3.1.25 \(\int (a+b \coth ^2(x))^{3/2} \, dx\) [25]

Optimal. Leaf size=88 \[ -\frac {1}{2} \sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-\frac {1}{2} b \coth (x) \sqrt {a+b \coth ^2(x)} \]

[Out]

(a+b)^(3/2)*arctanh(coth(x)*(a+b)^(1/2)/(a+b*coth(x)^2)^(1/2))-1/2*(3*a+2*b)*arctanh(coth(x)*b^(1/2)/(a+b*coth
(x)^2)^(1/2))*b^(1/2)-1/2*b*coth(x)*(a+b*coth(x)^2)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3742, 427, 537, 223, 212, 385} \begin {gather*} -\frac {1}{2} b \coth (x) \sqrt {a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-\frac {1}{2} \sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Coth[x]^2)^(3/2),x]

[Out]

-1/2*(Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]) + (a + b)^(3/2)*ArcTanh[(Sqrt[a +
b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] - (b*Coth[x]*Sqrt[a + b*Coth[x]^2])/2

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 3742

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[c*(ff/f), Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps

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

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Mathematica [A]
time = 0.44, size = 111, normalized size = 1.26 \begin {gather*} \frac {1}{2} \left (-2 (-a-b)^{3/2} \text {ArcTan}\left (\frac {\coth (x) \sqrt {a+b \coth ^2(x)}-\sqrt {b} \text {csch}^2(x)}{\sqrt {-a-b}}\right )-b \coth (x) \sqrt {a+b \coth ^2(x)}+\sqrt {b} (3 a+2 b) \log \left (-\sqrt {b} \coth (x)+\sqrt {a+b \coth ^2(x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Coth[x]^2)^(3/2),x]

[Out]

(-2*(-a - b)^(3/2)*ArcTan[(Coth[x]*Sqrt[a + b*Coth[x]^2] - Sqrt[b]*Csch[x]^2)/Sqrt[-a - b]] - b*Coth[x]*Sqrt[a
+ b*Coth[x]^2] + Sqrt[b]*(3*a + 2*b)*Log[-(Sqrt[b]*Coth[x]) + Sqrt[a + b*Coth[x]^2]])/2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(472\) vs. \(2(70)=140\).
time = 0.77, size = 473, normalized size = 5.38

