∫ (a + b coth2(x))3∕2 tanh2(x) dx [27]

3.1.27 \(\int (a+b \coth ^2(x))^{3/2} \tanh ^2(x) \, dx\) [27]

Optimal. Leaf size=77 \[ -b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-a \sqrt {a+b \coth ^2(x)} \tanh (x) \]

[Out]

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

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

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Rubi [A]
time = 0.09, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3751, 485, 537, 223, 212, 385} \begin {gather*} b^{3/2} \left (-\tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )\right )+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-a \tanh (x) \sqrt {a+b \coth ^2(x)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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 385

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 485

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[c*(e*x)^(

m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(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 - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1) + a*d*(q - 1)) + d*((c*b - a*

d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]

&& GtQ[q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I

nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(180\) vs. \(2(77)=154\).
time = 0.35, size = 180, normalized size = 2.34 \begin {gather*} \frac {\left (-\sqrt {2} b^{3/2} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \cosh (x)}{\sqrt {-a+b+(a+b) \cosh (2 x)}}\right ) \cosh (x)+\sqrt {2} (a+b)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a+b} \cosh (x)}{\sqrt {-a+b+(a+b) \cosh (2 x)}}\right ) \cosh (x)-a \sqrt {a+b} \sqrt {-a+b+(a+b) \cosh (2 x)}\right ) \sqrt {(-a+b+(a+b) \cosh (2 x)) \text {csch}^2(x)} \tanh (x)}{\sqrt {2} \sqrt {a+b} \sqrt {-a+b+(a+b) \cosh (2 x)}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

+ b]*Sqrt[-a + b + (a + b)*Cosh[2*x]])*Sqrt[(-a + b + (a + b)*Cosh[2*x])*Csch[x]^2]*Tanh[x])/(Sqrt[2]*Sqrt[a +

b]*Sqrt[-a + b + (a + b)*Cosh[2*x]])

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Maple [F]
time = 1.88, size = 0, normalized size = 0.00 \[\int \left (a +b \left (\coth ^{2}\left (x \right )\right )\right )^{\frac {3}{2}} \left (\tanh ^{2}\left (x \right )\right )\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (63) = 126\).
time = 0.53, size = 4025, normalized size = 52.27 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

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

sh(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)*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)))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1), 1/4*(4*(b*cosh(x)^2 + 2*b*co

sh(x)*sinh(x) + b*sinh(x)^2 + b)*sqrt(-b)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(

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

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

a + b)*log(-((a + b)*cosh(x)^4 + 4*(a + b)*cosh...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

sage0*x

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

[Out]

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

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