3.217 \(\int \coth ^4(x) \sqrt{a+b \tanh ^2(x)} \, dx\)
Optimal. Leaf size=78 \[ \sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right )-\frac{1}{3} \coth ^3(x) \sqrt{a+b \tanh ^2(x)}-\frac{(3 a+b) \coth (x) \sqrt{a+b \tanh ^2(x)}}{3 a} \]
[Out]
Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] - ((3*a + b)*Coth[x]*Sqrt[a + b*Tanh[x]^2])/(
3*a) - (Coth[x]^3*Sqrt[a + b*Tanh[x]^2])/3
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Rubi [A] time = 0.145969, antiderivative size = 78, normalized size of antiderivative = 1.,
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 =
{3670, 475, 583, 12, 377, 206} \[ \sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right )-\frac{1}{3} \coth ^3(x) \sqrt{a+b \tanh ^2(x)}-\frac{(3 a+b) \coth (x) \sqrt{a+b \tanh ^2(x)}}{3 a} \]
Antiderivative was successfully verified.
[In]
Int[Coth[x]^4*Sqrt[a + b*Tanh[x]^2],x]
[Out]
Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] - ((3*a + b)*Coth[x]*Sqrt[a + b*Tanh[x]^2])/(
3*a) - (Coth[x]^3*Sqrt[a + b*Tanh[x]^2])/3
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 475
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*e*(m + 1)), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*
x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + b*n*(p + q + 1))*x^n, x
], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] &&
IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 583
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 377
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \coth ^4(x) \sqrt{a+b \tanh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{x^4 \left (1-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=-\frac{1}{3} \coth ^3(x) \sqrt{a+b \tanh ^2(x)}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{3 a+b+2 b x^2}{x^2 \left (1-x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tanh (x)\right )\\ &=-\frac{(3 a+b) \coth (x) \sqrt{a+b \tanh ^2(x)}}{3 a}-\frac{1}{3} \coth ^3(x) \sqrt{a+b \tanh ^2(x)}-\frac{\operatorname{Subst}\left (\int -\frac{3 a (a+b)}{\left (1-x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tanh (x)\right )}{3 a}\\ &=-\frac{(3 a+b) \coth (x) \sqrt{a+b \tanh ^2(x)}}{3 a}-\frac{1}{3} \coth ^3(x) \sqrt{a+b \tanh ^2(x)}-(-a-b) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tanh (x)\right )\\ &=-\frac{(3 a+b) \coth (x) \sqrt{a+b \tanh ^2(x)}}{3 a}-\frac{1}{3} \coth ^3(x) \sqrt{a+b \tanh ^2(x)}-(-a-b) \operatorname{Subst}\left (\int \frac{1}{1-(a+b) x^2} \, dx,x,\frac{\tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right )\\ &=\sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right )-\frac{(3 a+b) \coth (x) \sqrt{a+b \tanh ^2(x)}}{3 a}-\frac{1}{3} \coth ^3(x) \sqrt{a+b \tanh ^2(x)}\\ \end{align*}
Mathematica [C] time = 8.83605, size = 235, normalized size = 3.01 \[ -\frac{\tanh (x) \left (-12 \sqrt{2} a (a+b) \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}}}{\sqrt{2}}\right ),1\right )+\text{csch}^4(x) \left (4 \left (a^2-3 a b-b^2\right ) \cosh (2 x)+\left (4 a^2+5 a b+b^2\right ) \cosh (4 x)+b (7 a+3 b)\right )+12 \sqrt{2} a^2 \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}} \Pi \left (\frac{b}{a+b};\left .\sin ^{-1}\left (\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right )\right |1\right )\right )}{12 \sqrt{2} a \sqrt{\text{sech}^2(x) ((a+b) \cosh (2 x)+a-b)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[x]^4*Sqrt[a + b*Tanh[x]^2],x]
[Out]
-(((b*(7*a + 3*b) + 4*(a^2 - 3*a*b - b^2)*Cosh[2*x] + (4*a^2 + 5*a*b + b^2)*Cosh[4*x])*Csch[x]^4 - 12*Sqrt[2]*
a*(a + b)*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*C
sch[x]^2)/b]/Sqrt[2]], 1] + 12*Sqrt[2]*a^2*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*EllipticPi[b/(a + b
), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1])*Tanh[x])/(12*Sqrt[2]*a*Sqrt[(a - b + (
a + b)*Cosh[2*x])*Sech[x]^2])
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Maple [F] time = 0.152, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (x\right ) \right ) ^{4}\sqrt{a+b \left ( \tanh \left ( x \right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)^4*(a+b*tanh(x)^2)^(1/2),x)