method	result	size
derivativedivides	\(\frac {\left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}{6}-\frac {b \left (\frac {\left (2 b \left (1+\coth \left (x \right )\right )-2 b \right ) \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{4 b}+\frac {\left (4 b \left (a +b \right )-4 b^{2}\right ) \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}+\frac {\left (a +b \right ) \left (\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}-\sqrt {b}\, \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )-\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )\right )}{2}-\frac {\left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}{6}-\frac {b \left (\frac {\left (2 b \left (\coth \left (x \right )-1\right )+2 b \right ) \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{4 b}+\frac {\left (4 b \left (a +b \right )-4 b^{2}\right ) \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}-\frac {\left (a +b \right ) \left (\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}+\sqrt {b}\, \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )-\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )\right )}{2}\)	\(473\)
default	\(\frac {\left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}{6}-\frac {b \left (\frac {\left (2 b \left (1+\coth \left (x \right )\right )-2 b \right ) \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{4 b}+\frac {\left (4 b \left (a +b \right )-4 b^{2}\right ) \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}+\frac {\left (a +b \right ) \left (\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}-\sqrt {b}\, \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )-\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )\right )}{2}-\frac {\left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}{6}-\frac {b \left (\frac {\left (2 b \left (\coth \left (x \right )-1\right )+2 b \right ) \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{4 b}+\frac {\left (4 b \left (a +b \right )-4 b^{2}\right ) \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}-\frac {\left (a +b \right ) \left (\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}+\sqrt {b}\, \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )-\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )\right )}{2}\)	\(473\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(3/2)-1/2*b*(1/4*(2*b*(1+coth(x))-2*b)/b*(b*(1+coth(x))^2-2*b*(1+cot
h(x))+a+b)^(1/2)+1/8*(4*b*(a+b)-4*b^2)/b^(3/2)*ln((b*(1+coth(x))-b)/b^(1/2)+(b*(1+coth(x))^2-2*b*(1+coth(x))+a
+b)^(1/2)))+1/2*(a+b)*((b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2)-b^(1/2)*ln((b*(1+coth(x))-b)/b^(1/2)+(b*(1+
coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2))-(a+b)^(1/2)*ln((2*a+2*b-2*b*(1+coth(x))+2*(a+b)^(1/2)*(b*(1+coth(x))^2-
2*b*(1+coth(x))+a+b)^(1/2))/(1+coth(x))))-1/6*(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(3/2)-1/2*b*(1/4*(2*b*(cot
h(x)-1)+2*b)/b*(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2)+1/8*(4*b*(a+b)-4*b^2)/b^(3/2)*ln((b*(coth(x)-1)+b)/
b^(1/2)+(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2)))-1/2*(a+b)*((b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2)+b
^(1/2)*ln((b*(coth(x)-1)+b)/b^(1/2)+(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2))-(a+b)^(1/2)*ln((2*a+2*b+2*b*(
coth(x)-1)+2*(a+b)^(1/2)*(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2))/(coth(x)-1)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*coth(x)^2 + a)^(3/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 947 vs. \(2 (70) = 140\).
time = 0.56, size = 5037, normalized size = 57.24 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 - 2*(a + b)*cosh(x)^2 + 2*(3*(a + b
)*cosh(x)^2 - a - b)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 - (a + b)*cosh(x))*sinh(x) + a + b)*sqrt(a + b)*log(((a*
b^2 + b^3)*cosh(x)^8 + 8*(a*b^2 + b^3)*cosh(x)*sinh(x)^7 + (a*b^2 + b^3)*sinh(x)^8 + 2*(a*b^2 + 2*b^3)*cosh(x)
^6 + 2*(a*b^2 + 2*b^3 + 14*(a*b^2 + b^3)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a*b^2 + b^3)*cosh(x)^3 + 3*(a*b^2 + 2*b
^3)*cosh(x))*sinh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^4 + (70*(a*b^2 + b^3)*cosh(x)^4 + a^3 - a^2*b
+ 4*a*b^2 + 6*b^3 + 30*(a*b^2 + 2*b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a*b^2 + b^3)*cosh(x)^5 + 10*(a*b^2 + 2*b
^3)*cosh(x)^3 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x))*sinh(x)^3 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(a^3 -
3*a*b^2 - 2*b^3)*cosh(x)^2 + 2*(14*(a*b^2 + b^3)*cosh(x)^6 + 15*(a*b^2 + 2*b^3)*cosh(x)^4 - a^3 + 3*a*b^2 + 2*
b^3 + 3*(a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(b^2*cosh(x)^6 + 6*b^2*cosh(x)*sinh(x)^
5 + b^2*sinh(x)^6 + 3*b^2*cosh(x)^4 + 3*(5*b^2*cosh(x)^2 + b^2)*sinh(x)^4 + 4*(5*b^2*cosh(x)^3 + 3*b^2*cosh(x)
)*sinh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x)^2 + (15*b^2*cosh(x)^4 + 18*b^2*cosh(x)^2 - a^2 + 2*a*b + 3*b^2)*si
nh(x)^2 + a^2 + 2*a*b + b^2 + 2*(3*b^2*cosh(x)^5 + 6*b^2*cosh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x))*sinh(x))*s
qrt(a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) +
4*(2*(a*b^2 + b^3)*cosh(x)^7 + 3*(a*b^2 + 2*b^3)*cosh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^3 - (a^3
- 3*a*b^2 - 2*b^3)*cosh(x))*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3
*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + ((3*a + 2*b)*cosh(x)^4 + 4*(3*a + 2*
b)*cosh(x)*sinh(x)^3 + (3*a + 2*b)*sinh(x)^4 - 2*(3*a + 2*b)*cosh(x)^2 + 2*(3*(3*a + 2*b)*cosh(x)^2 - 3*a - 2*
b)*sinh(x)^2 + 4*((3*a + 2*b)*cosh(x)^3 - (3*a + 2*b)*cosh(x))*sinh(x) + 3*a + 2*b)*sqrt(b)*log(-((a + 2*b)*co
sh(x)^4 + 4*(a + 2*b)*cosh(x)*sinh(x)^3 + (a + 2*b)*sinh(x)^4 - 2*(a - 2*b)*cosh(x)^2 + 2*(3*(a + 2*b)*cosh(x)
^2 - a + 2*b)*sinh(x)^2 - 2*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(b)*sqrt(((a + b)*cosh
(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*((a + 2*b)*cosh(x)^3 - (a
- 2*b)*cosh(x))*sinh(x) + a + 2*b)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^
2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)) + ((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (
a + b)*sinh(x)^4 - 2*(a + b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 - a - b)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 - (a
+ b)*cosh(x))*sinh(x) + a + b)*sqrt(a + b)*log(-((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*si
nh(x)^4 - 2*a*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 - a)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sin
h(x)^2 - 1)*sqrt(a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) +
sinh(x)^2)) + 4*((a + b)*cosh(x)^3 - a*cosh(x))*sinh(x) + a + b)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2))
- 2*sqrt(2)*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2
- a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cos
h(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1), 1/4*(2*((3*a + 2*b)*cosh(x)^4 + 4*
(3*a + 2*b)*cosh(x)*sinh(x)^3 + (3*a + 2*b)*sinh(x)^4 - 2*(3*a + 2*b)*cosh(x)^2 + 2*(3*(3*a + 2*b)*cosh(x)^2 -
3*a - 2*b)*sinh(x)^2 + 4*((3*a + 2*b)*cosh(x)^3 - (3*a + 2*b)*cosh(x))*sinh(x) + 3*a + 2*b)*sqrt(-b)*arctan(s
qrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(-b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 -
a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)
*sinh(x)^4 - 2*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 - a + b)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 - (a - b)*
cosh(x))*sinh(x) + a + b)) + ((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 - 2*(a + b)*
cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 - a - b)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 - (a + b)*cosh(x))*sinh(x) + a +
b)*sqrt(a + b)*log(((a*b^2 + b^3)*cosh(x)^8 + 8*(a*b^2 + b^3)*cosh(x)*sinh(x)^7 + (a*b^2 + b^3)*sinh(x)^8 + 2*
(a*b^2 + 2*b^3)*cosh(x)^6 + 2*(a*b^2 + 2*b^3 + 14*(a*b^2 + b^3)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a*b^2 + b^3)*cos
h(x)^3 + 3*(a*b^2 + 2*b^3)*cosh(x))*sinh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^4 + (70*(a*b^2 + b^3)*
cosh(x)^4 + a^3 - a^2*b + 4*a*b^2 + 6*b^3 + 30*(a*b^2 + 2*b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a*b^2 + b^3)*cosh
(x)^5 + 10*(a*b^2 + 2*b^3)*cosh(x)^3 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x))*sinh(x)^3 + a^3 + 3*a^2*b + 3*
a*b^2 + b^3 - 2*(a^3 - 3*a*b^2 - 2*b^3)*cosh(x)^2 + 2*(14*(a*b^2 + b^3)*cosh(x)^6 + 15*(a*b^2 + 2*b^3)*cosh(x)
^4 - a^3 + 3*a*b^2 + 2*b^3 + 3*(a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(b^2*cosh(x)^6 +
6*b^2*cosh(x)*sinh(x)^5 + b^2*sinh(x)^6 + 3*b^...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(x)**2)**(3/2),x)

[Out]

Integral((a + b*coth(x)**2)**(3/2), x)

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Giac [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((a+b*coth(x)^2)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(ex

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*coth(x)^2)^(3/2),x)

[Out]

int((a + b*coth(x)^2)^(3/2), x)

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