3.13 \(\int \frac{1}{(a+b \text{csch}^2(c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=82 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{a^{3/2} d}+\frac{b \coth (c+d x)}{a d (a-b) \sqrt{a+b \coth ^2(c+d x)-b}} \]

[Out]

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

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Rubi [A]  time = 0.0560802, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4128, 382, 377, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{a^{3/2} d}+\frac{b \coth (c+d x)}{a d (a-b) \sqrt{a+b \coth ^2(c+d x)-b}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Csch[c + d*x]^2)^(-3/2),x]

[Out]

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

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

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

Mathematica [A]  time = 0.339284, size = 148, normalized size = 1.8 \[ \frac{\text{csch}^2(c+d x) \left (\frac{2 \sqrt{a} b \coth (c+d x) (a \cosh (2 (c+d x))-a+2 b)}{a-b}+\sqrt{2} \text{csch}(c+d x) (a \cosh (2 (c+d x))-a+2 b)^{3/2} \log \left (\sqrt{a \cosh (2 (c+d x))-a+2 b}+\sqrt{2} \sqrt{a} \cosh (c+d x)\right )\right )}{4 a^{3/2} d \left (a+b \text{csch}^2(c+d x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Csch[c + d*x]^2)^(-3/2),x]

[Out]

(Csch[c + d*x]^2*((2*Sqrt[a]*b*(-a + 2*b + a*Cosh[2*(c + d*x)])*Coth[c + d*x])/(a - b) + Sqrt[2]*(-a + 2*b + a
*Cosh[2*(c + d*x)])^(3/2)*Csch[c + d*x]*Log[Sqrt[2]*Sqrt[a]*Cosh[c + d*x] + Sqrt[-a + 2*b + a*Cosh[2*(c + d*x)
]]]))/(4*a^(3/2)*d*(a + b*Csch[c + d*x]^2)^(3/2))

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Maple [F]  time = 0.141, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ({\rm csch} \left (dx+c\right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*csch(d*x+c)^2)^(3/2),x)

[Out]

int(1/(a+b*csch(d*x+c)^2)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{csch}\left (d x + c\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csch(d*x+c)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*csch(d*x + c)^2 + a)^(-3/2), x)