[Out]
int(coth(x)^4*(a+b*tanh(x)^2)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \tanh \left (x\right )^{2} + a} \coth \left (x\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^4*(a+b*tanh(x)^2)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(b*tanh(x)^2 + a)*coth(x)^4, x)
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Fricas [B] time = 3.91429, size = 6839, normalized size = 87.68 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^4*(a+b*tanh(x)^2)^(1/2),x, algorithm="fricas")
[Out]
[1/12*(3*(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 - 3*a*cosh(x)^4 + 3*(5*a*cosh(x)^2 - a)*sinh(x)^4
+ 4*(5*a*cosh(x)^3 - 3*a*cosh(x))*sinh(x)^3 + 3*a*cosh(x)^2 + 3*(5*a*cosh(x)^4 - 6*a*cosh(x)^2 + a)*sinh(x)^2
+ 6*(a*cosh(x)^5 - 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) - a)*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 + (70*(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))*s
inh(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*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 - 3*a*cosh(x)
^4 + 3*(5*a*cosh(x)^2 - a)*sinh(x)^4 + 4*(5*a*cosh(x)^3 - 3*a*cosh(x))*sinh(x)^3 + 3*a*cosh(x)^2 + 3*(5*a*cosh
(x)^4 - 6*a*cosh(x)^2 + a)*sinh(x)^2 + 6*(a*cosh(x)^5 - 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) - a)*sqrt(a + b)*lo
g(((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 + 2*a*cosh(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)*((4*a + b)*cosh(x)^4 + 4*(4*a + b)*
cosh(x)*sinh(x)^3 + (4*a + b)*sinh(x)^4 - 2*(2*a + b)*cosh(x)^2 + 2*(3*(4*a + b)*cosh(x)^2 - 2*a - b)*sinh(x)^
2 + 4*((4*a + b)*cosh(x)^3 - (2*a + b)*cosh(x))*sinh(x) + 4*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*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 - 3
*a*cosh(x)^4 + 3*(5*a*cosh(x)^2 - a)*sinh(x)^4 + 4*(5*a*cosh(x)^3 - 3*a*cosh(x))*sinh(x)^3 + 3*a*cosh(x)^2 + 3
*(5*a*cosh(x)^4 - 6*a*cosh(x)^2 + a)*sinh(x)^2 + 6*(a*cosh(x)^5 - 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) - a), -1/
6*(3*(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 - 3*a*cosh(x)^4 + 3*(5*a*cosh(x)^2 - a)*sinh(x)^4 + 4*
(5*a*cosh(x)^3 - 3*a*cosh(x))*sinh(x)^3 + 3*a*cosh(x)^2 + 3*(5*a*cosh(x)^4 - 6*a*cosh(x)^2 + a)*sinh(x)^2 + 6*
(a*cosh(x)^5 - 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) - a)*sqrt(-a - b)*arctan(sqrt(2)*(b*cosh(x)^2 + 2*b*cosh(x)*
sinh(x) + b*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*b + b^2)*cosh(x)^4 + 4*(a*b + b^2)*cosh(x)*sinh(x)^3 + (a*b + b^2)*sinh(x)
^4 + (a^2 - a*b - 2*b^2)*cosh(x)^2 + (6*(a*b + b^2)*cosh(x)^2 + a^2 - a*b - 2*b^2)*sinh(x)^2 + a^2 + 2*a*b + b
^2 + 2*(2*(a*b + b^2)*cosh(x)^3 + (a^2 - a*b - 2*b^2)*cosh(x))*sinh(x))) + 3*(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x
)^5 + a*sinh(x)^6 - 3*a*cosh(x)^4 + 3*(5*a*cosh(x)^2 - a)*sinh(x)^4 + 4*(5*a*cosh(x)^3 - 3*a*cosh(x))*sinh(x)^
3 + 3*a*cosh(x)^2 + 3*(5*a*cosh(x)^4 - 6*a*cosh(x)^2 + a)*sinh(x)^2 + 6*(a*cosh(x)^5 - 2*a*cosh(x)^3 + a*cosh(
x))*sinh(x) - a)*sqrt(-a - b)*arctan(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))/((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)*s
inh(x)^2 + 4*((a + b)*cosh(x)^3 + (a - b)*cosh(x))*sinh(x) + a + b)) + 2*sqrt(2)*((4*a + b)*cosh(x)^4 + 4*(4*a
+ b)*cosh(x)*sinh(x)^3 + (4*a + b)*sinh(x)^4 - 2*(2*a + b)*cosh(x)^2 + 2*(3*(4*a + b)*cosh(x)^2 - 2*a - b)*si
nh(x)^2 + 4*((4*a + b)*cosh(x)^3 - (2*a + b)*cosh(x))*sinh(x) + 4*a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sin
h(x)^2 + a - b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)
^6 - 3*a*cosh(x)^4 + 3*(5*a*cosh(x)^2 - a)*sinh(x)^4 + 4*(5*a*cosh(x)^3 - 3*a*cosh(x))*sinh(x)^3 + 3*a*cosh(x)
^2 + 3*(5*a*cosh(x)^4 - 6*a*cosh(x)^2 + a)*sinh(x)^2 + 6*(a*cosh(x)^5 - 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) - a
)]
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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)**4*(a+b*tanh(x)**2)**(1/2),x)
[Out]
Timed out
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^4*(a+b*tanh(x)^2)^(1/2),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError