∫ tanh(x)____ (a+b coth2(x))3∕2 dx [41]

3.1.41 \(\int \frac {\tanh (x)}{(a+b \coth ^2(x))^{3/2}} \, dx\) [41]

Optimal. Leaf size=78 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}+\frac {b}{a (a+b) \sqrt {a+b \coth ^2(x)}} \]

[Out]

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

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

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Rubi [A]
time = 0.10, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3751, 457, 87, 162, 65, 214} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {b}{a (a+b) \sqrt {a+b \coth ^2(x)}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{3/2}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

b/(a*(a + b)*Sqrt[a + b*Coth[x]^2])

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 87

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[f*((e + f*x)^(p +

1)/((p + 1)*(b*e - a*f)*(d*e - c*f))), x] + Dist[1/((b*e - a*f)*(d*e - c*f)), Int[(b*d*e - b*c*f - a*d*f - b*

d*f*x)*((e + f*x)^(p + 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[p, -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 \frac {\tanh (x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx &=\text {Subst}\left (\int \frac {1}{x \left (1-x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\coth (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1-x) x (a+b x)^{3/2}} \, dx,x,\coth ^2(x)\right )\\ &=\frac {b}{a (a+b) \sqrt {a+b \coth ^2(x)}}-\frac {\text {Subst}\left (\int \frac {-a-b+b x}{(1-x) x \sqrt {a+b x}} \, dx,x,\coth ^2(x)\right )}{2 a (a+b)}\\ &=\frac {b}{a (a+b) \sqrt {a+b \coth ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\coth ^2(x)\right )}{2 a}+\frac {\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x}} \, dx,x,\coth ^2(x)\right )}{2 (a+b)}\\ &=\frac {b}{a (a+b) \sqrt {a+b \coth ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \coth ^2(x)}\right )}{a b}+\frac {\text {Subst}\left (\int \frac {1}{1+\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \coth ^2(x)}\right )}{b (a+b)}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}+\frac {b}{a (a+b) \sqrt {a+b \coth ^2(x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.04, size = 70, normalized size = 0.90 \begin {gather*} \frac {-a \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \coth ^2(x)}{a+b}\right )+(a+b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {b \coth ^2(x)}{a}\right )}{a (a+b) \sqrt {a+b \coth ^2(x)}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-(a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Coth[x]^2)/(a + b)]) + (a + b)*Hypergeometric2F1[-1/2, 1, 1/2, 1 +

(b*Coth[x]^2)/a])/(a*(a + b)*Sqrt[a + b*Coth[x]^2])

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Maple [F]
time = 2.02, size = 0, normalized size = 0.00 \[\int \frac {\tanh \left (x \right )}{\left (a +b \left (\coth ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1437 vs. \(2 (64) = 128\).
time = 0.61, size = 6991, normalized size = 89.63 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(((a^3 + a^2*b)*cosh(x)^4 + 4*(a^3 + a^2*b)*cosh(x)*sinh(x)^3 + (a^3 + a^2*b)*sinh(x)^4 + a^3 + a^2*b - 2

*(a^3 - a^2*b)*cosh(x)^2 - 2*(a^3 - a^2*b - 3*(a^3 + a^2*b)*cosh(x)^2)*sinh(x)^2 + 4*((a^3 + a^2*b)*cosh(x)^3

- (a^3 - a^2*b)*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)*sin

h(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*co

sh(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)) + 2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^4 + 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)*si

nh(x)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(x)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(a^3 + a^2*b - a*b^2 -

b^3)*cosh(x)^2 - 2*(a^3 + a^2*b - a*b^2 - b^3 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^2)*sinh(x)^2 + 4*((

a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^3 - (a^3 + a^2*b - a*b^2 - b^3)*cosh(x))*sinh(x))*sqrt(a)*log(-((2*a +

b)*cosh(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)*co

sh(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)

*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)*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)*sin

h(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)) + ((a^3 + a^2*b)*cosh(x)^4 + 4*(a^3 + a^2*b)*cosh

(x)*sinh(x)^3 + (a^3 + a^2*b)*sinh(x)^4 + a^3 + a^2*b - 2*(a^3 - a^2*b)*cosh(x)^2 - 2*(a^3 - a^2*b - 3*(a^3 +

a^2*b)*cosh(x)^2)*sinh(x)^2 + 4*((a^3 + a^2*b)*cosh(x)^3 - (a^3 - a^2*b)*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)) - 4*sqrt(2)*(a^2*b + a*b^2 - (a^2*b + a*b^2)*cosh(x)

^2 - 2*(a^2*b + a*b^2)*cosh(x)*sinh(x) - (a^2*b + a*b^2)*sinh(x)^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)))/(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3 + (a^5 + 3*a^4*b

+ 3*a^3*b^2 + a^2*b^3)*cosh(x)^4 + 4*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)*sinh(x)^3 + (a^5 + 3*a^4*b

+ 3*a^3*b^2 + a^2*b^3)*sinh(x)^4 - 2*(a^5 + a^4*b - a^3*b^2 - a^2*b^3)*cosh(x)^2 - 2*(a^5 + a^4*b - a^3*b^2 -

a^2*b^3 - 3*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(x)^2)*sinh(x)^2 + 4*((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*

b^3)*cosh(x)^3 - (a^5 + a^4*b - a^3*b^2 - a^2*b^3)*cosh(x))*sinh(x)), 1/4*(4*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*

cosh(x)^4 + 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)*sinh(x)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(x)^4 +

a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(a^3 + a^2*b - a*b^2 - b^3)*cosh(x)^2 - 2*(a^3 + a^2*b - a*b^2 - b^3 - 3*(a^

3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^2)*sinh(x)^2 + 4*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^3 - (a^3 + a^2*

b - a*b^2 - b^3)*cosh(x))*sinh(x))*sqrt(-a)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqr

t(-a)*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)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 - (a - b)*cosh(x))*sinh(x) + a + b)) + ((a^3 + a^2*b)*cosh(x)^4 + 4*

(a^3 + a^2*b)*cosh(x)*sinh(x)^3 + (a^3 + a^2*b)*sinh(x)^4 + a^3 + a^2*b - 2*(a^3 - a^2*b)*cosh(x)^2 - 2*(a^3 -

a^2*b - 3*(a^3 + a^2*b)*cosh(x)^2)*sinh(x)^2 +...

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP

UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch

eck [abs(ex

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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