3.1.53 \(\int \frac {\coth (x)}{(a+b \coth ^4(x))^{3/2}} \, dx\) [53]

Optimal. Leaf size=74 \[ \frac {\tanh ^{-1}\left (\frac {a+b \coth ^2(x)}{\sqrt {a+b} \sqrt {a+b \coth ^4(x)}}\right )}{2 (a+b)^{3/2}}-\frac {a-b \coth ^2(x)}{2 a (a+b) \sqrt {a+b \coth ^4(x)}} \]

[Out]

1/2*arctanh((a+b*coth(x)^2)/(a+b)^(1/2)/(a+b*coth(x)^4)^(1/2))/(a+b)^(3/2)+1/2*(-a+b*coth(x)^2)/a/(a+b)/(a+b*c
oth(x)^4)^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3751, 1262, 755, 12, 739, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {a+b \coth ^2(x)}{\sqrt {a+b} \sqrt {a+b \coth ^4(x)}}\right )}{2 (a+b)^{3/2}}-\frac {a-b \coth ^2(x)}{2 a (a+b) \sqrt {a+b \coth ^4(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(a + b*Coth[x]^4)^(3/2),x]

[Out]

ArcTanh[(a + b*Coth[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Coth[x]^4])]/(2*(a + b)^(3/2)) - (a - b*Coth[x]^2)/(2*a*(a +
b)*Sqrt[a + b*Coth[x]^4])

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 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 755

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 1262

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

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

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Mathematica [A]
time = 0.37, size = 73, normalized size = 0.99 \begin {gather*} \frac {1}{2} \left (\frac {\tanh ^{-1}\left (\frac {a+b \coth ^2(x)}{\sqrt {a+b} \sqrt {a+b \coth ^4(x)}}\right )}{(a+b)^{3/2}}-\frac {a-b \coth ^2(x)}{a (a+b) \sqrt {a+b \coth ^4(x)}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/(a + b*Coth[x]^4)^(3/2),x]

[Out]

(ArcTanh[(a + b*Coth[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Coth[x]^4])]/(a + b)^(3/2) - (a - b*Coth[x]^2)/(a*(a + b)*S
qrt[a + b*Coth[x]^4]))/2

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 3.04, size = 431, normalized size = 5.82

method	result	size
derivativedivides	\(\frac {b \left (-\frac {\coth ^{3}\left (x \right )}{4 a \left (a +b \right )}+\frac {\coth ^{2}\left (x \right )}{4 a \left (a +b \right )}-\frac {\coth \left (x \right )}{4 a \left (a +b \right )}-\frac {1}{4 \left (a +b \right ) b}\right )}{\sqrt {\left (\coth ^{4}\left (x \right )+\frac {a}{b}\right ) b}}-\frac {-\frac {\arctanh \left (\frac {2 b \left (\coth ^{2}\left (x \right )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \EllipticPi \left (\coth \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}}{2 \left (a +b \right )}+\frac {b \left (\frac {\coth ^{3}\left (x \right )}{4 a \left (a +b \right )}+\frac {\coth ^{2}\left (x \right )}{4 a \left (a +b \right )}+\frac {\coth \left (x \right )}{4 a \left (a +b \right )}-\frac {1}{4 \left (a +b \right ) b}\right )}{\sqrt {\left (\coth ^{4}\left (x \right )+\frac {a}{b}\right ) b}}-\frac {-\frac {\arctanh \left (\frac {2 b \left (\coth ^{2}\left (x \right )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b}}-\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \EllipticPi \left (\coth \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}}{2 \left (a +b \right )}\)	\(431\)
default	\(\frac {b \left (-\frac {\coth ^{3}\left (x \right )}{4 a \left (a +b \right )}+\frac {\coth ^{2}\left (x \right )}{4 a \left (a +b \right )}-\frac {\coth \left (x \right )}{4 a \left (a +b \right )}-\frac {1}{4 \left (a +b \right ) b}\right )}{\sqrt {\left (\coth ^{4}\left (x \right )+\frac {a}{b}\right ) b}}-\frac {-\frac {\arctanh \left (\frac {2 b \left (\coth ^{2}\left (x \right )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \EllipticPi \left (\coth \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}}{2 \left (a +b \right )}+\frac {b \left (\frac {\coth ^{3}\left (x \right )}{4 a \left (a +b \right )}+\frac {\coth ^{2}\left (x \right )}{4 a \left (a +b \right )}+\frac {\coth \left (x \right )}{4 a \left (a +b \right )}-\frac {1}{4 \left (a +b \right ) b}\right )}{\sqrt {\left (\coth ^{4}\left (x \right )+\frac {a}{b}\right ) b}}-\frac {-\frac {\arctanh \left (\frac {2 b \left (\coth ^{2}\left (x \right )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b}}-\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \EllipticPi \left (\coth \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}}{2 \left (a +b \right )}\)	\(431\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)/(a+b*coth(x)^4)^(3/2),x,method=_RETURNVERBOSE)

