Integrand size = 27, antiderivative size = 113 \[ \int \frac {\text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\text {arctanh}(\cosh (c+d x))}{a d}+\frac {2 b^3 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {\text {sech}(c+d x)}{a d}-\frac {b \text {sech}(c+d x) (b+a \sinh (c+d x))}{a \left (a^2+b^2\right ) d} \]
-arctanh(cosh(d*x+c))/a/d+2*b^3*arctanh((b-a*tanh(1/2*d*x+1/2*c))/(a^2+b^2 )^(1/2))/a/(a^2+b^2)^(3/2)/d+sech(d*x+c)/a/d-b*sech(d*x+c)*(b+a*sinh(d*x+c ))/a/(a^2+b^2)/d
Leaf count is larger than twice the leaf count of optimal. \(233\) vs. \(2(113)=226\).
Time = 1.11 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.06 \[ \int \frac {\text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {-2 b^3 \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )-a^2 \sqrt {-a^2-b^2} \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-b^2 \sqrt {-a^2-b^2} \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+a^2 \sqrt {-a^2-b^2} \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+b^2 \sqrt {-a^2-b^2} \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+a^2 \sqrt {-a^2-b^2} \text {sech}(c+d x)-a b \sqrt {-a^2-b^2} \tanh (c+d x)}{a \left (-a^2-b^2\right )^{3/2} d} \]
-((-2*b^3*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]] - a^2*Sqrt[-a ^2 - b^2]*Log[Cosh[(c + d*x)/2]] - b^2*Sqrt[-a^2 - b^2]*Log[Cosh[(c + d*x) /2]] + a^2*Sqrt[-a^2 - b^2]*Log[Sinh[(c + d*x)/2]] + b^2*Sqrt[-a^2 - b^2]* Log[Sinh[(c + d*x)/2]] + a^2*Sqrt[-a^2 - b^2]*Sech[c + d*x] - a*b*Sqrt[-a^ 2 - b^2]*Tanh[c + d*x])/(a*(-a^2 - b^2)^(3/2)*d))
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3042, 26, 3377, 2009}
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.
| \(\displaystyle \int \frac {\text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\sin (i c+i d x) \cos (i c+i d x)^2 (a-i b \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\cos (i c+i d x)^2 \sin (i c+i d x) (a-i b \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 3377 |
| \(\displaystyle i \int \left (\frac {i b \text {sech}^2(c+d x)}{a (a+b \sinh (c+d x))}-\frac {i \text {csch}(c+d x) \text {sech}^2(c+d x)}{a}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (-\frac {2 i b^3 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a d \left (a^2+b^2\right )^{3/2}}+\frac {i b \text {sech}(c+d x) (a \sinh (c+d x)+b)}{a d \left (a^2+b^2\right )}+\frac {i \text {arctanh}(\cosh (c+d x))}{a d}-\frac {i \text {sech}(c+d x)}{a d}\right )\) |
I*((I*ArcTanh[Cosh[c + d*x]])/(a*d) - ((2*I)*b^3*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a*(a^2 + b^2)^(3/2)*d) - (I*Sech[c + d*x])/(a* d) + (I*b*Sech[c + d*x]*(b + a*Sinh[c + d*x]))/(a*(a^2 + b^2)*d))
3.5.43.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*sin[(e_.) + (f_.)*(x_)]^(n_))/((a _) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, sin[e + f*x]^n/(a + b*sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[n, 0] || IGtQ[p + 1/ 2, 0])
Time = 4.80 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 b^{3} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {2 \left (b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}}{d}\) | \(106\) |
| default | \(\frac {-\frac {2 b^{3} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {2 \left (b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}}{d}\) | \(106\) |
| risch | \(\frac {2 a \,{\mathrm e}^{d x +c}+2 b}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}+\frac {b^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d a}-\frac {b^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d a}\) | \(209\) |
1/d*(-2/a*b^3/(a^2+b^2)^(3/2)*arctanh(1/2*(2*a*tanh(1/2*d*x+1/2*c)-2*b)/(a ^2+b^2)^(1/2))+1/a*ln(tanh(1/2*d*x+1/2*c))-2/(a^2+b^2)*(b*tanh(1/2*d*x+1/2 *c)-a)/(1+tanh(1/2*d*x+1/2*c)^2))
Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (110) = 220\).
Time = 0.33 (sec) , antiderivative size = 581, normalized size of antiderivative = 5.14 \[ \int \frac {\text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 \, a^{3} b + 2 \, a b^{3} + {\left (b^{3} \cosh \left (d x + c\right )^{2} + 2 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{3} \sinh \left (d x + c\right )^{2} + b^{3}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right ) + 2 \, {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (d x + c\right ) - {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (d x + c\right )^{2}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (d x + c\right )^{2}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \, {\left (a^{4} + a^{2} b^{2}\right )} \sinh \left (d x + c\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \sinh \left (d x + c\right )^{2} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d} \]
(2*a^3*b + 2*a*b^3 + (b^3*cosh(d*x + c)^2 + 2*b^3*cosh(d*x + c)*sinh(d*x + c) + b^3*sinh(d*x + c)^2 + b^3)*sqrt(a^2 + b^2)*log((b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c) + 2*a^2 + b^2 + 2*(b^2*cosh(d* x + c) + a*b)*sinh(d*x + c) + 2*sqrt(a^2 + b^2)*(b*cosh(d*x + c) + b*sinh( d*x + c) + a))/(b*cosh(d*x + c)^2 + b*sinh(d*x + c)^2 + 2*a*cosh(d*x + c) + 2*(b*cosh(d*x + c) + a)*sinh(d*x + c) - b)) + 2*(a^4 + a^2*b^2)*cosh(d*x + c) - (a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*cosh(d*x + c)*sinh(d*x + c) + (a^4 + 2*a^2*b^2 + b^4)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + (a^4 + 2* a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b ^2 + b^4)*cosh(d*x + c)*sinh(d*x + c) + (a^4 + 2*a^2*b^2 + b^4)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(a^4 + a^2*b^2)*sinh(d*x + c))/((a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(d*x + c)^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(d*x + c)*sinh(d*x + c) + (a^5 + 2*a^3*b^2 + a*b^4)*d*sinh(d *x + c)^2 + (a^5 + 2*a^3*b^2 + a*b^4)*d)
\[ \int \frac {\text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\operatorname {csch}{\left (c + d x \right )} \operatorname {sech}^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.49 \[ \int \frac {\text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b^{3} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{3} + a b^{2}\right )} \sqrt {a^{2} + b^{2}} d} + \frac {2 \, {\left (a e^{\left (-d x - c\right )} - b\right )}}{{\left (a^{2} + b^{2} + {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} - \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} \]
-b^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt (a^2 + b^2)))/((a^3 + a*b^2)*sqrt(a^2 + b^2)*d) + 2*(a*e^(-d*x - c) - b)/( (a^2 + b^2 + (a^2 + b^2)*e^(-2*d*x - 2*c))*d) - log(e^(-d*x - c) + 1)/(a*d ) + log(e^(-d*x - c) - 1)/(a*d)
Time = 0.31 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.29 \[ \int \frac {\text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {b^{3} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{3} + a b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {\log \left (e^{\left (d x + c\right )} + 1\right )}{a} - \frac {\log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a} - \frac {2 \, {\left (a e^{\left (d x + c\right )} + b\right )}}{{\left (a^{2} + b^{2}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}}}{d} \]
-(b^3*log(abs(2*b*e^(d*x + c) + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^(d*x + c) + 2*a + 2*sqrt(a^2 + b^2)))/((a^3 + a*b^2)*sqrt(a^2 + b^2)) + log(e^(d* x + c) + 1)/a - log(abs(e^(d*x + c) - 1))/a - 2*(a*e^(d*x + c) + b)/((a^2 + b^2)*(e^(2*d*x + 2*c) + 1)))/d
Time = 6.87 (sec) , antiderivative size = 668, normalized size of antiderivative = 5.91 \[ \int \frac {\text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {2\,b}{d\,\left (a^2+b^2\right )}+\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2+b^2\right )}}{{\mathrm {e}}^{2\,c+2\,d\,x}+1}+\frac {\ln \left ({\mathrm {e}}^{c+d\,x}-1\right )}{a\,d}-\frac {\ln \left ({\mathrm {e}}^{c+d\,x}+1\right )}{a\,d}-\frac {b^3\,\ln \left (\frac {32\,\left (-4\,{\mathrm {e}}^{c+d\,x}\,a^3+2\,a^2\,b-5\,{\mathrm {e}}^{c+d\,x}\,a\,b^2+2\,b^3\right )}{b^2\,{\left (a^2+b^2\right )}^2}-\frac {128\,a^{10}\,{\mathrm {e}}^{c+d\,x}-64\,a^9\,b-96\,a\,b^9+64\,b^7\,\sqrt {{\left (a^2+b^2\right )}^3}-384\,a^3\,b^7-512\,a^5\,b^5-288\,a^7\,b^3+288\,a^2\,b^8\,{\mathrm {e}}^{c+d\,x}+960\,a^4\,b^6\,{\mathrm {e}}^{c+d\,x}+1152\,a^6\,b^4\,{\mathrm {e}}^{c+d\,x}+608\,a^8\,b^2\,{\mathrm {e}}^{c+d\,x}-64\,a\,b^6\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}+32\,a^3\,b^4\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}}{b^2\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}\,\left (a^2+b^2\right )}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}}{d\,a^7+3\,d\,a^5\,b^2+3\,d\,a^3\,b^4+d\,a\,b^6}+\frac {b^3\,\ln \left (\frac {32\,\left (-4\,{\mathrm {e}}^{c+d\,x}\,a^3+2\,a^2\,b-5\,{\mathrm {e}}^{c+d\,x}\,a\,b^2+2\,b^3\right )}{b^2\,{\left (a^2+b^2\right )}^2}-\frac {96\,a\,b^9+64\,a^9\,b-128\,a^{10}\,{\mathrm {e}}^{c+d\,x}+64\,b^7\,\sqrt {{\left (a^2+b^2\right )}^3}+384\,a^3\,b^7+512\,a^5\,b^5+288\,a^7\,b^3-288\,a^2\,b^8\,{\mathrm {e}}^{c+d\,x}-960\,a^4\,b^6\,{\mathrm {e}}^{c+d\,x}-1152\,a^6\,b^4\,{\mathrm {e}}^{c+d\,x}-608\,a^8\,b^2\,{\mathrm {e}}^{c+d\,x}-64\,a\,b^6\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}+32\,a^3\,b^4\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}}{b^2\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}\,\left (a^2+b^2\right )}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}}{d\,a^7+3\,d\,a^5\,b^2+3\,d\,a^3\,b^4+d\,a\,b^6} \]
((2*b)/(d*(a^2 + b^2)) + (2*a*exp(c + d*x))/(d*(a^2 + b^2)))/(exp(2*c + 2* d*x) + 1) + log(exp(c + d*x) - 1)/(a*d) - log(exp(c + d*x) + 1)/(a*d) - (b ^3*log((32*(2*a^2*b - 4*a^3*exp(c + d*x) + 2*b^3 - 5*a*b^2*exp(c + d*x)))/ (b^2*(a^2 + b^2)^2) - (128*a^10*exp(c + d*x) - 64*a^9*b - 96*a*b^9 + 64*b^ 7*((a^2 + b^2)^3)^(1/2) - 384*a^3*b^7 - 512*a^5*b^5 - 288*a^7*b^3 + 288*a^ 2*b^8*exp(c + d*x) + 960*a^4*b^6*exp(c + d*x) + 1152*a^6*b^4*exp(c + d*x) + 608*a^8*b^2*exp(c + d*x) - 64*a*b^6*exp(c + d*x)*((a^2 + b^2)^3)^(1/2) + 32*a^3*b^4*exp(c + d*x)*((a^2 + b^2)^3)^(1/2))/(b^2*((a^2 + b^2)^3)^(3/2) *(a^2 + b^2)))*((a^2 + b^2)^3)^(1/2))/(a^7*d + 3*a^3*b^4*d + 3*a^5*b^2*d + a*b^6*d) + (b^3*log((32*(2*a^2*b - 4*a^3*exp(c + d*x) + 2*b^3 - 5*a*b^2*e xp(c + d*x)))/(b^2*(a^2 + b^2)^2) - (96*a*b^9 + 64*a^9*b - 128*a^10*exp(c + d*x) + 64*b^7*((a^2 + b^2)^3)^(1/2) + 384*a^3*b^7 + 512*a^5*b^5 + 288*a^ 7*b^3 - 288*a^2*b^8*exp(c + d*x) - 960*a^4*b^6*exp(c + d*x) - 1152*a^6*b^4 *exp(c + d*x) - 608*a^8*b^2*exp(c + d*x) - 64*a*b^6*exp(c + d*x)*((a^2 + b ^2)^3)^(1/2) + 32*a^3*b^4*exp(c + d*x)*((a^2 + b^2)^3)^(1/2))/(b^2*((a^2 + b^2)^3)^(3/2)*(a^2 + b^2)))*((a^2 + b^2)^3)^(1/2))/(a^7*d + 3*a^3*b^4*d + 3*a^5*b^2*d + a*b^6*d)