3.1.51 \(\int \coth (x) \sqrt {a+b \coth ^4(x)} \, dx\) [51]
Optimal. Leaf size=89 \[ -\frac {1}{2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \coth ^2(x)}{\sqrt {a+b \coth ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b} \tanh ^{-1}\left (\frac {a+b \coth ^2(x)}{\sqrt {a+b} \sqrt {a+b \coth ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b \coth ^4(x)} \]
[Out]
-1/2*arctanh(coth(x)^2*b^(1/2)/(a+b*coth(x)^4)^(1/2))*b^(1/2)+1/2*arctanh((a+b*coth(x)^2)/(a+b)^(1/2)/(a+b*cot
h(x)^4)^(1/2))*(a+b)^(1/2)-1/2*(a+b*coth(x)^4)^(1/2)
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Rubi [A]
time = 0.10, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3751, 1262,
749, 858, 223, 212, 739} \begin {gather*} -\frac {1}{2} \sqrt {a+b \coth ^4(x)}-\frac {1}{2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \coth ^2(x)}{\sqrt {a+b \coth ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b} \tanh ^{-1}\left (\frac {a+b \coth ^2(x)}{\sqrt {a+b} \sqrt {a+b \coth ^4(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Coth[x]*Sqrt[a + b*Coth[x]^4],x]
[Out]
-1/2*(Sqrt[b]*ArcTanh[(Sqrt[b]*Coth[x]^2)/Sqrt[a + b*Coth[x]^4]]) + (Sqrt[a + b]*ArcTanh[(a + b*Coth[x]^2)/(Sq
rt[a + b]*Sqrt[a + b*Coth[x]^4])])/2 - Sqrt[a + b*Coth[x]^4]/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 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 749
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + c*x^2)^p/(e
*(m + 2*p + 1))), x] + Dist[2*(p/(e*(m + 2*p + 1))), Int[(d + e*x)^m*Simp[a*e - c*d*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] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !Ration
alQ[m] || LtQ[m, 1]) && !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
Rule 858
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && !IGtQ[m, 0]
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 \coth (x) \sqrt {a+b \coth ^4(x)} \, dx &=\text {Subst}\left (\int \frac {x \sqrt {a+b x^4}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{1-x} \, dx,x,\coth ^2(x)\right )\\ &=-\frac {1}{2} \sqrt {a+b \coth ^4(x)}-\frac {1}{2} \text {Subst}\left (\int \frac {-a-b x}{(1-x) \sqrt {a+b x^2}} \, dx,x,\coth ^2(x)\right )\\ &=-\frac {1}{2} \sqrt {a+b \coth ^4(x)}-\frac {1}{2} (-a-b) \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x^2}} \, dx,x,\coth ^2(x)\right )-\frac {1}{2} b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\coth ^2(x)\right )\\ &=-\frac {1}{2} \sqrt {a+b \coth ^4(x)}-\frac {1}{2} b \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\coth ^2(x)}{\sqrt {a+b \coth ^4(x)}}\right )-\frac {1}{2} (a+b) \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 )\\ &=-\frac {1}{2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \coth ^2(x)}{\sqrt {a+b \coth ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b} \tanh ^{-1}\left (\frac {a+b \coth ^2(x)}{\sqrt {a+b} \sqrt {a+b \coth ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b \coth ^4(x)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 86, normalized size = 0.97 \begin {gather*} \frac {1}{2} \left (-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \coth ^2(x)}{\sqrt {a+b \coth ^4(x)}}\right )+\sqrt {a+b} \tanh ^{-1}\left (\frac {a+b \coth ^2(x)}{\sqrt {a+b} \sqrt {a+b \coth ^4(x)}}\right )-\sqrt {a+b \coth ^4(x)}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Coth[x]*Sqrt[a + b*Coth[x]^4],x]
[Out]
(-(Sqrt[b]*ArcTanh[(Sqrt[b]*Coth[x]^2)/Sqrt[a + b*Coth[x]^4]]) + Sqrt[a + b]*ArcTanh[(a + b*Coth[x]^2)/(Sqrt[a
+ b]*Sqrt[a + b*Coth[x]^4])] - Sqrt[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 = 3.25, size = 333, normalized size = 3.74
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(-\frac {\sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}{2}-\frac {\sqrt {b}\, \ln \left (2 \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )+2 \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}\right )}{2}-\frac {\left (a +b \right ) \left (-\frac {\arctanh \left (\frac {2 b \left (\coth ^{2}\left (x \right )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \EllipticPi \left (\coth \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2}-\frac {\left (a +b \right ) \left (-\frac {\arctanh \left (\frac {2 b \left (\coth ^{2}\left (x \right )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b}}-\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \EllipticPi \left (\coth \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2}\) |
\(333\) |
| default |
\(-\frac {\sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}{2}-\frac {\sqrt {b}\, \ln \left (2 \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )+2 \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}\right )}{2}-\frac {\left (a +b \right ) \left (-\frac {\arctanh \left (\frac {2 b \left (\coth ^{2}\left (x \right )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \EllipticPi \left (\coth \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2}-\frac {\left (a +b \right ) \left (-\frac {\arctanh \left (\frac {2 b \left (\coth ^{2}\left (x \right )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b}}-\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \EllipticPi \left (\coth \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2}\) |
\(333\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)*(a+b*coth(x)^4)^(1/2),x,method=_RETURNVERBOSE)
[Out]
-1/2*(a+b*coth(x)^4)^(1/2)-1/2*b^(1/2)*ln(2*b^(1/2)*coth(x)^2+2*(a+b*coth(x)^4)^(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)))-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(c
oth(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)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(b*coth(x)^4 + a)*coth(x), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1053 vs.
\(2 (69) = 138\).
time = 0.52, size = 5172, normalized size = 58.11 \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)^(1/2),x, algorithm="fricas")
[Out]
[1/4*((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)*sqrt(b)*log(-((a + 2*b)*cosh(x)^8 + 8*(a + 2*b)*cosh(x)*sinh(x)^7 + (a + 2*b)*sinh(x
)^8 - 4*(a - 2*b)*cosh(x)^6 + 4*(7*(a + 2*b)*cosh(x)^2 - a + 2*b)*sinh(x)^6 + 8*(7*(a + 2*b)*cosh(x)^3 - 3*(a
- 2*b)*cosh(x))*sinh(x)^5 + 6*(a + 2*b)*cosh(x)^4 + 2*(35*(a + 2*b)*cosh(x)^4 - 30*(a - 2*b)*cosh(x)^2 + 3*a +
6*b)*sinh(x)^4 + 8*(7*(a + 2*b)*cosh(x)^5 - 10*(a - 2*b)*cosh(x)^3 + 3*(a + 2*b)*cosh(x))*sinh(x)^3 - 4*(a -
2*b)*cosh(x)^2 + 4*(7*(a + 2*b)*cosh(x)^6 - 15*(a - 2*b)*cosh(x)^4 + 9*(a + 2*b)*cosh(x)^2 - a + 2*b)*sinh(x)^
2 - 2*sqrt(2)*(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)*sqrt(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)) + 8*((a + 2*b)*cosh(x)^7 - 3*(a - 2*b)*cosh(x)^5 + 3*(a + 2*b)*cosh(
x)^3 - (a - 2*b)*cosh(x))*sinh(x) + a + 2*b)/(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 - 1
)*sinh(x)^6 - 4*cosh(x)^6 + 8*(7*cosh(x)^3 - 3*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 - 30*cosh(x)^2 + 3)*sinh(x
)^4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 - 10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 - 15*cosh(x)^4 + 9*c
osh(x)^2 - 1)*sinh(x)^2 - 4*cosh(x)^2 + 8*(cosh(x)^7 - 3*cosh(x)^5 + 3*cosh(x)^3 - cosh(x))*sinh(x) + 1)) + (c
osh(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)*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)*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*c
osh(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)*cosh(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*cos
h(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4)) - 2*sqrt(2)*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*sin
h(x) + 6*cosh(x)^2*sinh(x)^2 - 4*cosh(x)*sinh(x)^3 + sinh(x)^4)))/(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), -1/4*(2*(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)*sqrt(-a - b)*arctan(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)*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)*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 + 6*(a^2 + 2*a*b + 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 + 6*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 + 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 + 9*(a^2 + 2*a*b + b^2)*cosh(x)^2 - a^2 + b^2)
*sinh(x)^2 + a^2 + 2*a*b + b^2 + 8*((a^2 + 2*a*b + b^2)*cosh(x)^7 - 3*(a^2 - b^2)*cosh(x)^5 + 3*(a^2 + 2*a*b +
b^2)*cosh(x)^3 - (a^2 - b^2)*cosh(x))*sinh(x))) - (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)*sqrt(b)*log(-((a + 2*b)*cosh(x)^8 + 8*(
a + 2*b)*cosh(x)*sinh(x)^7 + (a + 2*b)*sinh(x)^...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \coth ^{4}{\left (x \right )}} \coth {\left (x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)*(a+b*coth(x)**4)**(1/2),x)
[Out]
Integral(sqrt(a + b*coth(x)**4)*coth(x), 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)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(b*coth(x)^4 + a)*coth(x), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {coth}\left (x\right )\,\sqrt {b\,{\mathrm {coth}\left (x\right )}^4+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)*(a + b*coth(x)^4)^(1/2),x)
[Out]
int(coth(x)*(a + b*coth(x)^4)^(1/2), x)
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