[Out]

b*(-1/4/a/(a+b)*coth(x)^3+1/4/a/(a+b)*coth(x)^2-1/4/a/(a+b)*coth(x)-1/4/(a+b)/b)/((coth(x)^4+a/b)*b)^(1/2)-1/2
/(a+b)*(-1/2/(a+b)^(1/2)*arctanh(1/2*(2*b*coth(x)^2+2*a)/(a+b)^(1/2)/(a+b*coth(x)^4)^(1/2))+1/(I/a^(1/2)*b^(1/
2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*coth(x)^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*coth(x)^2)^(1/2)/(a+b*coth(x)^4)^(1/2)*El
lipticPi(coth(x)*(I/a^(1/2)*b^(1/2))^(1/2),-I*a^(1/2)/b^(1/2),(-I/a^(1/2)*b^(1/2))^(1/2)/(I/a^(1/2)*b^(1/2))^(
1/2)))+b*(1/4/a/(a+b)*coth(x)^3+1/4/a/(a+b)*coth(x)^2+1/4/a/(a+b)*coth(x)-1/4/(a+b)/b)/((coth(x)^4+a/b)*b)^(1/
2)-1/2/(a+b)*(-1/2/(a+b)^(1/2)*arctanh(1/2*(2*b*coth(x)^2+2*a)/(a+b)^(1/2)/(a+b*coth(x)^4)^(1/2))-1/(I/a^(1/2)
*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*coth(x)^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*coth(x)^2)^(1/2)/(a+b*coth(x)^4)^(1
/2)*EllipticPi(coth(x)*(I/a^(1/2)*b^(1/2))^(1/2),-I*a^(1/2)/b^(1/2),(-I/a^(1/2)*b^(1/2))^(1/2)/(I/a^(1/2)*b^(1
/2))^(1/2)))

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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)/(a+b*coth(x)^4)^(3/2),x, algorithm="maxima")

[Out]

