3.22 \(\int \coth ^3(x) (a+b \coth ^2(x))^{3/2} \, dx\)
Optimal. Leaf size=82 \[ -\frac{\left (a+b \coth ^2(x)\right )^{5/2}}{5 b}-\frac{1}{3} \left (a+b \coth ^2(x)\right )^{3/2}-(a+b) \sqrt{a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right ) \]
[Out]
(a + b)^(3/2)*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - (a + b)*Sqrt[a + b*Coth[x]^2] - (a + b*Coth[x]^2)^(
3/2)/3 - (a + b*Coth[x]^2)^(5/2)/(5*b)
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Rubi [A] time = 0.148513, antiderivative size = 82, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used =
{3670, 446, 80, 50, 63, 208} \[ -\frac{\left (a+b \coth ^2(x)\right )^{5/2}}{5 b}-\frac{1}{3} \left (a+b \coth ^2(x)\right )^{3/2}-(a+b) \sqrt{a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right ) \]
Antiderivative was successfully verified.
[In]
Int[Coth[x]^3*(a + b*Coth[x]^2)^(3/2),x]
[Out]
(a + b)^(3/2)*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - (a + b)*Sqrt[a + b*Coth[x]^2] - (a + b*Coth[x]^2)^(
3/2)/3 - (a + b*Coth[x]^2)^(5/2)/(5*b)
Rule 3670
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]))
Rule 446
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rule 80
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]
Rule 50
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \coth ^3(x) \left (a+b \coth ^2(x)\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int \frac{x^3 \left (a+b x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (a+b x)^{3/2}}{1-x} \, dx,x,\coth ^2(x)\right )\\ &=-\frac{\left (a+b \coth ^2(x)\right )^{5/2}}{5 b}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{1-x} \, dx,x,\coth ^2(x)\right )\\ &=-\frac{1}{3} \left (a+b \coth ^2(x)\right )^{3/2}-\frac{\left (a+b \coth ^2(x)\right )^{5/2}}{5 b}+\frac{1}{2} (a+b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{1-x} \, dx,x,\coth ^2(x)\right )\\ &=-(a+b) \sqrt{a+b \coth ^2(x)}-\frac{1}{3} \left (a+b \coth ^2(x)\right )^{3/2}-\frac{\left (a+b \coth ^2(x)\right )^{5/2}}{5 b}+\frac{1}{2} (a+b)^2 \operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{a+b x}} \, dx,x,\coth ^2(x)\right )\\ &=-(a+b) \sqrt{a+b \coth ^2(x)}-\frac{1}{3} \left (a+b \coth ^2(x)\right )^{3/2}-\frac{\left (a+b \coth ^2(x)\right )^{5/2}}{5 b}+\frac{(a+b)^2 \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \coth ^2(x)}\right )}{b}\\ &=(a+b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right )-(a+b) \sqrt{a+b \coth ^2(x)}-\frac{1}{3} \left (a+b \coth ^2(x)\right )^{3/2}-\frac{\left (a+b \coth ^2(x)\right )^{5/2}}{5 b}\\ \end{align*}
Mathematica [A] time = 0.407963, size = 86, normalized size = 1.05 \[ (a+b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right )-\frac{\sqrt{a+b \coth ^2(x)} \left (3 a^2+b (6 a+5 b) \coth ^2(x)+20 a b+3 b^2 \coth ^4(x)+15 b^2\right )}{15 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[x]^3*(a + b*Coth[x]^2)^(3/2),x]
[Out]
(a + b)^(3/2)*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - (Sqrt[a + b*Coth[x]^2]*(3*a^2 + 20*a*b + 15*b^2 + b
*(6*a + 5*b)*Coth[x]^2 + 3*b^2*Coth[x]^4))/(15*b)
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Maple [B] time = 0.017, size = 593, normalized size = 7.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)^3*(a+b*coth(x)^2)^(3/2),x)
[Out]
-1/5*(a+b*coth(x)^2)^(5/2)/b-1/6*((1+coth(x))^2*b-2*(1+coth(x))*b+a+b)^(3/2)+1/4*b*((1+coth(x))^2*b-2*(1+coth(
x))*b+a+b)^(1/2)*coth(x)+3/4*b^(1/2)*ln(((1+coth(x))*b-b)/b^(1/2)+((1+coth(x))^2*b-2*(1+coth(x))*b+a+b)^(1/2))
*a+1/2*(a+b)^(1/2)*ln((2*a+2*b-2*(1+coth(x))*b+2*(a+b)^(1/2)*((1+coth(x))^2*b-2*(1+coth(x))*b+a+b)^(1/2))/(1+c
oth(x)))*a-1/2*((1+coth(x))^2*b-2*(1+coth(x))*b+a+b)^(1/2)*a+1/2*(a+b)^(1/2)*ln((2*a+2*b-2*(1+coth(x))*b+2*(a+
b)^(1/2)*((1+coth(x))^2*b-2*(1+coth(x))*b+a+b)^(1/2))/(1+coth(x)))*b+1/2*b^(3/2)*ln(((1+coth(x))*b-b)/b^(1/2)+
((1+coth(x))^2*b-2*(1+coth(x))*b+a+b)^(1/2))-1/2*((1+coth(x))^2*b-2*(1+coth(x))*b+a+b)^(1/2)*b-1/6*((coth(x)-1
)^2*b+2*(coth(x)-1)*b+a+b)^(3/2)-1/4*b*((coth(x)-1)^2*b+2*(coth(x)-1)*b+a+b)^(1/2)*coth(x)-3/4*b^(1/2)*ln(((co
th(x)-1)*b+b)/b^(1/2)+((coth(x)-1)^2*b+2*(coth(x)-1)*b+a+b)^(1/2))*a+1/2*ln((2*a+2*b+2*(coth(x)-1)*b+2*(a+b)^(
1/2)*((coth(x)-1)^2*b+2*(coth(x)-1)*b+a+b)^(1/2))/(coth(x)-1))*(a+b)^(1/2)*a-1/2*((coth(x)-1)^2*b+2*(coth(x)-1
)*b+a+b)^(1/2)*a+1/2*ln((2*a+2*b+2*(coth(x)-1)*b+2*(a+b)^(1/2)*((coth(x)-1)^2*b+2*(coth(x)-1)*b+a+b)^(1/2))/(c
oth(x)-1))*(a+b)^(1/2)*b-1/2*ln(((coth(x)-1)*b+b)/b^(1/2)+((coth(x)-1)^2*b+2*(coth(x)-1)*b+a+b)^(1/2))*b^(3/2)
-1/2*((coth(x)-1)^2*b+2*(coth(x)-1)*b+a+b)^(1/2)*b
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \coth \left (x\right )^{2} + a\right )}^{\frac{3}{2}} \coth \left (x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^3*(a+b*coth(x)^2)^(3/2),x, algorithm="maxima")
[Out]
integrate((b*coth(x)^2 + a)^(3/2)*coth(x)^3, x)
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Fricas [B] time = 7.68422, size = 13650, normalized size = 166.46 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^3*(a+b*coth(x)^2)^(3/2),x, algorithm="fricas")
