3.1.23 \(\int \coth ^2(x) (a+b \coth ^2(x))^{3/2} \, dx\) [23]

Optimal. Leaf size=123 \[ -\frac {\left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{8 \sqrt {b}}+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-\frac {1}{8} (5 a+4 b) \coth (x) \sqrt {a+b \coth ^2(x)}-\frac {1}{4} b \coth ^3(x) \sqrt {a+b \coth ^2(x)} \]

[Out]

(a+b)^(3/2)*arctanh(coth(x)*(a+b)^(1/2)/(a+b*coth(x)^2)^(1/2))-1/8*(3*a^2+12*a*b+8*b^2)*arctanh(coth(x)*b^(1/2
)/(a+b*coth(x)^2)^(1/2))/b^(1/2)-1/8*(5*a+4*b)*coth(x)*(a+b*coth(x)^2)^(1/2)-1/4*b*coth(x)^3*(a+b*coth(x)^2)^(
1/2)

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Rubi [A]
time = 0.17, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3751, 488, 596, 537, 223, 212, 385} \begin {gather*} -\frac {\left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{8 \sqrt {b}}-\frac {1}{8} (5 a+4 b) \coth (x) \sqrt {a+b \coth ^2(x)}-\frac {1}{4} b \coth ^3(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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]^2*(a + b*Coth[x]^2)^(3/2),x]

[Out]

-1/8*((3*a^2 + 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]])/Sqrt[b] + (a + b)^(3/2)*ArcTa
nh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] - ((5*a + 4*b)*Coth[x]*Sqrt[a + b*Coth[x]^2])/8 - (b*Coth[x]^3
*Sqrt[a + b*Coth[x]^2])/4

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 488

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*(e*x)^
(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Dist[1/(b*(m + n*(p + q) + 1
)), Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d
)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && N
eQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, 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 596

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q +
1) + 1))), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((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[(d*ff*(x/c))^m*((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, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

\begin {align*} \int \coth ^2(x) \left (a+b \coth ^2(x)\right )^{3/2} \, dx &=\text {Subst}\left (\int \frac {x^2 \left (a+b x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{4} b \coth ^3(x) \sqrt {a+b \coth ^2(x)}-\frac {1}{4} \text {Subst}\left (\int \frac {x^2 \left (-a (4 a+3 b)-b (5 a+4 b) x^2\right )}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{8} (5 a+4 b) \coth (x) \sqrt {a+b \coth ^2(x)}-\frac {1}{4} b \coth ^3(x) \sqrt {a+b \coth ^2(x)}-\frac {\text {Subst}\left (\int \frac {-a b (5 a+4 b)-b \left (3 a^2+12 a b+8 b^2\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{8 b}\\ &=-\frac {1}{8} (5 a+4 b) \coth (x) \sqrt {a+b \coth ^2(x)}-\frac {1}{4} b \coth ^3(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}{8} \left (3 a^2+12 a b+8 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{8} (5 a+4 b) \coth (x) \sqrt {a+b \coth ^2(x)}-\frac {1}{4} b \coth ^3(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}{8} \left (3 a^2+12 a b+8 b^2\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )\\ &=-\frac {\left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{8 \sqrt {b}}+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-\frac {1}{8} (5 a+4 b) \coth (x) \sqrt {a+b \coth ^2(x)}-\frac {1}{4} b \coth ^3(x) \sqrt {a+b \coth ^2(x)}\\ \end {align*}

