3.1.44 \(\int \frac {\coth ^2(x)}{(a+b \coth ^2(x))^{5/2}} \, dx\) [44]
Optimal. Leaf size=88 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {(2 a-b) \coth (x)}{3 a (a+b)^2 \sqrt {a+b \coth ^2(x)}} \]
[Out]
arctanh(coth(x)*(a+b)^(1/2)/(a+b*coth(x)^2)^(1/2))/(a+b)^(5/2)-1/3*coth(x)/(a+b)/(a+b*coth(x)^2)^(3/2)-1/3*(2*
a-b)*coth(x)/a/(a+b)^2/(a+b*coth(x)^2)^(1/2)
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Rubi [A]
time = 0.09, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3751, 482, 541,
12, 385, 212} \begin {gather*} -\frac {(2 a-b) \coth (x)}{3 a (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Coth[x]^2/(a + b*Coth[x]^2)^(5/2),x]
[Out]
ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(5/2) - Coth[x]/(3*(a + b)*(a + b*Coth[x]^2)^(3/2
)) - ((2*a - b)*Coth[x])/(3*a*(a + b)^2*Sqrt[a + b*Coth[x]^2])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 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 482
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 541
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -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 \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx &=\text {Subst}\left (\int \frac {x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\coth (x)\right )\\ &=-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {1+2 x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\coth (x)\right )}{3 (a+b)}\\ &=-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {(2 a-b) \coth (x)}{3 a (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {\text {Subst}\left (\int -\frac {3 a}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{3 a (a+b)^2}\\ &=-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {(2 a-b) \coth (x)}{3 a (a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{(a+b)^2}\\ &=-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {(2 a-b) \coth (x)}{3 a (a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^2}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {(2 a-b) \coth (x)}{3 a (a+b)^2 \sqrt {a+b \coth ^2(x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 5.29, size = 190, normalized size = 2.16 \begin {gather*} \frac {\left (-12 (a+b)^3 \cosh ^4(x) \coth ^2(x) \left (a+b \coth ^2(x)\right ) \, _2F_1\left (2,2;\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )+\frac {35 a \left (-5 a-2 b \coth ^2(x)\right ) \sinh ^2(x) \left (3 \text {ArcSin}\left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \left (a+b \coth ^2(x)\right )^2+a \left (3 a+(a+4 b) \coth ^2(x)\right ) \text {csch}^2(x) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}\right )}{\sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}\right ) \tanh (x)}{315 a^3 (a+b)^2 \left (a+b \coth ^2(x)\right )^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Coth[x]^2/(a + b*Coth[x]^2)^(5/2),x]
[Out]
((-12*(a + b)^3*Cosh[x]^4*Coth[x]^2*(a + b*Coth[x]^2)*Hypergeometric2F1[2, 2, 9/2, ((a + b)*Cosh[x]^2)/a] + (3
5*a*(-5*a - 2*b*Coth[x]^2)*Sinh[x]^2*(3*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*(a + b*Coth[x]^2)^2 + a*(3*a + (a
+ 4*b)*Coth[x]^2)*Csch[x]^2*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)]))/Sqrt[-(((a + b)*Cos
h[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)])*Tanh[x])/(315*a^3*(a + b)^2*(a + b*Coth[x]^2)^(3/2))
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(583\) vs.
\(2(74)=148\).
time = 0.73, size = 584, normalized size = 6.64 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)^2/(a+b*coth(x)^2)^(5/2),x,method=_RETURNVERBOSE)
[Out]
-1/3*coth(x)/a/(a+b*coth(x)^2)^(3/2)-2/3/a^2*coth(x)/(a+b*coth(x)^2)^(1/2)+1/6/(a+b)/(b*(1+coth(x))^2-2*b*(1+c
oth(x))+a+b)^(3/2)+1/2*b/(a+b)*(2/3*(2*b*(1+coth(x))-2*b)/(4*b*(a+b)-4*b^2)/(b*(1+coth(x))^2-2*b*(1+coth(x))+a
+b)^(3/2)+16/3*b/(4*b*(a+b)-4*b^2)^2*(2*b*(1+coth(x))-2*b)/(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2))+1/2/(a
+b)*(1/(a+b)/(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2)+2*b/(a+b)*(2*b*(1+coth(x))-2*b)/(4*b*(a+b)-4*b^2)/(b*
(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2)-1/(a+b)^(3/2)*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/(a+b)/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(3/2)+1/2*b/(a+b
)*(2/3*(2*b*(coth(x)-1)+2*b)/(4*b*(a+b)-4*b^2)/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(3/2)+16/3*b/(4*b*(a+b)-4
*b^2)^2*(2*b*(coth(x)-1)+2*b)/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2))-1/2/(a+b)*(1/(a+b)/(b*(coth(x)-1)^2
+2*b*(coth(x)-1)+a+b)^(1/2)-2*b/(a+b)*(2*b*(coth(x)-1)+2*b)/(4*b*(a+b)-4*b^2)/(b*(coth(x)-1)^2+2*b*(coth(x)-1)
+a+b)^(1/2)-1/(a+b)^(3/2)*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)^(5/2),x, algorithm="maxima")
[Out]
integrate(coth(x)^2/(b*coth(x)^2 + a)^(5/2), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3016 vs.