[Out]
[1/60*(15*((a*b + b^2)*cosh(x)^10 + 10*(a*b + b^2)*cosh(x)*sinh(x)^9 + (a*b + b^2)*sinh(x)^10 - 5*(a*b + b^2)*
cosh(x)^8 + 5*(9*(a*b + b^2)*cosh(x)^2 - a*b - b^2)*sinh(x)^8 + 40*(3*(a*b + b^2)*cosh(x)^3 - (a*b + b^2)*cosh
(x))*sinh(x)^7 + 10*(a*b + b^2)*cosh(x)^6 + 10*(21*(a*b + b^2)*cosh(x)^4 - 14*(a*b + b^2)*cosh(x)^2 + a*b + b^
2)*sinh(x)^6 + 4*(63*(a*b + b^2)*cosh(x)^5 - 70*(a*b + b^2)*cosh(x)^3 + 15*(a*b + b^2)*cosh(x))*sinh(x)^5 - 10
*(a*b + b^2)*cosh(x)^4 + 10*(21*(a*b + b^2)*cosh(x)^6 - 35*(a*b + b^2)*cosh(x)^4 + 15*(a*b + b^2)*cosh(x)^2 -
a*b - b^2)*sinh(x)^4 + 40*(3*(a*b + b^2)*cosh(x)^7 - 7*(a*b + b^2)*cosh(x)^5 + 5*(a*b + b^2)*cosh(x)^3 - (a*b
+ b^2)*cosh(x))*sinh(x)^3 + 5*(a*b + b^2)*cosh(x)^2 + 5*(9*(a*b + b^2)*cosh(x)^8 - 28*(a*b + b^2)*cosh(x)^6 +
30*(a*b + b^2)*cosh(x)^4 - 12*(a*b + b^2)*cosh(x)^2 + a*b + b^2)*sinh(x)^2 - a*b - b^2 + 10*((a*b + b^2)*cosh(
x)^9 - 4*(a*b + b^2)*cosh(x)^7 + 6*(a*b + b^2)*cosh(x)^5 - 4*(a*b + b^2)*cosh(x)^3 + (a*b + b^2)*cosh(x))*sinh
(x))*sqrt(a + b)*log(-((a^3 + a^2*b)*cosh(x)^8 + 8*(a^3 + a^2*b)*cosh(x)*sinh(x)^7 + (a^3 + a^2*b)*sinh(x)^8 -
2*(2*a^3 + a^2*b)*cosh(x)^6 - 2*(2*a^3 + a^2*b - 14*(a^3 + a^2*b)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a^3 + a^2*b)*
cosh(x)^3 - 3*(2*a^3 + a^2*b)*cosh(x))*sinh(x)^5 + (6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x)^4 + (70*(a^3 + a^2*
b)*cosh(x)^4 + 6*a^3 + 4*a^2*b - a*b^2 + b^3 - 30*(2*a^3 + a^2*b)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a^3 + a^2*b)*c
osh(x)^5 - 10*(2*a^3 + a^2*b)*cosh(x)^3 + (6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x))*sinh(x)^3 + a^3 + 3*a^2*b +
3*a*b^2 + b^3 - 2*(2*a^3 + 3*a^2*b - b^3)*cosh(x)^2 + 2*(14*(a^3 + a^2*b)*cosh(x)^6 - 15*(2*a^3 + a^2*b)*cosh
(x)^4 - 2*a^3 - 3*a^2*b + b^3 + 3*(6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(a^2*cosh(x)^
6 + 6*a^2*cosh(x)*sinh(x)^5 + a^2*sinh(x)^6 - 3*a^2*cosh(x)^4 + 3*(5*a^2*cosh(x)^2 - a^2)*sinh(x)^4 + 4*(5*a^2
*cosh(x)^3 - 3*a^2*cosh(x))*sinh(x)^3 + (3*a^2 + 2*a*b - b^2)*cosh(x)^2 + (15*a^2*cosh(x)^4 - 18*a^2*cosh(x)^2
+ 3*a^2 + 2*a*b - b^2)*sinh(x)^2 - a^2 - 2*a*b - b^2 + 2*(3*a^2*cosh(x)^5 - 6*a^2*cosh(x)^3 + (3*a^2 + 2*a*b
- 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^3 + a^2*b)*cosh(x)^7 - 3*(2*a^3 + a^2*b)*cosh(x)^5 + (6*a^3 + 4*a^2*b - a*b
^2 + b^3)*cosh(x)^3 - (2*a^3 + 3*a^2*b - 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)) + 15*((a*b +
