\(\int \frac {1}{a+b \text {csch}(c+d x)} \, dx\) [74]

Optimal result

Integrand size = 12, antiderivative size = 54 \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=\frac {x}{a}+\frac {2 b \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3868, 2739, 632, 210} \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=\frac {2 b \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a d \sqrt {a^2+b^2}}+\frac {x}{a} \]

[In]

Int[(a + b*Csch[c + d*x])^(-1),x]

[Out]

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 3868

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(-1), x_Symbol] :> Simp[x/a, x] - Dist[1/a, Int[1/(1 + (a/b)*Sin[c
+ d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {x}{a}-\frac {\int \frac {1}{1+\frac {a \sinh (c+d x)}{b}} \, dx}{a} \\ & = \frac {x}{a}+\frac {(2 i) \text {Subst}\left (\int \frac {1}{1-\frac {2 i a x}{b}+x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a d} \\ & = \frac {x}{a}-\frac {(4 i) \text {Subst}\left (\int \frac {1}{-4 \left (1+\frac {a^2}{b^2}\right )-x^2} \, dx,x,-\frac {2 i a}{b}+2 \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a d} \\ & = \frac {x}{a}+\frac {2 b \text {arctanh}\left (\frac {b \left (\frac {a}{b}-\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.19 \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=\frac {\frac {c}{d}+x-\frac {2 b \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2} d}}{a} \]

[In]

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

[Out]

(c/d + x - (2*b*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/(Sqrt[-a^2 - b^2]*d))/a

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.52

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 b \,\operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}}{d}\)	\(82\)
default	\(\frac {\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 b \,\operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}}{d}\)	\(82\)
risch	\(\frac {x}{a}+\frac {b \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, d a}-\frac {b \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, d a}\)	\(124\)

[In]

int(1/(a+b*csch(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (51) = 102\).

Time = 0.27 (sec) , antiderivative size = 186, normalized size of antiderivative = 3.44 \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=\frac {{\left (a^{2} + b^{2}\right )} d x + \sqrt {a^{2} + b^{2}} b \log \left (\frac {a^{2} \cosh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + b\right )}}{a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) + 2 \, {\left (a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) - a}\right )}{{\left (a^{3} + a b^{2}\right )} d} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=\int \frac {1}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

Integral(1/(a + b*csch(c + d*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.57 \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=-\frac {b \log \left (\frac {a e^{\left (-d x - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d x - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a d} + \frac {d x + c}{a d} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.56 \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=-\frac {\frac {b \log \left (\frac {{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a} - \frac {d x + c}{a}}{d} \]

[In]

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

[Out]

-(b*log(abs(2*a*e^(d*x + c) + 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*e^(d*x + c) + 2*b + 2*sqrt(a^2 + b^2)))/(sqrt(a
^2 + b^2)*a) - (d*x + c)/a)/d

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.24 \[ \int \frac {1}{a+b \text {csch}(c+d x)} \, dx=\frac {x}{a}-\frac {b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{a^2}-\frac {2\,b\,\left (a-b\,{\mathrm {e}}^{c+d\,x}\right )}{a^2\,\sqrt {a^2+b^2}}\right )}{a\,d\,\sqrt {a^2+b^2}}+\frac {b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{a^2}+\frac {2\,b\,\left (a-b\,{\mathrm {e}}^{c+d\,x}\right )}{a^2\,\sqrt {a^2+b^2}}\right )}{a\,d\,\sqrt {a^2+b^2}} \]

[In]

int(1/(a + b/sinh(c + d*x)),x)

[Out]

x/a - (b*log((2*b*exp(c + d*x))/a^2 - (2*b*(a - b*exp(c + d*x)))/(a^2*(a^2 + b^2)^(1/2))))/(a*d*(a^2 + b^2)^(1
/2)) + (b*log((2*b*exp(c + d*x))/a^2 + (2*b*(a - b*exp(c + d*x)))/(a^2*(a^2 + b^2)^(1/2))))/(a*d*(a^2 + b^2)^(
1/2))