integrate(coth(x)/(b*coth(x)^4 + a)^(3/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1947 vs. \(2 (63) = 126\).
time = 0.59, size = 3938, normalized size = 53.22 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*coth(x)^4)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(((a^2 + a*b)*cosh(x)^8 + 8*(a^2 + a*b)*cosh(x)*sinh(x)^7 + (a^2 + a*b)*sinh(x)^8 - 4*(a^2 - a*b)*cosh(x)
^6 + 4*(7*(a^2 + a*b)*cosh(x)^2 - a^2 + a*b)*sinh(x)^6 + 8*(7*(a^2 + a*b)*cosh(x)^3 - 3*(a^2 - a*b)*cosh(x))*s
inh(x)^5 + 6*(a^2 + a*b)*cosh(x)^4 + 2*(35*(a^2 + a*b)*cosh(x)^4 - 30*(a^2 - a*b)*cosh(x)^2 + 3*a^2 + 3*a*b)*s
inh(x)^4 + 8*(7*(a^2 + a*b)*cosh(x)^5 - 10*(a^2 - a*b)*cosh(x)^3 + 3*(a^2 + a*b)*cosh(x))*sinh(x)^3 - 4*(a^2 -
a*b)*cosh(x)^2 + 4*(7*(a^2 + a*b)*cosh(x)^6 - 15*(a^2 - a*b)*cosh(x)^4 + 9*(a^2 + a*b)*cosh(x)^2 - a^2 + a*b)
*sinh(x)^2 + a^2 + a*b + 8*((a^2 + a*b)*cosh(x)^7 - 3*(a^2 - a*b)*cosh(x)^5 + 3*(a^2 + a*b)*cosh(x)^3 - (a^2 -
a*b)*cosh(x))*sinh(x))*sqrt(a + b)*log(((a^2 + 2*a*b + b^2)*cosh(x)^8 + 8*(a^2 + 2*a*b + b^2)*cosh(x)*sinh(x)
^7 + (a^2 + 2*a*b + b^2)*sinh(x)^8 - 4*(a^2 - b^2)*cosh(x)^6 + 4*(7*(a^2 + 2*a*b + b^2)*cosh(x)^2 - a^2 + b^2)
*sinh(x)^6 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(x)^3 - 3*(a^2 - b^2)*cosh(x))*sinh(x)^5 + 2*(3*a^2 + 2*a*b + 3*b^2)
*cosh(x)^4 + 2*(35*(a^2 + 2*a*b + b^2)*cosh(x)^4 - 30*(a^2 - b^2)*cosh(x)^2 + 3*a^2 + 2*a*b + 3*b^2)*sinh(x)^4
+ 8*(7*(a^2 + 2*a*b + b^2)*cosh(x)^5 - 10*(a^2 - b^2)*cosh(x)^3 + (3*a^2 + 2*a*b + 3*b^2)*cosh(x))*sinh(x)^3
- 4*(a^2 - b^2)*cosh(x)^2 + 4*(7*(a^2 + 2*a*b + b^2)*cosh(x)^6 - 15*(a^2 - b^2)*cosh(x)^4 + 3*(3*a^2 + 2*a*b +
3*b^2)*cosh(x)^2 - a^2 + b^2)*sinh(x)^2 + sqrt(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)*c
osh(x))*sinh(x) + a + b)*sqrt(a + b)*sqrt(((a + b)*cosh(x)^4 + (a + b)*sinh(x)^4 - 4*(a - b)*cosh(x)^2 + 2*(3*
(a + b)*cosh(x)^2 - 2*a + 2*b)*sinh(x)^2 + 3*a + 3*b)/(cosh(x)^4 - 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2
- 4*cosh(x)*sinh(x)^3 + sinh(x)^4)) + a^2 + 2*a*b + b^2 + 8*((a^2 + 2*a*b + b^2)*cosh(x)^7 - 3*(a^2 - b^2)*co
sh(x)^5 + (3*a^2 + 2*a*b + 3*b^2)*cosh(x)^3 - (a^2 - b^2)*cosh(x))*sinh(x))/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) +
6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4)) - 2*sqrt(2)*((a^2 - b^2)*cosh(x)^4 + 4*(a^2 - b^2)*
cosh(x)*sinh(x)^3 + (a^2 - b^2)*sinh(x)^4 - 2*(a^2 + 2*a*b + b^2)*cosh(x)^2 + 2*(3*(a^2 - b^2)*cosh(x)^2 - a^2
- 2*a*b - b^2)*sinh(x)^2 + a^2 - b^2 + 4*((a^2 - b^2)*cosh(x)^3 - (a^2 + 2*a*b + b^2)*cosh(x))*sinh(x))*sqrt(
((a + b)*cosh(x)^4 + (a + b)*sinh(x)^4 - 4*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 - 2*a + 2*b)*sinh(x)^2 +
3*a + 3*b)/(cosh(x)^4 - 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 - 4*cosh(x)*sinh(x)^3 + sinh(x)^4)))/((a^
4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^8 + 8*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)*sinh(x)^7 + (a^4 +
3*a^3*b + 3*a^2*b^2 + a*b^3)*sinh(x)^8 - 4*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^6 - 4*(a^4 + a^3*b - a^2*b^