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Mathematica [A]
time = 0.67, size = 219, normalized size = 1.78 \begin {gather*} \frac {\sqrt {(-a+b+(a+b) \cosh (2 x)) \text {csch}^2(x)} \left (-\sqrt {2} \sqrt {a+b} \left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \cosh (x)}{\sqrt {-a+b+(a+b) \cosh (2 x)}}\right )+\sqrt {b} \left (8 \sqrt {2} (a+b)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a+b} \cosh (x)}{\sqrt {-a+b+(a+b) \cosh (2 x)}}\right )-\sqrt {a+b} \sqrt {-a+b+(a+b) \cosh (2 x)} \coth (x) \text {csch}(x) \left (5 a+6 b+2 b \text {csch}^2(x)\right )\right )\right ) \sinh (x)}{8 \sqrt {2} \sqrt {b} \sqrt {a+b} \sqrt {-a+b+(a+b) \cosh (2 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(-a + b + (a + b)*Cosh[2*x])*Csch[x]^2]*(-(Sqrt[2]*Sqrt[a + b]*(3*a^2 + 12*a*b + 8*b^2)*ArcTanh[(Sqrt[2]
*Sqrt[b]*Cosh[x])/Sqrt[-a + b + (a + b)*Cosh[2*x]]]) + Sqrt[b]*(8*Sqrt[2]*(a + b)^2*ArcTanh[(Sqrt[2]*Sqrt[a +
b]*Cosh[x])/Sqrt[-a + b + (a + b)*Cosh[2*x]]] - Sqrt[a + b]*Sqrt[-a + b + (a + b)*Cosh[2*x]]*Coth[x]*Csch[x]*(
5*a + 6*b + 2*b*Csch[x]^2)))*Sinh[x])/(8*Sqrt[2]*Sqrt[b]*Sqrt[a + b]*Sqrt[-a + b + (a + b)*Cosh[2*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(528\) vs. \(2(101)=202\).
time = 0.63, size = 529, normalized size = 4.30 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*coth(x)*(a+b*coth(x)^2)^(3/2)-3/4*a*(1/2*coth(x)*(a+b*coth(x)^2)^(1/2)+1/2*a/b^(1/2)*ln(b^(1/2)*coth(x)+(
a+b*coth(x)^2)^(1/2)))+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+coth(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+cot
h(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*(coth(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(coth(x)^2*(a+b*coth(x)^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2260 vs. \(2 (101) = 202\).
time = 0.77, size = 10286, normalized size = 83.63 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(4*((a*b + b^2)*cosh(x)^8 + 8*(a*b + b^2)*cosh(x)*sinh(x)^7 + (a*b + b^2)*sinh(x)^8 - 4*(a*b + b^2)*cosh
(x)^6 + 4*(7*(a*b + b^2)*cosh(x)^2 - a*b - b^2)*sinh(x)^6 + 8*(7*(a*b + b^2)*cosh(x)^3 - 3*(a*b + b^2)*cosh(x)
)*sinh(x)^5 + 6*(a*b + b^2)*cosh(x)^4 + 2*(35*(a*b + b^2)*cosh(x)^4 - 30*(a*b + b^2)*cosh(x)^2 + 3*a*b + 3*b^2
)*sinh(x)^4 + 8*(7*(a*b + b^2)*cosh(x)^5 - 10*(a*b + b^2)*cosh(x)^3 + 3*(a*b + b^2)*cosh(x))*sinh(x)^3 - 4*(a*
b + b^2)*cosh(x)^2 + 4*(7*(a*b + b^2)*cosh(x)^6 - 15*(a*b + b^2)*cosh(x)^4 + 9*(a*b + b^2)*cosh(x)^2 - a*b - b
^2)*sinh(x)^2 + a*b + b^2 + 8*((a*b + b^2)*cosh(x)^7 - 3*(a*b + b^2)*cosh(x)^5 + 3*(a*b + b^2)*cosh(x)^3 - (a*
b + b^2)*cosh(x))*sinh(x))*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)*sinh(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))*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*(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 + sin