b^2)*cosh(x)^10 + 10*(a*b + b^2)*cosh(x)*sinh(x)^9 + (a*b + b^2)*sinh(x)^10 - 5*(a*b + b^2)*cosh(x)^8 + 5*(9*
(a*b + b^2)*cosh(x)^2 - a*b - b^2)*sinh(x)^8 + 40*(3*(a*b + b^2)*cosh(x)^3 - (a*b + b^2)*cosh(x))*sinh(x)^7 +
10*(a*b + b^2)*cosh(x)^6 + 10*(21*(a*b + b^2)*cosh(x)^4 - 14*(a*b + b^2)*cosh(x)^2 + a*b + b^2)*sinh(x)^6 + 4*
(63*(a*b + b^2)*cosh(x)^5 - 70*(a*b + b^2)*cosh(x)^3 + 15*(a*b + b^2)*cosh(x))*sinh(x)^5 - 10*(a*b + b^2)*cosh
(x)^4 + 10*(21*(a*b + b^2)*cosh(x)^6 - 35*(a*b + b^2)*cosh(x)^4 + 15*(a*b + b^2)*cosh(x)^2 - a*b - b^2)*sinh(x
)^4 + 40*(3*(a*b + b^2)*cosh(x)^7 - 7*(a*b + b^2)*cosh(x)^5 + 5*(a*b + b^2)*cosh(x)^3 - (a*b + b^2)*cosh(x))*s
inh(x)^3 + 5*(a*b + b^2)*cosh(x)^2 + 5*(9*(a*b + b^2)*cosh(x)^8 - 28*(a*b + b^2)*cosh(x)^6 + 30*(a*b + b^2)*co
sh(x)^4 - 12*(a*b + b^2)*cosh(x)^2 + a*b + b^2)*sinh(x)^2 - a*b - b^2 + 10*((a*b + b^2)*cosh(x)^9 - 4*(a*b + b
^2)*cosh(x)^7 + 6*(a*b + b^2)*cosh(x)^5 - 4*(a*b + b^2)*cosh(x)^3 + (a*b + b^2)*cosh(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*b*cosh(x)^2 + 2*(3*(a + b)*cosh(x
)^2 + b)*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 + b*cosh(x
))*sinh(x) + a + b)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) - 4*sqrt(2)*((3*a^2 + 26*a*b + 23*b^2)*cosh(x
)^8 + 8*(3*a^2 + 26*a*b + 23*b^2)*cosh(x)*sinh(x)^7 + (3*a^2 + 26*a*b + 23*b^2)*sinh(x)^8 - 4*(3*a^2 + 20*a*b
+ 12*b^2)*cosh(x)^6 + 4*(7*(3*a^2 + 26*a*b + 23*b^2)*cosh(x)^2 - 3*a^2 - 20*a*b - 12*b^2)*sinh(x)^6 + 8*(7*(3*
a^2 + 26*a*b + 23*b^2)*cosh(x)^3 - 3*(3*a^2 + 20*a*b + 12*b^2)*cosh(x))*sinh(x)^5 + 2*(9*a^2 + 54*a*b + 49*b^2
)*cosh(x)^4 + 2*(35*(3*a^2 + 26*a*b + 23*b^2)*cosh(x)^4 - 30*(3*a^2 + 20*a*b + 12*b^2)*cosh(x)^2 + 9*a^2 + 54*
a*b + 49*b^2)*sinh(x)^4 + 8*(7*(3*a^2 + 26*a*b + 23*b^2)*cosh(x)^5 - 10*(3*a^2 + 20*a*b + 12*b^2)*cosh(x)^3 +
(9*a^2 + 54*a*b + 49*b^2)*cosh(x))*sinh(x)^3 - 4*(3*a^2 + 20*a*b + 12*b^2)*cosh(x)^2 + 4*(7*(3*a^2 + 26*a*b +
23*b^2)*cosh(x)^6 - 15*(3*a^2 + 20*a*b + 12*b^2)*cosh(x)^4 + 3*(9*a^2 + 54*a*b + 49*b^2)*cosh(x)^2 - 3*a^2 - 2
0*a*b - 12*b^2)*sinh(x)^2 + 3*a^2 + 26*a*b + 23*b^2 + 8*((3*a^2 + 26*a*b + 23*b^2)*cosh(x)^7 - 3*(3*a^2 + 20*a
*b + 12*b^2)*cosh(x)^5 + (9*a^2 + 54*a*b + 49*b^2)*cosh(x)^3 - (3*a^2 + 20*a*b + 12*b^2)*cosh(x))*sinh(x))*sqr
t(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(b*cosh(x)^10
+ 10*b*cosh(x)*sinh(x)^9 + b*sinh(x)^10 - 5*b*cosh(x)^8 + 5*(9*b*cosh(x)^2 - b)*sinh(x)^8 + 40*(3*b*cosh(x)^3
- b*cosh(x))*sinh(x)^7 + 10*b*cosh(x)^6 + 10*(21*b*cosh(x)^4 - 14*b*cosh(x)^2 + b)*sinh(x)^6 + 4*(63*b*cosh(x)
