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

Optimal. Leaf size=50 \[ \frac{x}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )}{a d \sqrt{a-b}} \]

[Out]

x/a - (Sqrt[b]*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]])/(a*Sqrt[a - b]*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0553109, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4127, 3181, 205} \[ \frac{x}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )}{a d \sqrt{a-b}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Csch[c + d*x]^2)^(-1),x]

[Out]

x/a - (Sqrt[b]*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]])/(a*Sqrt[a - b]*d)

Rule 4127

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> Simp[x/a, x] - Dist[b/a, Int[1/(b + a*Cos[e +
f*x]^2), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0]

Rule 3181

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

Rule 205

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

Rubi steps

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

Mathematica [A]  time = 0.197123, size = 52, normalized size = 1.04 \[ \frac{-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )}{d \sqrt{a-b}}+\frac{c}{d}+x}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Csch[c + d*x]^2)^(-1),x]

[Out]

(c/d + x - (Sqrt[b]*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]])/(Sqrt[a - b]*d))/a

________________________________________________________________________________________

Maple [B]  time = 0.07, size = 302, normalized size = 6. \begin{align*}{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{b}{da}\arctan \left ({b\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{a \left ( a-b \right ) }+2\,a-b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{a \left ( a-b \right ) }+2\,a-b \right ) b}}}}+{\frac{b}{d}\arctan \left ({b\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{a \left ( a-b \right ) }+2\,a-b \right ) b}}}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}{\frac{1}{\sqrt{ \left ( 2\,\sqrt{a \left ( a-b \right ) }+2\,a-b \right ) b}}}}-{\frac{b}{da}{\it Artanh} \left ({b\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) b}}}}+{\frac{b}{d}{\it Artanh} \left ({b\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) b}}}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}{\frac{1}{\sqrt{ \left ( 2\,\sqrt{a \left ( a-b \right ) }-2\,a+b \right ) b}}}}-{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d/a*ln(tanh(1/2*d*x+1/2*c)+1)+1/d/a*b/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arctan(tanh(1/2*d*x+1/2*c)*b/((2*(
a*(a-b))^(1/2)+2*a-b)*b)^(1/2))+1/d*b/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arctan(tanh(1/2*d*x+
1/2*c)*b/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2))-1/d/a*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)*arctanh(tanh(1/2*d*x
+1/2*c)*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))+1/d*b/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)*arcta
nh(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))-1/d/a*ln(tanh(1/2*d*x+1/2*c)-1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.79702, size = 1122, normalized size = 22.44 \begin{align*} \left [\frac{2 \, d x + \sqrt{-\frac{b}{a - b}} \log \left (\frac{a^{2} \cosh \left (d x + c\right )^{4} + 4 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{2} \sinh \left (d x + c\right )^{4} - 2 \,{\left (a^{2} - 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} - a^{2} + 2 \, a b\right )} \sinh \left (d x + c\right )^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \,{\left (a^{2} \cosh \left (d x + c\right )^{3} -{\left (a^{2} - 2 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \,{\left ({\left (a^{2} - a b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (a^{2} - a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (a^{2} - a b\right )} \sinh \left (d x + c\right )^{2} - a^{2} + 3 \, a b - 2 \, b^{2}\right )} \sqrt{-\frac{b}{a - b}}}{a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \,{\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, a \cosh \left (d x + c\right )^{2} - a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (a \cosh \left (d x + c\right )^{3} -{\left (a - 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a}\right )}{2 \, a d}, \frac{d x - \sqrt{\frac{b}{a - b}} \arctan \left (\frac{{\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} - a + 2 \, b\right )} \sqrt{\frac{b}{a - b}}}{2 \, b}\right )}{a d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*d*x + sqrt(-b/(a - b))*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x +
c)^4 - 2*(a^2 - 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 - a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 - 8*a*
b + 8*b^2 + 4*(a^2*cosh(d*x + c)^3 - (a^2 - 2*a*b)*cosh(d*x + c))*sinh(d*x + c) - 4*((a^2 - a*b)*cosh(d*x + c)
^2 + 2*(a^2 - a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 - a*b)*sinh(d*x + c)^2 - a^2 + 3*a*b - 2*b^2)*sqrt(-b/(a
- b)))/(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)))/(a*d), (d*x - sqrt(b/(a - b))*arctan(1/2*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c)
+ a*sinh(d*x + c)^2 - a + 2*b)*sqrt(b/(a - b))/b))/(a*d)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.18768, size = 88, normalized size = 1.76 \begin{align*} -\frac{b \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} - a + 2 \, b}{2 \, \sqrt{a b - b^{2}}}\right )}{\sqrt{a b - b^{2}} a d} + \frac{d x + c}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*arctan(1/2*(a*e^(2*d*x + 2*c) - a + 2*b)/sqrt(a*b - b^2))/(sqrt(a*b - b^2)*a*d) + (d*x + c)/(a*d)