h(x)^6)) + ((3*a^2 + 12*a*b + 8*b^2)*cosh(x)^8 + 8*(3*a^2 + 12*a*b + 8*b^2)*cosh(x)*sinh(x)^7 + (3*a^2 + 12*a*
b + 8*b^2)*sinh(x)^8 - 4*(3*a^2 + 12*a*b + 8*b^2)*cosh(x)^6 + 4*(7*(3*a^2 + 12*a*b + 8*b^2)*cosh(x)^2 - 3*a^2
- 12*a*b - 8*b^2)*sinh(x)^6 + 8*(7*(3*a^2 + 12*a*b + 8*b^2)*cosh(x)^3 - 3*(3*a^2 + 12*a*b + 8*b^2)*cosh(x))*si
nh(x)^5 + 6*(3*a^2 + 12*a*b + 8*b^2)*cosh(x)^4 + 2*(35*(3*a^2 + 12*a*b + 8*b^2)*cosh(x)^4 - 30*(3*a^2 + 12*a*b
+ 8*b^2)*cosh(x)^2 + 9*a^2 + 36*a*b + 24*b^2)*sinh(x)^4 + 8*(7*(3*a^2 + 12*a*b + 8*b^2)*cosh(x)^5 - 10*(3*a^2
+ 12*a*b + 8*b^2)*cosh(x)^3 + 3*(3*a^2 + 12*a*b + 8*b^2)*cosh(x))*sinh(x)^3 - 4*(3*a^2 + 12*a*b + 8*b^2)*cosh
(x)^2 + 4*(7*(3*a^2 + 12*a*b + 8*b^2)*cosh(x)^6 - 15*(3*a^2 + 12*a*b + 8*b^2)*cosh(x)^4 + 9*(3*a^2 + 12*a*b +
8*b^2)*cosh(x)^2 - 3*a^2 - 12*a*b - 8*b^2)*sinh(x)^2 + 3*a^2 + 12*a*b + 8*b^2 + 8*((3*a^2 + 12*a*b + 8*b^2)*co
sh(x)^7 - 3*(3*a^2 + 12*a*b + 8*b^2)*cosh(x)^5 + 3*(3*a^2 + 12*a*b + 8*b^2)*cosh(x)^3 - (3*a^2 + 12*a*b + 8*b^
2)*cosh(x))*sinh(x))*sqrt(b)*log(-((a + 2*b)*cosh(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)) + 4*((a
*b + b^2)*cosh(x)^8 + 8*(a*b + b^2)*cosh(x)*sinh(x)^7 + (a*b + b^2)*sinh(x)^8 - 4*(a*b + b^2)*cosh(x)^6 + 4*(7
*(a*b + b^2)*cosh(x)^2 - a*b - b^2)*sinh(x)^6 + 8*(7*(a*b + b^2)*cosh(x)^3 - 3*(a*b + b^2)*cosh(x))*sinh(x)^5
+ 6*(a*b + b^2)*cosh(x)^4 + 2*(35*(a*b + b^2)*cosh(x)^4 - 30*(a*b + b^2)*cosh(x)^2 + 3*a*b + 3*b^2)*sinh(x)^4
+ 8*(7*(a*b + b^2)*cosh(x)^5 - 10*(a*b + b^2)*cosh(x)^3 + 3*(a*b + b^2)*cosh(x))*sinh(x)^3 - 4*(a*b + b^2)*cos
h(x)^2 + 4*(7*(a*b + b^2)*cosh(x)^6 - 15*(a*b + b^2)*cosh(x)^4 + 9*(a*b + b^2)*cosh(x)^2 - a*b - b^2)*sinh(x)^
2 + a*b + b^2 + 8*((a*b + b^2)*cosh(x)^7 - 3*(a*b + b^2)*cosh(x)^5 + 3*(a*b + b^2)*cosh(x)^3 - (a*b + b^2)*cos
h(x))*sinh(x))*sqrt(a + b)*log(-((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 - 2*a*cos
h(x)^2 + 2*(3*(a + b)*cosh(x)^2 - a)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(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)*((5*a
*b + 6*b^2)*cosh(x)^6 + 6*(5*a*b + 6*b^2)*cosh(x)*sinh(x)^5 + (5*a*b + 6*b^2)*sinh(x)^6 - (5*a*b - 2*b^2)*cosh
(x)^4 + (15*(5*a*b + 6*b^2)*cosh(x)^2 - 5*a*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}} \coth ^{2}{\left (x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*coth(x)**2)**(3/2)*coth(x)**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(coth(x)^2*(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 {\mathrm {coth}\left (x\right )}^2\,{\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(coth(x)^2*(a + b*coth(x)^2)^(3/2),x)

[Out]

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

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