^5 - 70*b*cosh(x)^3 + 15*b*cosh(x))*sinh(x)^5 - 10*b*cosh(x)^4 + 10*(21*b*cosh(x)^6 - 35*b*cosh(x)^4 + 15*b*co
sh(x)^2 - b)*sinh(x)^4 + 40*(3*b*cosh(x)^7 - 7*b*cosh(x)^5 + 5*b*cosh(x)^3 - b*cosh(x))*sinh(x)^3 + 5*b*cosh(x
)^2 + 5*(9*b*cosh(x)^8 - 28*b*cosh(x)^6 + 30*b*cosh(x)^4 - 12*b*cosh(x)^2 + b)*sinh(x)^2 + 10*(b*cosh(x)^9 - 4
*b*cosh(x)^7 + 6*b*cosh(x)^5 - 4*b*cosh(x)^3 + b*cosh(x))*sinh(x) - b), -1/30*(15*((a*b + b^2)*cosh(x)^10 + 10
*(a*b + b^2)*cosh(x)*sinh(x)^9 + (a*b + b^2)*sinh(x)^10 - 5*(a*b + b^2)*cosh(x)^8 + 5*(9*(a*b + b^2)*cosh(x)^2
- a*b - b^2)*sinh(x)^8 + 40*(3*(a*b + b^2)*cosh(x)^3 - (a*b + b^2)*cosh(x))*sinh(x)^7 + 10*(a*b + b^2)*cosh(x
)^6 + 10*(21*(a*b + b^2)*cosh(x)^4 - 14*(a*b + b^2)*cosh(x)^2 + a*b + b^2)*sinh(x)^6 + 4*(63*(a*b + b^2)*cosh(
x)^5 - 70*(a*b + b^2)*cosh(x)^3 + 15*(a*b + b^2)*cosh(x))*sinh(x)^5 - 10*(a*b + b^2)*cosh(x)^4 + 10*(21*(a*b +
b^2)*cosh(x)^6 - 35*(a*b + b^2)*cosh(x)^4 + 15*(a*b + b^2)*cosh(x)^2 - a*b - b^2)*sinh(x)^4 + 40*(3*(a*b + b^
2)*cosh(x)^7 - 7*(a*b + b^2)*cosh(x)^5 + 5*(a*b + b^2)*cosh(x)^3 - (a*b + b^2)*cosh(x))*sinh(x)^3 + 5*(a*b + b
^2)*cosh(x)^2 + 5*(9*(a*b + b^2)*cosh(x)^8 - 28*(a*b + b^2)*cosh(x)^6 + 30*(a*b + b^2)*cosh(x)^4 - 12*(a*b + b
^2)*cosh(x)^2 + a*b + b^2)*sinh(x)^2 - a*b - b^2 + 10*((a*b + b^2)*cosh(x)^9 - 4*(a*b + b^2)*cosh(x)^7 + 6*(a*
b + b^2)*cosh(x)^5 - 4*(a*b + b^2)*cosh(x)^3 + (a*b + b^2)*cosh(x))*sinh(x))*sqrt(-a - b)*arctan(sqrt(2)*(a*co
sh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 - a - b)*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))/((a^2 + a*b)*cosh(x)^4 + 4*(a^2 + a*b)*cosh(x)*sinh(x)^3
+ (a^2 + a*b)*sinh(x)^4 - (2*a^2 + a*b - b^2)*cosh(x)^2 + (6*(a^2 + a*b)*cosh(x)^2 - 2*a^2 - a*b + b^2)*sinh(
x)^2 + a^2 + 2*a*b + b^2 + 2*(2*(a^2 + a*b)*cosh(x)^3 - (2*a^2 + a*b - b^2)*cosh(x))*sinh(x))) + 15*((a*b + b^
2)*cosh(x)^10 + 10*(a*b + b^2)*cosh(x)*sinh(x)^9 + (a*b + b^2)*sinh(x)^10 - 5*(a*b + b^2)*cosh(x)^8 + 5*(9*(a*
b + b^2)*cosh(x)^2 - a*b - b^2)*sinh(x)^8 + 40*(3*(a*b + b^2)*cosh(x)^3 - (a*b + b^2)*cosh(x))*sinh(x)^7 + 10*
(a*b + b^2)*cosh(x)^6 + 10*(21*(a*b + b^2)*cosh(x)^4 - 14*(a*b + b^2)*cosh(x)^2 + a*b + b^2)*sinh(x)^6 + 4*(63
*(a*b + b^2)*cosh(x)^5 - 70*(a*b + b^2)*cosh(x)^3 + 15*(a*b + b^2)*cosh(x))*sinh(x)^5 - 10*(a*b + b^2)*cosh(x)
^4 + 10*(21*(a*b + b^2)*cosh(x)^6 - 35*(a*b + b^2)*cosh(x)^4 + 15*(a*b + b^2)*cosh(x)^2 - a*b - b^2)*sinh(x)^4
