3.18 \(\int \coth (x) \sqrt{a+b \coth ^2(x)} \, dx\)

Optimal. Leaf size=44 \[ \sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right )-\sqrt{a+b \coth ^2(x)} \]

[Out]

Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Coth[x]^2]

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Rubi [A]  time = 0.0770213, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3670, 444, 50, 63, 208} \[ \sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right )-\sqrt{a+b \coth ^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]*Sqrt[a + b*Coth[x]^2],x]

[Out]

Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Coth[x]^2]

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 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 208

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

Rubi steps

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

Mathematica [A]  time = 0.0313469, size = 44, normalized size = 1. \[ \sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right )-\sqrt{a+b \coth ^2(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Coth[x]^2]

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Maple [B]  time = 0.044, size = 238, normalized size = 5.4 \begin{align*} -{\frac{1}{2}\sqrt{ \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}b-2\, \left ( 1+{\rm coth} \left (x\right ) \right ) b+a+b}}+{\frac{1}{2}\sqrt{b}\ln \left ({( \left ( 1+{\rm coth} \left (x\right ) \right ) b-b){\frac{1}{\sqrt{b}}}}+\sqrt{ \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}b-2\, \left ( 1+{\rm coth} \left (x\right ) \right ) b+a+b} \right ) }+{\frac{1}{2}\sqrt{a+b}\ln \left ({\frac{1}{1+{\rm coth} \left (x\right )} \left ( 2\,a+2\,b-2\, \left ( 1+{\rm coth} \left (x\right ) \right ) b+2\,\sqrt{a+b}\sqrt{ \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}b-2\, \left ( 1+{\rm coth} \left (x\right ) \right ) b+a+b} \right ) } \right ) }-{\frac{1}{2}\sqrt{ \left ({\rm coth} \left (x\right )-1 \right ) ^{2}b+2\, \left ({\rm coth} \left (x\right )-1 \right ) b+a+b}}-{\frac{1}{2}\sqrt{b}\ln \left ({( \left ({\rm coth} \left (x\right )-1 \right ) b+b){\frac{1}{\sqrt{b}}}}+\sqrt{ \left ({\rm coth} \left (x\right )-1 \right ) ^{2}b+2\, \left ({\rm coth} \left (x\right )-1 \right ) b+a+b} \right ) }+{\frac{1}{2}\sqrt{a+b}\ln \left ({\frac{1}{{\rm coth} \left (x\right )-1} \left ( 2\,a+2\,b+2\, \left ({\rm coth} \left (x\right )-1 \right ) b+2\,\sqrt{a+b}\sqrt{ \left ({\rm coth} \left (x\right )-1 \right ) ^{2}b+2\, \left ({\rm coth} \left (x\right )-1 \right ) b+a+b} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((1+coth(x))^2*b-2*(1+coth(x))*b+a+b)^(1/2)+1/2*b^(1/2)*ln(((1+coth(x))*b-b)/b^(1/2)+((1+coth(x))^2*b-2*(
1+coth(x))*b+a+b)^(1/2))+1/2*(a+b)^(1/2)*ln((2*a+2*b-2*(1+coth(x))*b+2*(a+b)^(1/2)*((1+coth(x))^2*b-2*(1+coth(
x))*b+a+b)^(1/2))/(1+coth(x)))-1/2*((coth(x)-1)^2*b+2*(coth(x)-1)*b+a+b)^(1/2)-1/2*b^(1/2)*ln(((coth(x)-1)*b+b
)/b^(1/2)+((coth(x)-1)^2*b+2*(coth(x)-1)*b+a+b)^(1/2))+1/2*(a+b)^(1/2)*ln((2*a+2*b+2*(coth(x)-1)*b+2*(a+b)^(1/
2)*((coth(x)-1)^2*b+2*(coth(x)-1)*b+a+b)^(1/2))/(coth(x)-1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \coth \left (x\right )^{2} + a} \coth \left (x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 3.31521, size = 4456, normalized size = 101.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*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)
*cosh(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)) + (cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*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)*si
nh(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)) - 4*sqrt(2)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2
- a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1), -1/2*(
(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-a - b)*arctan(sqrt(2)*(a*cosh(x)^2 + 2*a*cosh(x)*sinh(x)
+ a*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^2 + a*b)*cosh(x)^4 + 4*(a^2 + a*b)*cosh(x)*sinh(x)^3 + (a^2 + a*b)*sinh(x)^4 - (2
*a^2 + a*b - b^2)*cosh(x)^2 + (6*(a^2 + a*b)*cosh(x)^2 - 2*a^2 - a*b + b^2)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 2*
(2*(a^2 + a*b)*cosh(x)^3 - (2*a^2 + a*b - b^2)*cosh(x))*sinh(x))) + (cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2
- 1)*sqrt(-a - b)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(-a - b)*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 + 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)) + 2*sqrt(2)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x
)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError