Integrand size = 14, antiderivative size = 50 \[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\frac {x}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{a \sqrt {a-b} d} \]
Time = 2.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.04 \[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\frac {\frac {c}{d}+x-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a-b} d}}{a} \]
Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3042, 4615, 3042, 3660, 218}
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.
\(\displaystyle \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{a-b \sec \left (i c+i d x+\frac {\pi }{2}\right )^2}dx\) |
\(\Big \downarrow \) 4615 |
\(\displaystyle \frac {b \int \frac {1}{-a \sinh ^2(c+d x)-b}dx}{a}+\frac {x}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x}{a}+\frac {b \int \frac {1}{a \sin (i c+i d x)^2-b}dx}{a}\) |
\(\Big \downarrow \) 3660 |
\(\displaystyle \frac {b \int \frac {1}{-\left ((a-b) \tanh ^2(c+d x)\right )-b}d\tanh (c+d x)}{a d}+\frac {x}{a}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {x}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{a d \sqrt {a-b}}\) |
3.1.5.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[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]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> Simp[x/a, x ] - Simp[b/a Int[1/(b + a*Cos[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(123\) vs. \(2(42)=84\).
Time = 0.42 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.48
method | result | size |
risch | \(\frac {x}{a}+\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a +2 \sqrt {-b \left (a -b \right )}-2 b}{a}\right )}{2 \left (a -b \right ) d a}-\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {-a +2 \sqrt {-b \left (a -b \right )}+2 b}{a}\right )}{2 \left (a -b \right ) d a}\) | \(124\) |
derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 b^{2} \left (\frac {\left (\sqrt {a \left (a -b \right )}+a \right ) \arctan \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 b \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}-\frac {\left (\sqrt {a \left (a -b \right )}-a \right ) \operatorname {arctanh}\left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 b \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{a}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(208\) |
default | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 b^{2} \left (\frac {\left (\sqrt {a \left (a -b \right )}+a \right ) \arctan \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 b \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}-\frac {\left (\sqrt {a \left (a -b \right )}-a \right ) \operatorname {arctanh}\left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 b \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{a}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(208\) |
x/a+1/2*(-b*(a-b))^(1/2)/(a-b)/d/a*ln(exp(2*d*x+2*c)-(a+2*(-b*(a-b))^(1/2) -2*b)/a)-1/2*(-b*(a-b))^(1/2)/(a-b)/d/a*ln(exp(2*d*x+2*c)+(-a+2*(-b*(a-b)) ^(1/2)+2*b)/a)
Time = 0.28 (sec) , antiderivative size = 457, normalized size of antiderivative = 9.14 \[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\left [\frac {2 \, d x + \sqrt {-\frac {b}{a - b}} \log \left (\frac {a^{2} \cosh \left (d x + c\right )^{4} + 4 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{2} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} - 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} - a^{2} + 2 \, a b\right )} \sinh \left (d x + c\right )^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} - {\left (a^{2} - 2 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left ({\left (a^{2} - a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} - a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} - a b\right )} \sinh \left (d x + c\right )^{2} - a^{2} + 3 \, a b - 2 \, b^{2}\right )} \sqrt {-\frac {b}{a - b}}}{a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, {\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} - {\left (a - 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a}\right )}{2 \, a d}, \frac {d x - \sqrt {\frac {b}{a - b}} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} - a + 2 \, b\right )} \sqrt {\frac {b}{a - b}}}{2 \, b}\right )}{a d}\right ] \]
[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)*sin h(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)]
\[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\int \frac {1}{a + b \operatorname {csch}^{2}{\left (c + d x \right )}}\, dx \]
Exception generated. \[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for more details)Is
\[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\int { \frac {1}{b \operatorname {csch}\left (d x + c\right )^{2} + a} \,d x } \]
Time = 2.79 (sec) , antiderivative size = 471, normalized size of antiderivative = 9.42 \[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\frac {x}{a}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\left (a^5\,\sqrt {a^3\,d^2-a^2\,b\,d^2}-a^4\,b\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\right )\,\left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (\frac {2\,\left (a^2-8\,a\,b+8\,b^2\right )\,\left (8\,b^{5/2}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}-8\,a\,b^{3/2}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}+a^2\,\sqrt {b}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\right )}{a^8\,d\,{\left (a-b\right )}^2\,\sqrt {a^3\,d^2-a^2\,b\,d^2}}+\frac {4\,\sqrt {b}\,\left (2\,a-4\,b\right )\,\left (4\,d\,a^3\,b-12\,d\,a^2\,b^2+8\,d\,a\,b^3\right )}{a^7\,\left (a-b\right )\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\,\sqrt {a^2\,d^2\,\left (a-b\right )}}\right )+\frac {2\,\left (2\,a\,b^{3/2}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}-a^2\,\sqrt {b}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\right )\,\left (a^2-8\,a\,b+8\,b^2\right )}{a^8\,d\,{\left (a-b\right )}^2\,\sqrt {a^3\,d^2-a^2\,b\,d^2}}+\frac {4\,\sqrt {b}\,\left (2\,a^2\,b^2\,d-2\,a^3\,b\,d\right )\,\left (2\,a-4\,b\right )}{a^7\,\left (a-b\right )\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\,\sqrt {a^2\,d^2\,\left (a-b\right )}}\right )}{4\,b}\right )}{\sqrt {a^3\,d^2-a^2\,b\,d^2}} \]
x/a - (b^(1/2)*atan(((a^5*(a^3*d^2 - a^2*b*d^2)^(1/2) - a^4*b*(a^3*d^2 - a ^2*b*d^2)^(1/2))*(exp(2*c)*exp(2*d*x)*((2*(a^2 - 8*a*b + 8*b^2)*(8*b^(5/2) *(a^3*d^2 - a^2*b*d^2)^(1/2) - 8*a*b^(3/2)*(a^3*d^2 - a^2*b*d^2)^(1/2) + a ^2*b^(1/2)*(a^3*d^2 - a^2*b*d^2)^(1/2)))/(a^8*d*(a - b)^2*(a^3*d^2 - a^2*b *d^2)^(1/2)) + (4*b^(1/2)*(2*a - 4*b)*(8*a*b^3*d - 12*a^2*b^2*d + 4*a^3*b* d))/(a^7*(a - b)*(a^3*d^2 - a^2*b*d^2)^(1/2)*(a^2*d^2*(a - b))^(1/2))) + ( 2*(2*a*b^(3/2)*(a^3*d^2 - a^2*b*d^2)^(1/2) - a^2*b^(1/2)*(a^3*d^2 - a^2*b* d^2)^(1/2))*(a^2 - 8*a*b + 8*b^2))/(a^8*d*(a - b)^2*(a^3*d^2 - a^2*b*d^2)^ (1/2)) + (4*b^(1/2)*(2*a^2*b^2*d - 2*a^3*b*d)*(2*a - 4*b))/(a^7*(a - b)*(a ^3*d^2 - a^2*b*d^2)^(1/2)*(a^2*d^2*(a - b))^(1/2))))/(4*b)))/(a^3*d^2 - a^ 2*b*d^2)^(1/2)