3.3.18 \(\int \coth ^5(x) \sqrt {a+b \tanh ^2(x)} \, dx\) [218]

Optimal. Leaf size=121 \[ -\frac {\left (8 a^2+4 a b-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a}}\right )}{8 a^{3/2}}+\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right )-\frac {(4 a+b) \coth ^2(x) \sqrt {a+b \tanh ^2(x)}}{8 a}-\frac {1}{4} \coth ^4(x) \sqrt {a+b \tanh ^2(x)} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3751, 457, 101, 156, 162, 65, 214} \begin {gather*} -\frac {\left (8 a^2+4 a b-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a}}\right )}{8 a^{3/2}}+\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right )-\frac {1}{4} \coth ^4(x) \sqrt {a+b \tanh ^2(x)}-\frac {(4 a+b) \coth ^2(x) \sqrt {a+b \tanh ^2(x)}}{8 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]^5*Sqrt[a + b*Tanh[x]^2],x]

[Out]

-1/8*((8*a^2 + 4*a*b - b^2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]])/a^(3/2) + Sqrt[a + b]*ArcTanh[Sqrt[a + b*T
anh[x]^2]/Sqrt[a + b]] - ((4*a + b)*Coth[x]^2*Sqrt[a + b*Tanh[x]^2])/(8*a) - (Coth[x]^4*Sqrt[a + b*Tanh[x]^2])
/4

Rule 65

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 101

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 457

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 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 ^5(x) \sqrt {a+b \tanh ^2(x)} \, dx &=\text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{x^5 \left (1-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{(1-x) x^3} \, dx,x,\tanh ^2(x)\right )\\ &=-\frac {1}{4} \coth ^4(x) \sqrt {a+b \tanh ^2(x)}+\frac {1}{4} \text {Subst}\left (\int \frac {\frac {1}{2} (4 a+b)+\frac {3 b x}{2}}{(1-x) x^2 \sqrt {a+b x}} \, dx,x,\tanh ^2(x)\right )\\ &=-\frac {(4 a+b) \coth ^2(x) \sqrt {a+b \tanh ^2(x)}}{8 a}-\frac {1}{4} \coth ^4(x) \sqrt {a+b \tanh ^2(x)}-\frac {\text {Subst}\left (\int \frac {\frac {1}{4} \left (-8 a^2-4 a b+b^2\right )-\frac {1}{4} b (4 a+b) x}{(1-x) x \sqrt {a+b x}} \, dx,x,\tanh ^2(x)\right )}{4 a}\\ &=-\frac {(4 a+b) \coth ^2(x) \sqrt {a+b \tanh ^2(x)}}{8 a}-\frac {1}{4} \coth ^4(x) \sqrt {a+b \tanh ^2(x)}-\frac {1}{2} (-a-b) \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x}} \, dx,x,\tanh ^2(x)\right )+\frac {\left (8 a^2+4 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tanh ^2(x)\right )}{16 a}\\ &=-\frac {(4 a+b) \coth ^2(x) \sqrt {a+b \tanh ^2(x)}}{8 a}-\frac {1}{4} \coth ^4(x) \sqrt {a+b \tanh ^2(x)}+\frac {(a+b) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tanh ^2(x)}\right )}{b}+\frac {1}{8} \left (4+\frac {8 a}{b}-\frac {b}{a}\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tanh ^2(x)}\right )\\ &=-\frac {\left (8 a^2+4 a b-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a}}\right )}{8 a^{3/2}}+\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right )-\frac {(4 a+b) \coth ^2(x) \sqrt {a+b \tanh ^2(x)}}{8 a}-\frac {1}{4} \coth ^4(x) \sqrt {a+b \tanh ^2(x)}\\ \end {align*}

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Mathematica [A]
time = 0.45, size = 111, normalized size = 0.92 \begin {gather*} \frac {\left (-8 a^2-4 a b+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a}}\right )+\sqrt {a} \left (8 a \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right )-\coth ^2(x) \left (4 a+b+2 a \coth ^2(x)\right ) \sqrt {a+b \tanh ^2(x)}\right )}{8 a^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^5*Sqrt[a + b*Tanh[x]^2],x]