2 - a*b^3 - 7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b
^3)*cosh(x)^3 - 3*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x))*sinh(x)^5 + 6*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*c
osh(x)^4 + 2*(35*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^4 + 3*a^4 + 9*a^3*b + 9*a^2*b^2 + 3*a*b^3 - 30*(a
^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^2)*sinh(x)^4 + a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 + 8*(7*(a^4 + 3*a^3*b +
3*a^2*b^2 + a*b^3)*cosh(x)^5 - 10*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^3 + 3*(a^4 + 3*a^3*b + 3*a^2*b^2 +
a*b^3)*cosh(x))*sinh(x)^3 - 4*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^2 + 4*(7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*
b^3)*cosh(x)^6 - 15*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cosh(x)^4 - a^4 - a^3*b + a^2*b^2 + a*b^3 + 9*(a^4 + 3*a^3
*b + 3*a^2*b^2 + a*b^3)*cosh(x)^2)*sinh(x)^2 + 8*((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^7 - 3*(a^4 + a^3
*b - a^2*b^2 - a*b^3)*cosh(x)^5 + 3*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cosh(x)^3 - (a^4 + a^3*b - a^2*b^2 - a
*b^3)*cosh(x))*sinh(x)), -1/2*(((a^2 + a*b)*cosh(x)^8 + 8*(a^2 + a*b)*cosh(x)*sinh(x)^7 + (a^2 + a*b)*sinh(x)^
8 - 4*(a^2 - a*b)*cosh(x)^6 + 4*(7*(a^2 + a*b)*cosh(x)^2 - a^2 + a*b)*sinh(x)^6 + 8*(7*(a^2 + a*b)*cosh(x)^3 -
3*(a^2 - a*b)*cosh(x))*sinh(x)^5 + 6*(a^2 + a*b)*cosh(x)^4 + 2*(35*(a^2 + a*b)*cosh(x)^4 - 30*(a^2 - a*b)*cos
h(x)^2 + 3*a^2 + 3*a*b)*sinh(x)^4 + 8*(7*(a^2 + a*b)*cosh(x)^5 - 10*(a^2 - a*b)*cosh(x)^3 + 3*(a^2 + a*b)*cosh
(x))*sinh(x)^3 - 4*(a^2 - a*b)*cosh(x)^2 + 4*(7*(a^2 + a*b)*cosh(x)^6 - 15*(a^2 - a*b)*cosh(x)^4 + 9*(a^2 + a*
b)*cosh(x)^2 - a^2 + a*b)*sinh(x)^2 + a^2 + a*b + 8*((a^2 + a*b)*cosh(x)^7 - 3*(a^2 - a*b)*cosh(x)^5 + 3*(a^2
+ a*b)*cosh(x)^3 - (a^2 - a*b)*cosh(x))*sinh(x))*sqrt(-a - b)*arctan(sqrt(2)*((a + b)*cosh(x)^4 + 4*(a + b)*co
sh(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)*sqrt(((a + b)*cosh(x)^4 + (a + b)*sinh(x)^4 -
4*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 - 2*a + 2*b)*sinh(x)^2 + 3*a + 3*b)/(cosh(x)^4 - 4*cosh(x)^3*sinh
(x) + 6*cosh(x)^2*sinh(x)^2 - 4*cosh(x)*sinh(x)^3 + sinh(x)^4))/((a^2 + 2*a*b + b^2)*cosh(x)^8 + 8*(a^2 + 2*a*
b + b^2)*cosh(x)*sinh(x)^7 + (a^2 + 2*a*b + b^2...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth {\left (x \right )}}{\left (a + b \coth ^{4}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*coth(x)**4)**(3/2),x)

[Out]

Integral(coth(x)/(a + b*coth(x)**4)**(3/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*coth(x)^4)^(3/2),x, algorithm="giac")

[Out]

integrate(coth(x)/(b*coth(x)^4 + a)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {coth}\left (x\right )}{{\left (b\,{\mathrm {coth}\left (x\right )}^4+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)/(a + b*coth(x)^4)^(3/2),x)

[Out]

int(coth(x)/(a + b*coth(x)^4)^(3/2), x)

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