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Fricas [B]  time = 3.08744, size = 7602, normalized size = 92.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csch(d*x+c)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(((a^2 - a*b)*cosh(d*x + c)^4 + 4*(a^2 - a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 - a*b)*sinh(d*x + c)^4
- 2*(a^2 - 3*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 - a*b)*cosh(d*x + c)^2 - a^2 + 3*a*b - 2*b^2)*sinh(d*x
+ c)^2 + a^2 - a*b + 4*((a^2 - a*b)*cosh(d*x + c)^3 - (a^2 - 3*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt
(a)*log((a*b^2*cosh(d*x + c)^8 + 8*a*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + a*b^2*sinh(d*x + c)^8 + 2*(a*b^2 + b^
3)*cosh(d*x + c)^6 + 2*(14*a*b^2*cosh(d*x + c)^2 + a*b^2 + b^3)*sinh(d*x + c)^6 + 4*(14*a*b^2*cosh(d*x + c)^3
+ 3*(a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^4 + (70*a*b^2*cosh(
d*x + c)^4 + a^3 - 4*a^2*b + 9*a*b^2 + 30*(a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(14*a*b^2*cosh(d*
x + c)^5 + 10*(a*b^2 + b^3)*cosh(d*x + c)^3 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 -
2*(a^3 - 3*a^2*b)*cosh(d*x + c)^2 + 2*(14*a*b^2*cosh(d*x + c)^6 + 15*(a*b^2 + b^3)*cosh(d*x + c)^4 - a^3 + 3*
a^2*b + 3*(a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + sqrt(2)*(b^2*cosh(d*x + c)^6 + 6*b^2*co
sh(d*x + c)*sinh(d*x + c)^5 + b^2*sinh(d*x + c)^6 + 3*b^2*cosh(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 + b^2)*si
nh(d*x + c)^4 + 4*(5*b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 - (a^2 - 4*a*b)*cosh(d*x + c)^
2 + (15*b^2*cosh(d*x + c)^4 + 18*b^2*cosh(d*x + c)^2 - a^2 + 4*a*b)*sinh(d*x + c)^2 + a^2 + 2*(3*b^2*cosh(d*x
+ c)^5 + 6*b^2*cosh(d*x + c)^3 - (a^2 - 4*a*b)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a)*sqrt((a*cosh(d*x + c)^2 +
a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 4*(2*a*b^
2*cosh(d*x + c)^7 + 3*(a*b^2 + b^3)*cosh(d*x + c)^5 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^3 - (a^3 - 3*a^2
*b)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x + c)^6 + 6*cosh(d*x + c)^5*sinh(d*x + c) + 15*cosh(d*x + c)^4*sinh
(d*x + c)^2 + 20*cosh(d*x + c)^3*sinh(d*x + c)^3 + 15*cosh(d*x + c)^2*sinh(d*x + c)^4 + 6*cosh(d*x + c)*sinh(d
*x + c)^5 + sinh(d*x + c)^6)) + ((a^2 - a*b)*cosh(d*x + c)^4 + 4*(a^2 - a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (
a^2 - a*b)*sinh(d*x + c)^4 - 2*(a^2 - 3*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 - a*b)*cosh(d*x + c)^2 - a^2
+ 3*a*b - 2*b^2)*sinh(d*x + c)^2 + a^2 - a*b + 4*((a^2 - a*b)*cosh(d*x + c)^3 - (a^2 - 3*a*b + 2*b^2)*cosh(d*x
+ c))*sinh(d*x + c))*sqrt(a)*log(-(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4
- 2*(a - b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - a + b)*sinh(d*x + c)^2 + sqrt(2)*(cosh(d*x + c)^2 + 2*c
osh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*
b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 4*(a*cosh(d*x + c)^3 - (a - b)*cosh(
d*x + c))*sinh(d*x + c) + a)/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 4*sqrt(2)*
(a*b*cosh(d*x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d*x + c)^2 + a*b)*sqrt((a*cosh(d*x + c)^2
+ a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)))/((a^4 - a
^3*b)*d*cosh(d*x + c)^4 + 4*(a^4 - a^3*b)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 - a^3*b)*d*sinh(d*x + c)^4 -
2*(a^4 - 3*a^3*b + 2*a^2*b^2)*d*cosh(d*x + c)^2 + 2*(3*(a^4 - a^3*b)*d*cosh(d*x + c)^2 - (a^4 - 3*a^3*b + 2*a^
2*b^2)*d)*sinh(d*x + c)^2 + (a^4 - a^3*b)*d + 4*((a^4 - a^3*b)*d*cosh(d*x + c)^3 - (a^4 - 3*a^3*b + 2*a^2*b^2)
*d*cosh(d*x + c))*sinh(d*x + c)), -1/2*(((a^2 - a*b)*cosh(d*x + c)^4 + 4*(a^2 - a*b)*cosh(d*x + c)*sinh(d*x +
c)^3 + (a^2 - a*b)*sinh(d*x + c)^4 - 2*(a^2 - 3*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 - a*b)*cosh(d*x + c)^
2 - a^2 + 3*a*b - 2*b^2)*sinh(d*x + c)^2 + a^2 - a*b + 4*((a^2 - a*b)*cosh(d*x + c)^3 - (a^2 - 3*a*b + 2*b^2)*
cosh(d*x + c))*sinh(d*x + c))*sqrt(-a)*arctan(sqrt(2)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b
*sinh(d*x + c)^2 + a)*sqrt(-a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cos
h(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2))/(a*b*cosh(d*x + c)^4 + 4*a*b*cosh(d*x + c)*sinh(d*x + c)^3 + a*b*
sinh(d*x + c)^4 - (a^2 - 3*a*b)*cosh(d*x + c)^2 + (6*a*b*cosh(d*x + c)^2 - a^2 + 3*a*b)*sinh(d*x + c)^2 + a^2
+ 2*(2*a*b*cosh(d*x + c)^3 - (a^2 - 3*a*b)*cosh(d*x + c))*sinh(d*x + c))) + ((a^2 - a*b)*cosh(d*x + c)^4 + 4*(
a^2 - a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 - a*b)*sinh(d*x + c)^4 - 2*(a^2 - 3*a*b + 2*b^2)*cosh(d*x + c)
^2 + 2*(3*(a^2 - a*b)*cosh(d*x + c)^2 - a^2 + 3*a*b - 2*b^2)*sinh(d*x + c)^2 + a^2 - a*b + 4*((a^2 - a*b)*cosh
(d*x + c)^3 - (a^2 - 3*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a)*arctan(sqrt(2)*(cosh(d*x + c)^2 + 2
*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(-a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a +
2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x +
c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(a - 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - a + 2*b)*sinh
(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (a - 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) - 2*sqrt(2)*(a*b*cosh(d*x +
c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d*x + c)^2 + a*b)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c
)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)))/((a^4 - a^3*b)*d*cosh(d*x
+ c)^4 + 4*(a^4 - a^3*b)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 - a^3*b)*d*sinh(d*x + c)^4 - 2*(a^4 - 3*a^3*b
+ 2*a^2*b^2)*d*cosh(d*x + c)^2 + 2*(3*(a^4 - a^3*b)*d*cosh(d*x + c)^2 - (a^4 - 3*a^3*b + 2*a^2*b^2)*d)*sinh(d
*x + c)^2 + (a^4 - a^3*b)*d + 4*((a^4 - a^3*b)*d*cosh(d*x + c)^3 - (a^4 - 3*a^3*b + 2*a^2*b^2)*d*cosh(d*x + c)
)*sinh(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{csch}^{2}{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csch(d*x+c)**2)**(3/2),x)

[Out]

Integral((a + b*csch(c + d*x)**2)**(-3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{csch}\left (d x + c\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*csch(d*x+c)^2)^(3/2),x, algorithm="giac")

[Out]

integrate((b*csch(d*x + c)^2 + a)^(-3/2), x)