[Out]

((-8*a^2 - 4*a*b + b^2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]] + Sqrt[a]*(8*a*Sqrt[a + b]*ArcTanh[Sqrt[a + b*T
anh[x]^2]/Sqrt[a + b]] - Coth[x]^2*(4*a + b + 2*a*Coth[x]^2)*Sqrt[a + b*Tanh[x]^2]))/(8*a^(3/2))

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Maple [F]
time = 1.57, size = 0, normalized size = 0.00 \[\int \left (\coth ^{5}\left (x \right )\right ) \sqrt {a +b \left (\tanh ^{2}\left (x \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^5*(a+b*tanh(x)^2)^(1/2),x)

[Out]

int(coth(x)^5*(a+b*tanh(x)^2)^(1/2),x)

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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)^5*(a+b*tanh(x)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*tanh(x)^2 + a)*coth(x)^5, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2094 vs. \(2 (99) = 198\).
time = 0.67, size = 9642, normalized size = 79.69 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^5*(a+b*tanh(x)^2)^(1/2),x, algorithm="fricas")

[Out]

[1/16*(4*(a^2*cosh(x)^8 + 8*a^2*cosh(x)*sinh(x)^7 + a^2*sinh(x)^8 - 4*a^2*cosh(x)^6 + 4*(7*a^2*cosh(x)^2 - a^2
)*sinh(x)^6 + 6*a^2*cosh(x)^4 + 8*(7*a^2*cosh(x)^3 - 3*a^2*cosh(x))*sinh(x)^5 + 2*(35*a^2*cosh(x)^4 - 30*a^2*c
osh(x)^2 + 3*a^2)*sinh(x)^4 - 4*a^2*cosh(x)^2 + 8*(7*a^2*cosh(x)^5 - 10*a^2*cosh(x)^3 + 3*a^2*cosh(x))*sinh(x)
^3 + 4*(7*a^2*cosh(x)^6 - 15*a^2*cosh(x)^4 + 9*a^2*cosh(x)^2 - a^2)*sinh(x)^2 + a^2 + 8*(a^2*cosh(x)^7 - 3*a^2
*cosh(x)^5 + 3*a^2*cosh(x)^3 - a^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)*cosh(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)*c
osh(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)) - ((8*a^2 + 4*a*b - b^2)*cosh(x)^8 + 8*(8*a^2 + 4*a*b - b^2)*cosh(x)*sinh(x
)^7 + (8*a^2 + 4*a*b - b^2)*sinh(x)^8 - 4*(8*a^2 + 4*a*b - b^2)*cosh(x)^6 + 4*(7*(8*a^2 + 4*a*b - b^2)*cosh(x)
^2 - 8*a^2 - 4*a*b + b^2)*sinh(x)^6 + 8*(7*(8*a^2 + 4*a*b - b^2)*cosh(x)^3 - 3*(8*a^2 + 4*a*b - b^2)*cosh(x))*
sinh(x)^5 + 6*(8*a^2 + 4*a*b - b^2)*cosh(x)^4 + 2*(35*(8*a^2 + 4*a*b - b^2)*cosh(x)^4 - 30*(8*a^2 + 4*a*b - b^
2)*cosh(x)^2 + 24*a^2 + 12*a*b - 3*b^2)*sinh(x)^4 + 8*(7*(8*a^2 + 4*a*b - b^2)*cosh(x)^5 - 10*(8*a^2 + 4*a*b -