\(2 (74) = 148\).
time = 0.69, size = 6591, normalized size = 74.90 \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)^(5/2),x, algorithm="fricas")
[Out]
[1/12*(3*((a^3 + 2*a^2*b + a*b^2)*cosh(x)^8 + 8*(a^3 + 2*a^2*b + a*b^2)*cosh(x)*sinh(x)^7 + (a^3 + 2*a^2*b + a
*b^2)*sinh(x)^8 - 4*(a^3 - a*b^2)*cosh(x)^6 - 4*(a^3 - a*b^2 - 7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^2)*sinh(x)^6
+ 8*(7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^3 - 3*(a^3 - a*b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^3 - 2*a^2*b + 3*a*b^2)*
cosh(x)^4 + 2*(35*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^4 + 3*a^3 - 2*a^2*b + 3*a*b^2 - 30*(a^3 - a*b^2)*cosh(x)^2)*
sinh(x)^4 + 8*(7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^5 - 10*(a^3 - a*b^2)*cosh(x)^3 + (3*a^3 - 2*a^2*b + 3*a*b^2)*
cosh(x))*sinh(x)^3 + a^3 + 2*a^2*b + a*b^2 - 4*(a^3 - a*b^2)*cosh(x)^2 + 4*(7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^
6 - 15*(a^3 - a*b^2)*cosh(x)^4 - a^3 + a*b^2 + 3*(3*a^3 - 2*a^2*b + 3*a*b^2)*cosh(x)^2)*sinh(x)^2 + 8*((a^3 +
2*a^2*b + a*b^2)*cosh(x)^7 - 3*(a^3 - a*b^2)*cosh(x)^5 + (3*a^3 - 2*a^2*b + 3*a*b^2)*cosh(x)^3 - (a^3 - a*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 + (7
0*(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 + sinh(x)^6))
+ 3*((a^3 + 2*a^2*b + a*b^2)*cosh(x)^8 + 8*(a^3 + 2*a^2*b + a*b^2)*cosh(x)*sinh(x)^7 + (a^3 + 2*a^2*b + a*b^2
)*sinh(x)^8 - 4*(a^3 - a*b^2)*cosh(x)^6 - 4*(a^3 - a*b^2 - 7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^2)*sinh(x)^6 + 8*
(7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^3 - 3*(a^3 - a*b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^3 - 2*a^2*b + 3*a*b^2)*cosh
(x)^4 + 2*(35*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^4 + 3*a^3 - 2*a^2*b + 3*a*b^2 - 30*(a^3 - a*b^2)*cosh(x)^2)*sinh
(x)^4 + 8*(7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^5 - 10*(a^3 - a*b^2)*cosh(x)^3 + (3*a^3 - 2*a^2*b + 3*a*b^2)*cosh
(x))*sinh(x)^3 + a^3 + 2*a^2*b + a*b^2 - 4*(a^3 - a*b^2)*cosh(x)^2 + 4*(7*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^6 -
15*(a^3 - a*b^2)*cosh(x)^4 - a^3 + a*b^2 + 3*(3*a^3 - 2*a^2*b + 3*a*b^2)*cosh(x)^2)*sinh(x)^2 + 8*((a^3 + 2*a^
2*b + a*b^2)*cosh(x)^7 - 3*(a^3 - a*b^2)*cosh(x)^5 + (3*a^3 - 2*a^2*b + 3*a*b^2)*cosh(x)^3 - (a^3 - a*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)) - 4*sqrt(2)*((3*a
^3 + 5*a^2*b + a*b^2 - b^3)*cosh(x)^6 + 6*(3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(x)*sinh(x)^5 + (3*a^3 + 5*a^2*b
+ a*b^2 - b^3)*sinh(x)^6 - 3*(a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^4 - 3*(a^3 - a^2*b - a*b^2 + b^3 - 5*(3*a^3
+ 5*a^2*b + a*b^2 - b^3)*cosh(x)^2)*sinh(x)^4 + 4*(5*(3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(x)^3 - 3*(a^3 - a^2*
b - a*b^2 + b^3)*cosh(x))*sinh(x)^3 + 3*a^3 + 5*a^2*b + a*b^2 - b^3 - 3*(a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^2
+ 3*(5*(3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(x)^4 - a^3 + a^2*b + a*b^2 - b^3 - 6*(a^3 - a^2*b - a*b^2 + b^3)*c
osh(x)^2)*sinh(x)^2 + 6*((3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(x)^5 - 2*(a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^3 -
(a^3 - a^2*b - a*b^2 + b^3)*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) + sinh(x)^2)))/((a^6 + 5*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*cosh(x)^8 +
8*(a^6 + 5*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*cosh(x)*sinh(x)^7 + (a^6 + 5*a^5*b + 10*a^4*b
^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*sinh(x)^8 - 4*(a^6 + 3*a^5*b + 2*a^4*b^2 - 2*a^3*b^3 - 3*a^2*b^4 - a*b^5)
*cosh(x)^6 - 4*(a^6 + 3*a^5*b + 2*a^4*b^2 - 2*a^3*b^3 - 3*a^2*b^4 - a*b^5 - 7*(a^6 + 5*a^5*b + 10*a^4*b^2 + 10
*a^3*b^3 + 5*a^2*b^4 + a*b^5)*cosh(x)^2)*sinh(x...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{2}{\left (x \right )}}{\left (a + b \coth ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)**2/(a+b*coth(x)**2)**(5/2),x)
[Out]
Integral(coth(x)**2/(a + b*coth(x)**2)**(5/2), 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)^(5/2),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 \frac {{\mathrm {coth}\left (x\right )}^2}{{\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)^2/(a + b*coth(x)^2)^(5/2),x)
[Out]
int(coth(x)^2/(a + b*coth(x)^2)^(5/2), x)
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