Integrand size = 16, antiderivative size = 60 \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {x^2}{2 a}+\frac {b \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d} \] Output:
1/2*x^2/a+b*arctanh((a-b*tanh(1/2*d*x^2+1/2*c))/(a^2+b^2)^(1/2))/a/(a^2+b^ 2)^(1/2)/d
Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.18 \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {\frac {c}{d}+x^2-\frac {2 b \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2} d}}{2 a} \] Input:
Integrate[x/(a + b*Csch[c + d*x^2]),x]
Output:
(c/d + x^2 - (2*b*ArcTan[(a - b*Tanh[(c + d*x^2)/2])/Sqrt[-a^2 - b^2]])/(S qrt[-a^2 - b^2]*d))/(2*a)
Time = 0.41 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5960, 3042, 4270, 3042, 3139, 1083, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx\) |
\(\Big \downarrow \) 5960 |
\(\displaystyle \frac {1}{2} \int \frac {1}{a+b \text {csch}\left (d x^2+c\right )}dx^2\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \frac {1}{a+i b \csc \left (i d x^2+i c\right )}dx^2\) |
\(\Big \downarrow \) 4270 |
\(\displaystyle \frac {1}{2} \left (\frac {x^2}{a}-\frac {\int \frac {1}{\frac {a \sinh \left (d x^2+c\right )}{b}+1}dx^2}{a}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (\frac {x^2}{a}-\frac {\int \frac {1}{1-\frac {i a \sin \left (i d x^2+i c\right )}{b}}dx^2}{a}\right )\) |
\(\Big \downarrow \) 3139 |
\(\displaystyle \frac {1}{2} \left (\frac {x^2}{a}+\frac {2 i \int \frac {1}{x^4+\frac {2 a \tanh \left (\frac {1}{2} \left (d x^2+c\right )\right )}{b}+1}d\left (i \tanh \left (\frac {1}{2} \left (d x^2+c\right )\right )\right )}{a d}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{2} \left (\frac {x^2}{a}-\frac {4 i \int \frac {1}{-x^4-4 \left (\frac {a^2}{b^2}+1\right )}d\left (2 i \tanh \left (\frac {1}{2} \left (d x^2+c\right )\right )-\frac {2 i a}{b}\right )}{a d}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (\frac {x^2}{a}-\frac {2 b \text {arctanh}\left (\frac {b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{2 \sqrt {a^2+b^2}}\right )}{a d \sqrt {a^2+b^2}}\right )\) |
Input:
Int[x/(a + b*Csch[c + d*x^2]),x]
Output:
(x^2/a - (2*b*ArcTanh[(b*Tanh[(c + d*x^2)/2])/(2*Sqrt[a^2 + b^2])])/(a*Sqr t[a^2 + b^2]*d))/2
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[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]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(-1), x_Symbol] :> Simp[x/a, x] - Simp[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]
Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo l] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csch[c + d*x] )^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.48
method | result | size |
derivativedivides | \(\frac {\frac {2 b \,\operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}-\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {\ln \left (1+\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{a}}{2 d}\) | \(89\) |
default | \(\frac {\frac {2 b \,\operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}-\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {\ln \left (1+\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{a}}{2 d}\) | \(89\) |
risch | \(\frac {x^{2}}{2 a}+\frac {b \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {b \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{2 \sqrt {a^{2}+b^{2}}\, d a}-\frac {b \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {b \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{2 \sqrt {a^{2}+b^{2}}\, d a}\) | \(132\) |
Input:
int(x/(a+b*csch(d*x^2+c)),x,method=_RETURNVERBOSE)
Output:
1/2/d*(2/a*b/(a^2+b^2)^(1/2)*arctanh(1/2*(-2*b*tanh(1/2*d*x^2+1/2*c)+2*a)/ (a^2+b^2)^(1/2))-1/a*ln(tanh(1/2*d*x^2+1/2*c)-1)+1/a*ln(1+tanh(1/2*d*x^2+1 /2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (55) = 110\).
Time = 0.09 (sec) , antiderivative size = 213, normalized size of antiderivative = 3.55 \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {{\left (a^{2} + b^{2}\right )} d x^{2} + \sqrt {a^{2} + b^{2}} b \log \left (\frac {a^{2} \cosh \left (d x^{2} + c\right )^{2} + a^{2} \sinh \left (d x^{2} + c\right )^{2} + 2 \, a b \cosh \left (d x^{2} + c\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (d x^{2} + c\right ) + a b\right )} \sinh \left (d x^{2} + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right ) + b\right )}}{a \cosh \left (d x^{2} + c\right )^{2} + a \sinh \left (d x^{2} + c\right )^{2} + 2 \, b \cosh \left (d x^{2} + c\right ) + 2 \, {\left (a \cosh \left (d x^{2} + c\right ) + b\right )} \sinh \left (d x^{2} + c\right ) - a}\right )}{2 \, {\left (a^{3} + a b^{2}\right )} d} \] Input:
integrate(x/(a+b*csch(d*x^2+c)),x, algorithm="fricas")
Output:
1/2*((a^2 + b^2)*d*x^2 + sqrt(a^2 + b^2)*b*log((a^2*cosh(d*x^2 + c)^2 + a^ 2*sinh(d*x^2 + c)^2 + 2*a*b*cosh(d*x^2 + c) + a^2 + 2*b^2 + 2*(a^2*cosh(d* x^2 + c) + a*b)*sinh(d*x^2 + c) + 2*sqrt(a^2 + b^2)*(a*cosh(d*x^2 + c) + a *sinh(d*x^2 + c) + b))/(a*cosh(d*x^2 + c)^2 + a*sinh(d*x^2 + c)^2 + 2*b*co sh(d*x^2 + c) + 2*(a*cosh(d*x^2 + c) + b)*sinh(d*x^2 + c) - a)))/((a^3 + a *b^2)*d)
\[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\int \frac {x}{a + b \operatorname {csch}{\left (c + d x^{2} \right )}}\, dx \] Input:
integrate(x/(a+b*csch(d*x**2+c)),x)
Output:
Integral(x/(a + b*csch(c + d*x**2)), x)
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.53 \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=-\frac {b \log \left (\frac {a e^{\left (-d x^{2} - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d x^{2} - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{2 \, \sqrt {a^{2} + b^{2}} a d} + \frac {d x^{2} + c}{2 \, a d} \] Input:
integrate(x/(a+b*csch(d*x^2+c)),x, algorithm="maxima")
Output:
-1/2*b*log((a*e^(-d*x^2 - c) - b - sqrt(a^2 + b^2))/(a*e^(-d*x^2 - c) - b + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a*d) + 1/2*(d*x^2 + c)/(a*d)
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.53 \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=-\frac {b \log \left (\frac {{\left | 2 \, a e^{\left (d x^{2} + c\right )} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d x^{2} + c\right )} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, \sqrt {a^{2} + b^{2}} a d} + \frac {d x^{2} + c}{2 \, a d} \] Input:
integrate(x/(a+b*csch(d*x^2+c)),x, algorithm="giac")
Output:
-1/2*b*log(abs(2*a*e^(d*x^2 + c) + 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*e^(d*x ^2 + c) + 2*b + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a*d) + 1/2*(d*x^2 + c )/(a*d)
Time = 3.11 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.92 \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {x^2}{2\,a}-\frac {\mathrm {atan}\left (\frac {a\,d\,\sqrt {b^2}}{\sqrt {-a^4\,d^2-a^2\,b^2\,d^2}}-\frac {b\,{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\sqrt {-a^4\,d^2-a^2\,b^2\,d^2}}{a^2\,d\,\sqrt {b^2}}+\frac {a^2\,b\,d\,{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\sqrt {b^2}\,\sqrt {-a^4\,d^2-a^2\,b^2\,d^2}}{a^6\,d^2+a^4\,b^2\,d^2}\right )\,\sqrt {b^2}}{\sqrt {-a^4\,d^2-a^2\,b^2\,d^2}} \] Input:
int(x/(a + b/sinh(c + d*x^2)),x)
Output:
x^2/(2*a) - (atan((a*d*(b^2)^(1/2))/(- a^4*d^2 - a^2*b^2*d^2)^(1/2) - (b*e xp(d*x^2)*exp(c)*(- a^4*d^2 - a^2*b^2*d^2)^(1/2))/(a^2*d*(b^2)^(1/2)) + (a ^2*b*d*exp(d*x^2)*exp(c)*(b^2)^(1/2)*(- a^4*d^2 - a^2*b^2*d^2)^(1/2))/(a^6 *d^2 + a^4*b^2*d^2))*(b^2)^(1/2))/(- a^4*d^2 - a^2*b^2*d^2)^(1/2)
Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.23 \[ \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx=\frac {-2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d \,x^{2}+c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) b i +a^{2} d \,x^{2}+b^{2} d \,x^{2}}{2 a d \left (a^{2}+b^{2}\right )} \] Input:
int(x/(a+b*csch(d*x^2+c)),x)
Output:
( - 2*sqrt(a**2 + b**2)*atan((e**(c + d*x**2)*a*i + b*i)/sqrt(a**2 + b**2) )*b*i + a**2*d*x**2 + b**2*d*x**2)/(2*a*d*(a**2 + b**2))