b^2)*cosh(x)^3 + 3*(8*a^2 + 4*a*b - b^2)*cosh(x))*sinh(x)^3 - 4*(8*a^2 + 4*a*b - b^2)*cosh(x)^2 + 4*(7*(8*a^2
+ 4*a*b - b^2)*cosh(x)^6 - 15*(8*a^2 + 4*a*b - b^2)*cosh(x)^4 + 9*(8*a^2 + 4*a*b - b^2)*cosh(x)^2 - 8*a^2 - 4
*a*b + b^2)*sinh(x)^2 + 8*a^2 + 4*a*b - b^2 + 8*((8*a^2 + 4*a*b - b^2)*cosh(x)^7 - 3*(8*a^2 + 4*a*b - b^2)*cos
h(x)^5 + 3*(8*a^2 + 4*a*b - b^2)*cosh(x)^3 - (8*a^2 + 4*a*b - b^2)*cosh(x))*sinh(x))*sqrt(a)*log(-((2*a + b)*c
osh(x)^4 + 4*(2*a + b)*cosh(x)*sinh(x)^3 + (2*a + b)*sinh(x)^4 + 2*(2*a - b)*cosh(x)^2 + 2*(3*(2*a + b)*cosh(x
)^2 + 2*a - b)*sinh(x)^2 + 2*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(a)*sqrt(((a + b)*cos
h(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*((2*a + b)*cosh(x)^3 + (2
*a - b)*cosh(x))*sinh(x) + 2*a + 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^2*cosh(x)^8 + 8*a^2*cosh(x)*sinh(x)^7 + a^2*si
nh(x)^8 - 4*a^2*cosh(x)^6 + 4*(7*a^2*cosh(x)^2 - a^2)*sinh(x)^6 + 6*a^2*cosh(x)^4 + 8*(7*a^2*cosh(x)^3 - 3*a^2
*cosh(x))*sinh(x)^5 + 2*(35*a^2*cosh(x)^4 - 30*a^2*cosh(x)^2 + 3*a^2)*sinh(x)^4 - 4*a^2*cosh(x)^2 + 8*(7*a^2*c
osh(x)^5 - 10*a^2*cosh(x)^3 + 3*a^2*cosh(x))*sinh(x)^3 + 4*(7*a^2*cosh(x)^6 - 15*a^2*cosh(x)^4 + 9*a^2*cosh(x)
^2 - a^2)*sinh(x)^2 + a^2 + 8*(a^2*cosh(x)^7 - 3*a^2*cosh(x)^5 + 3*a^2*cosh(x)^3 - a^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)) - 2*sqrt(2)*((6*a^2 + a*b)*cosh(x)^6
+ 6*(6*a^2 + a*b)*cosh(x)*sinh(x)^5 + (6*a^2 + a*b)*sinh(x)^6 + (2*a^2 - a*b)*cosh(x)^4 + (15*(6*a^2 + a*b)*co
sh(x)^2 + 2*a^2 - a*b)*sinh(x)^4 + 4*(5*(6*a^2 + a*b)*cosh(x)^3 + (2*a^2 - a*b)*cosh(x))*sinh(x)^3 + (2*a^2 -
a*b)*cosh(x)^2 + (15*(6*a^2 + a*b)*cosh(x)^4 + 6*(2*a^2 - a*b)*cosh(x)^2 + 2*a^2 - a*b)*sinh(x)^2 + 6*a^2 + a*
b + 2*(3*(6*a^2 + a*b)*cosh(x)^5 + 2*(2*a^2 - a*b)*cosh(x)^3 + (2*a^2 - a*b)*cosh(x))*sinh(x))*sqrt(((a + b)*c
osh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(a^2*cosh(x)^8 + 8*a^2*cos
h(x)*sinh(x)^7 + a^2*sinh(x)^8 - 4*a^2*cosh(x)^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \tanh ^{2}{\left (x \right )}} \coth ^{5}{\left (x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**5*(a+b*tanh(x)**2)**(1/2),x)

[Out]

Integral(sqrt(a + b*tanh(x)**2)*coth(x)**5, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 947 vs. \(2 (99) = 198\).
time = 1.00, size = 947, normalized size = 7.83 \begin {gather*} -\frac {1}{2} \, \sqrt {a + b} \log \left ({\left | -{\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} {\left (a + b\right )} - \sqrt {a + b} {\left (a - b\right )} \right |}\right ) + \frac {1}{2} \, \sqrt {a + b} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} + \sqrt {a + b} \right |}\right ) - \frac {1}{2} \, \sqrt {a + b} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b} \right |}\right ) + \frac {{\left (8 \, a^{2} + 4 \, a b - b^{2}\right )} \arctan \left (-\frac {\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b}}{2 \, \sqrt {-a}}\right )}{4 \, \sqrt {-a} a} + \frac {{\left (16 \, a^{2} + 12 \, a b + b^{2}\right )} {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )}^{7} - {\left (16 \, a^{2} + 52 \, a b + 7 \, b^{2}\right )} {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )}^{6} \sqrt {a + b} - {\left (48 \, a^{3} - 28 \, a^{2} b - 109 \, a b^{2} - 21 \, b^{3}\right )} {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )}^{5} + {\left (176 \, a^{3} + 156 \, a^{2} b - 115 \, a b^{2} - 35 \, b^{3}\right )} {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )}^{4} \sqrt {a + b} + {\left (304 \, a^{4} - 156 \, a^{3} b - 317 \, a^{2} b^{2} + 130 \, a b^{3} + 35 \, b^{4}\right )} {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )}^{3} - {\left (48 \, a^{4} + 476 \, a^{3} b - 379 \, a^{2} b^{2} + 94 \, a b^{3} + 21 \, b^{4}\right )} {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )}^{2} \sqrt {a + b} - {\left (272 \, a^{5} + 140 \, a^{4} b - 271 \, a^{3} b^{2} + 135 \, a^{2} b^{3} - 53 \, a b^{4} - 7 \, b^{5}\right )} {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} - {\left (112 \, a^{5} - 116 \, a^{4} b + 65 \, a^{3} b^{2} - 17 \, a^{2} b^{3} + 11 \, a b^{4} + b^{5}\right )} \sqrt {a + b}}{2 \, {\left ({\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )}^{2} - 2 \, {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} \sqrt {a + b} - 3 \, a + b\right )}^{4} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^5*(a+b*tanh(x)^2)^(1/2),x, algorithm="giac")

[Out]

-1/2*sqrt(a + b)*log(abs(-(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a +
b))*(a + b) - sqrt(a + b)*(a - b))) + 1/2*sqrt(a + b)*log(abs(-sqrt(a + b)*e^(2*x) + sqrt(a*e^(4*x) + b*e^(4*x
) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b) + sqrt(a + b))) - 1/2*sqrt(a + b)*log(abs(-sqrt(a + b)*e^(2*x) + sqrt(a
*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b) - sqrt(a + b))) + 1/4*(8*a^2 + 4*a*b - b^2)*arctan(-
1/2*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b) - sqrt(a + b))/sqrt
(-a))/(sqrt(-a)*a) + 1/2*((16*a^2 + 12*a*b + b^2)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2
*x) - 2*b*e^(2*x) + a + b))^7 - (16*a^2 + 52*a*b + 7*b^2)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) +
2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^6*sqrt(a + b) - (48*a^3 - 28*a^2*b - 109*a*b^2 - 21*b^3)*(sqrt(a + b)*e^(2
*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^5 + (176*a^3 + 156*a^2*b - 115*a*b^2 -
35*b^3)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^4*sqrt(a + b)
+ (304*a^4 - 156*a^3*b - 317*a^2*b^2 + 130*a*b^3 + 35*b^4)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) +
2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^3 - (48*a^4 + 476*a^3*b - 379*a^2*b^2 + 94*a*b^3 + 21*b^4)*(sqrt(a + b)*e
^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^2*sqrt(a + b) - (272*a^5 + 140*a^4*b
- 271*a^3*b^2 + 135*a^2*b^3 - 53*a*b^4 - 7*b^5)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*
x) - 2*b*e^(2*x) + a + b)) - (112*a^5 - 116*a^4*b + 65*a^3*b^2 - 17*a^2*b^3 + 11*a*b^4 + b^5)*sqrt(a + b))/(((
sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^2 - 2*(sqrt(a + b)*e^(2
*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b))*sqrt(a + b) - 3*a + b)^4*a)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {coth}\left (x\right )}^5\,\sqrt {b\,{\mathrm {tanh}\left (x\right )}^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^5*(a + b*tanh(x)^2)^(1/2),x)

[Out]

int(coth(x)^5*(a + b*tanh(x)^2)^(1/2), x)

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