 5*(135*d*x*cosh(d*x + c)^8 + 84*(5*d*x + 6)*cosh(d*x + c)
^6 + 90*(5*d*x + 6)*cosh(d*x + c)^4 + 12*(15*d*x + 28)*cosh(d*x + c)^2 + 15*d*x + (135*d*x*cosh(d*x + c)^8 + 8
4*(5*d*x + 6)*cosh(d*x + c)^6 + 90*(5*d*x + 6)*cosh(d*x + c)^4 + 12*(15*d*x + 28)*cosh(d*x + c)^2 + 15*d*x + 2
8)*e^(4*d*x + 4*c) - 2*(135*d*x*cosh(d*x + c)^8 + 84*(5*d*x + 6)*cosh(d*x + c)^6 + 90*(5*d*x + 6)*cosh(d*x + c
)^4 + 12*(15*d*x + 28)*cosh(d*x + c)^2 + 15*d*x + 28)*e^(2*d*x + 2*c) + 28)*sinh(d*x + c)^2 + 15*d*x + (15*d*x
*cosh(d*x + c)^10 + 15*(5*d*x + 6)*cosh(d*x + c)^8 + 30*(5*d*x + 6)*cosh(d*x + c)^6 + 10*(15*d*x + 28)*cosh(d*
x + c)^4 + 5*(15*d*x + 28)*cosh(d*x + c)^2 + 15*d*x + 46)*e^(4*d*x + 4*c) - 2*(15*d*x*cosh(d*x + c)^10 + 15*(5
*d*x + 6)*cosh(d*x + c)^8 + 30*(5*d*x + 6)*cosh(d*x + c)^6 + 10*(15*d*x + 28)*cosh(d*x + c)^4 + 5*(15*d*x + 28
)*cosh(d*x + c)^2 + 15*d*x + 46)*e^(2*d*x + 2*c) + 10*(15*d*x*cosh(d*x + c)^9 + 12*(5*d*x + 6)*cosh(d*x + c)^7
+ 18*(5*d*x + 6)*cosh(d*x + c)^5 + 4*(15*d*x + 28)*cosh(d*x + c)^3 + (15*d*x + 28)*cosh(d*x + c) + (15*d*x*co
sh(d*x + c)^9 + 12*(5*d*x + 6)*cosh(d*x + c)^7 + 18*(5*d*x + 6)*cosh(d*x + c)^5 + 4*(15*d*x + 28)*cosh(d*x + c
)^3 + (15*d*x + 28)*cosh(d*x + c))*e^(4*d*x + 4*c) - 2*(15*d*x*cosh(d*x + c)^9 + 12*(5*d*x + 6)*cosh(d*x + c)^
7 + 18*(5*d*x + 6)*cosh(d*x + c)^5 + 4*(15*d*x + 28)*cosh(d*x + c)^3 + (15*d*x + 28)*cosh(d*x + c))*e^(2*d*x +
2*c))*sinh(d*x + c) + 46)*sqrt((b*e^(8*d*x + 8*c) + 4*b*e^(6*d*x + 6*c) + 6*b*e^(4*d*x + 4*c) + 4*b*e^(2*d*x
+ 2*c) + b)/(e^(8*d*x + 8*c) - 4*e^(6*d*x + 6*c) + 6*e^(4*d*x + 4*c) - 4*e^(2*d*x + 2*c) + 1))/(b^2*d*cosh(d*x
+ c)^10 + 5*b^2*d*cosh(d*x + c)^8 + (b^2*d*e^(4*d*x + 4*c) + 2*b^2*d*e^(2*d*x + 2*c) + b^2*d)*sinh(d*x + c)^1
0 + 10*(b^2*d*cosh(d*x + c)*e^(4*d*x + 4*c) + 2*b^2*d*cosh(d*x + c)*e^(2*d*x + 2*c) + b^2*d*cosh(d*x + c))*sin
h(d*x + c)^9 + 10*b^2*d*cosh(d*x + c)^6 + 5*(9*b^2*d*cosh(d*x + c)^2 + b^2*d + (9*b^2*d*cosh(d*x + c)^2 + b^2*
d)*e^(4*d*x + 4*c) + 2*(9*b^2*d*cosh(d*x + c)^2 + b^2*d)*e^(2*d*x + 2*c))*sinh(d*x + c)^8 + 40*(3*b^2*d*cosh(d
*x + c)^3 + b^2*d*cosh(d*x + c) + (3*b^2*d*cosh(d*x + c)^3 + b^2*d*cosh(d*x + c))*e^(4*d*x + 4*c) + 2*(3*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)^7 + 10*b^2*d*cosh(d*x + c)^4 + 10*(21*b
^2*d*cosh(d*x + c)^4 + 14*b^2*d*cosh(d*x + c)^2 + b^2*d + (21*b^2*d*cosh(d*x + c)^4 + 14*b^2*d*cosh(d*x + c)^2
+ b^2*d)*e^(4*d*x + 4*c) + 2*(21*b^2*d*cosh(d*x + c)^4 + 14*b^2*d*cosh(d*x + c)^2 + b^2*d)*e^(2*d*x + 2*c))*s
inh(d*x + c)^6 + 4*(63*b^2*d*cosh(d*x + c)^5 + 70*b^2*d*cosh(d*x + c)^3 + 15*b^2*d*cosh(d*x + c) + (63*b^2*d*c
osh(d*x + c)^5 + 70*b^2*d*cosh(d*x + c)^3 + 15*b^2*d*cosh(d*x + c))*e^(4*d*x + 4*c) + 2*(63*b^2*d*cosh(d*x + c
)^5 + 70*b^2*d*cosh(d*x + c)^3 + 15*b^2*d*cosh(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x + c)^5 + 5*b^2*d*cosh(d*x +