+ 40*(3*(a*b + b^2)*cosh(x)^7 - 7*(a*b + b^2)*cosh(x)^5 + 5*(a*b + b^2)*cosh(x)^3 - (a*b + b^2)*cosh(x))*sinh
(x)^3 + 5*(a*b + b^2)*cosh(x)^2 + 5*(9*(a*b + b^2)*cosh(x)^8 - 28*(a*b + b^2)*cosh(x)^6 + 30*(a*b + b^2)*cosh(
x)^4 - 12*(a*b + b^2)*cosh(x)^2 + a*b + b^2)*sinh(x)^2 - a*b - b^2 + 10*((a*b + b^2)*cosh(x)^9 - 4*(a*b + b^2)
*cosh(x)^7 + 6*(a*b + b^2)*cosh(x)^5 - 4*(a*b + b^2)*cosh(x)^3 + (a*b + b^2)*cosh(x))*sinh(x))*sqrt(-a - b)*ar
ctan(sqrt(2)*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))/((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a + b)) + 2*sqrt(2)*((3*a^2
+ 26*a*b + 23*b^2)*cosh(x)^8 + 8*(3*a^2 + 26*a*b + 23*b^2)*cosh(x)*sinh(x)^7 + (3*a^2 + 26*a*b + 23*b^2)*sinh
(x)^8 - 4*(3*a^2 + 20*a*b + 12*b^2)*cosh(x)^6 + 4*(7*(3*a^2 + 26*a*b + 23*b^2)*cosh(x)^2 - 3*a^2 - 20*a*b - 12
*b^2)*sinh(x)^6 + 8*(7*(3*a^2 + 26*a*b + 23*b^2)*cosh(x)^3 - 3*(3*a^2 + 20*a*b + 12*b^2)*cosh(x))*sinh(x)^5 +
2*(9*a^2 + 54*a*b + 49*b^2)*cosh(x)^4 + 2*(35*(3*a^2 + 26*a*b + 23*b^2)*cosh(x)^4 - 30*(3*a^2 + 20*a*b + 12*b^
2)*cosh(x)^2 + 9*a^2 + 54*a*b + 49*b^2)*sinh(x)^4 + 8*(7*(3*a^2 + 26*a*b + 23*b^2)*cosh(x)^5 - 10*(3*a^2 + 20*
a*b + 12*b^2)*cosh(x)^3 + (9*a^2 + 54*a*b + 49*b^2)*cosh(x))*sinh(x)^3 - 4*(3*a^2 + 20*a*b + 12*b^2)*cosh(x)^2
+ 4*(7*(3*a^2 + 26*a*b + 23*b^2)*cosh(x)^6 - 15*(3*a^2 + 20*a*b + 12*b^2)*cosh(x)^4 + 3*(9*a^2 + 54*a*b + 49*
b^2)*cosh(x)^2 - 3*a^2 - 20*a*b - 12*b^2)*sinh(x)^2 + 3*a^2 + 26*a*b + 23*b^2 + 8*((3*a^2 + 26*a*b + 23*b^2)*c
osh(x)^7 - 3*(3*a^2 + 20*a*b + 12*b^2)*cosh(x)^5 + (9*a^2 + 54*a*b + 49*b^2)*cosh(x)^3 - (3*a^2 + 20*a*b + 12*
b^2)*cosh(x))*sinh(x))*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + s
inh(x)^2)))/(b*cosh(x)^10 + 10*b*cosh(x)*sinh(x)^9 + b*sinh(x)^10 - 5*b*cosh(x)^8 + 5*(9*b*cosh(x)^2 - b)*sinh
(x)^8 + 40*(3*b*cosh(x)^3 - b*cosh(x))*sinh(x)^7 + 10*b*cosh(x)^6 + 10*(21*b*cosh(x)^4 - 14*b*cosh(x)^2 + b)*s
inh(x)^6 + 4*(63*b*cosh(x)^5 - 70*b*cosh(x)^3 + 15*b*cosh(x))*sinh(x)^5 - 10*b*cosh(x)^4 + 10*(21*b*cosh(x)^6
- 35*b*cosh(x)^4 + 15*b*cosh(x)^2 - b)*sinh(x)^4 + 40*(3*b*cosh(x)^7 - 7*b*cosh(x)^5 + 5*b*cosh(x)^3 - b*cosh(
x))*sinh(x)^3 + 5*b*cosh(x)^2 + 5*(9*b*cosh(x)^8 - 28*b*cosh(x)^6 + 30*b*cosh(x)^4 - 12*b*cosh(x)^2 + b)*sinh(
x)^2 + 10*(b*cosh(x)^9 - 4*b*cosh(x)^7 + 6*b*cosh(x)^5 - 4*b*cosh(x)^3 + b*cosh(x))*sinh(x) - b)]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)**3*(a+b*coth(x)**2)**(3/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^3*(a+b*coth(x)^2)^(3/2),x, algorithm="giac")
[Out]
Exception raised: TypeError