c)^2 + 10*(21*b^2*d*cosh(d*x + c)^6 + 35*b^2*d*cosh(d*x + c)^4 + 15*b^2*d*cosh(d*x + c)^2 + b^2*d + (21*b^2*d
*cosh(d*x + c)^6 + 35*b^2*d*cosh(d*x + c)^4 + 15*b^2*d*cosh(d*x + c)^2 + b^2*d)*e^(4*d*x + 4*c) + 2*(21*b^2*d*
cosh(d*x + c)^6 + 35*b^2*d*cosh(d*x + c)^4 + 15*b^2*d*cosh(d*x + c)^2 + b^2*d)*e^(2*d*x + 2*c))*sinh(d*x + c)^
4 + 40*(3*b^2*d*cosh(d*x + c)^7 + 7*b^2*d*cosh(d*x + c)^5 + 5*b^2*d*cosh(d*x + c)^3 + b^2*d*cosh(d*x + c) + (3
*b^2*d*cosh(d*x + c)^7 + 7*b^2*d*cosh(d*x + c)^5 + 5*b^2*d*cosh(d*x + c)^3 + b^2*d*cosh(d*x + c))*e^(4*d*x + 4
*c) + 2*(3*b^2*d*cosh(d*x + c)^7 + 7*b^2*d*cosh(d*x + c)^5 + 5*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)^3 + b^2*d + 5*(9*b^2*d*cosh(d*x + c)^8 + 28*b^2*d*cosh(d*x + c)^6 + 30*b^2*d*cosh
(d*x + c)^4 + 12*b^2*d*cosh(d*x + c)^2 + b^2*d + (9*b^2*d*cosh(d*x + c)^8 + 28*b^2*d*cosh(d*x + c)^6 + 30*b^2*
d*cosh(d*x + c)^4 + 12*b^2*d*cosh(d*x + c)^2 + b^2*d)*e^(4*d*x + 4*c) + 2*(9*b^2*d*cosh(d*x + c)^8 + 28*b^2*d*
cosh(d*x + c)^6 + 30*b^2*d*cosh(d*x + c)^4 + 12*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)^10 + 5*b^2*d*cosh(d*x + c)^8 + 10*b^2*d*cosh(d*x + c)^6 + 10*b^2*d*cosh(d*x + c)^4 +
5*b^2*d*cosh(d*x + c)^2 + b^2*d)*e^(4*d*x + 4*c) + 2*(b^2*d*cosh(d*x + c)^10 + 5*b^2*d*cosh(d*x + c)^8 + 10*b^
2*d*cosh(d*x + c)^6 + 10*b^2*d*cosh(d*x + c)^4 + 5*b^2*d*cosh(d*x + c)^2 + b^2*d)*e^(2*d*x + 2*c) + 10*(b^2*d*
cosh(d*x + c)^9 + 4*b^2*d*cosh(d*x + c)^7 + 6*b^2*d*cosh(d*x + c)^5 + 4*b^2*d*cosh(d*x + c)^3 + b^2*d*cosh(d*x
+ c) + (b^2*d*cosh(d*x + c)^9 + 4*b^2*d*cosh(d*x + c)^7 + 6*b^2*d*cosh(d*x + c)^5 + 4*b^2*d*cosh(d*x + c)^3 +
b^2*d*cosh(d*x + c))*e^(4*d*x + 4*c) + 2*(b^2*d*cosh(d*x + c)^9 + 4*b^2*d*cosh(d*x + c)^7 + 6*b^2*d*cosh(d*x
+ c)^5 + 4*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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \coth ^{4}{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*coth(d*x+c)**4)**(3/2),x)
[Out]
Integral((b*coth(c + d*x)**4)**(-3/2), x)
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Giac [A] time = 1.25118, size = 138, normalized size = 1.17 \begin{align*} \frac{\frac{15 \,{\left (d x + c\right )}}{\sqrt{b} d} + \frac{2 \,{\left (45 \, \sqrt{b} e^{\left (8 \, d x + 8 \, c\right )} + 90 \, \sqrt{b} e^{\left (6 \, d x + 6 \, c\right )} + 140 \, \sqrt{b} e^{\left (4 \, d x + 4 \, c\right )} + 70 \, \sqrt{b} e^{\left (2 \, d x + 2 \, c\right )} + 23 \, \sqrt{b}\right )}}{b d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{15 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*coth(d*x+c)^4)^(3/2),x, algorithm="giac")
[Out]
1/15*(15*(d*x + c)/(sqrt(b)*d) + 2*(45*sqrt(b)*e^(8*d*x + 8*c) + 90*sqrt(b)*e^(6*d*x + 6*c) + 140*sqrt(b)*e^(4
*d*x + 4*c) + 70*sqrt(b)*e^(2*d*x + 2*c) + 23*sqrt(b))/(b*d*(e^(2*d*x + 2*c) + 1)^5))/b