Integrand size = 21, antiderivative size = 47 \[ \int \frac {\sinh (c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{a^{3/2} d}+\frac {\cosh (c+d x)}{a d} \]
Result contains complex when optimal does not.
Time = 0.99 (sec) , antiderivative size = 328, normalized size of antiderivative = 6.98 \[ \int \frac {\sinh (c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {\left (-\frac {(a+4 b) \left (\arctan \left (\frac {\left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )+\arctan \left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )\right )}{\sqrt {b}}+\frac {a \left (\arctan \left (\frac {\sqrt {a}-i \sqrt {a+b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+\arctan \left (\frac {\sqrt {a}+i \sqrt {a+b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )\right )}{\sqrt {b}}+4 \sqrt {a} \cosh (c+d x)\right ) (a+2 b+a \cosh (2 (c+d x))) \text {sech}^2(c+d x)}{8 a^{3/2} d \left (a+b \text {sech}^2(c+d x)\right )} \]
((-(((a + 4*b)*(ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^ 2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + ArcTan[((Sqrt[a] + I*Sqrt[a + b] *Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I *Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]))/Sqrt[b ]) + (a*(ArcTan[(Sqrt[a] - I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]] + Arc Tan[(Sqrt[a] + I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]]))/Sqrt[b] + 4*Sqr t[a]*Cosh[c + d*x])*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2)/(8*a^ (3/2)*d*(a + b*Sech[c + d*x]^2))
Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 26, 4621, 262, 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.
| \(\displaystyle \int \frac {\sinh (c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i c+i d x)}{a+b \sec (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\sin (i c+i d x)}{b \sec (i c+i d x)^2+a}dx\) |
| \(\Big \downarrow \) 4621 |
| \(\displaystyle \frac {\int \frac {\cosh ^2(c+d x)}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\frac {\cosh (c+d x)}{a}-\frac {b \int \frac {1}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{a}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\cosh (c+d x)}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{a^{3/2}}}{d}\) |
3.1.28.3.1 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[((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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/ 2] && IntegerQ[n] && IntegerQ[p]
Time = 1.80 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(\frac {1}{d a \,\operatorname {sech}\left (d x +c \right )}+\frac {b \arctan \left (\frac {b \,\operatorname {sech}\left (d x +c \right )}{\sqrt {a b}}\right )}{d a \sqrt {a b}}\) | \(44\) |
| default | \(\frac {1}{d a \,\operatorname {sech}\left (d x +c \right )}+\frac {b \arctan \left (\frac {b \,\operatorname {sech}\left (d x +c \right )}{\sqrt {a b}}\right )}{d a \sqrt {a b}}\) | \(44\) |
| risch | \(\frac {{\mathrm e}^{d x +c}}{2 a d}+\frac {{\mathrm e}^{-d x -c}}{2 a d}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{2 a^{2} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{2 a^{2} d}\) | \(119\) |
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (39) = 78\).
Time = 0.27 (sec) , antiderivative size = 595, normalized size of antiderivative = 12.66 \[ \int \frac {\sinh (c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\left [\frac {\sqrt {-\frac {b}{a}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} \log \left (\frac {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 ) - 4 \, {\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3} + a \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} + a\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {b}{a}} + a}{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 ) + \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}{2 \, {\left (a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right )\right )}}, \frac {2 \, \sqrt {\frac {b}{a}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3} + {\left (a + 4 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} + a + 4 \, b\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{a}}}{2 \, b}\right ) - 2 \, \sqrt {\frac {b}{a}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{a}}}{2 \, b}\right ) + \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}{2 \, {\left (a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right )\right )}}\right ] \]
[1/2*(sqrt(-b/a)*(cosh(d*x + c) + sinh(d*x + c))*log((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) - 4*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + a*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + a)*sinh(d*x + c))*sqrt(-b/a) + a)/(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)) + cosh(d* x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)/(a*d*cosh( d*x + c) + a*d*sinh(d*x + c)), 1/2*(2*sqrt(b/a)*(cosh(d*x + c) + sinh(d*x + c))*arctan(1/2*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + (a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + a + 4 *b)*sinh(d*x + c))*sqrt(b/a)/b) - 2*sqrt(b/a)*(cosh(d*x + c) + sinh(d*x + c))*arctan(1/2*(a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt(b/a)/b) + cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)/(a*d*cosh(d *x + c) + a*d*sinh(d*x + c))]
\[ \int \frac {\sinh (c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int \frac {\sinh {\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\sinh (c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )}{b \operatorname {sech}\left (d x + c\right )^{2} + a} \,d x } \]
1/2*(e^(2*d*x + 2*c) + 1)*e^(-d*x - c)/(a*d) - 1/2*integrate(4*(b*e^(3*d*x + 3*c) - b*e^(d*x + c))/(a^2*e^(4*d*x + 4*c) + a^2 + 2*(a^2*e^(2*c) + 2*a *b*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\sinh (c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )}{b \operatorname {sech}\left (d x + c\right )^{2} + a} \,d x } \]
Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.89 \[ \int \frac {\sinh (c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {\mathrm {cosh}\left (c+d\,x\right )}{a\,d}-\frac {b\,\mathrm {atan}\left (\frac {a\,\mathrm {cosh}\left (c+d\,x\right )}{\sqrt {a\,b}}\right )}{a\,d\,\sqrt {a\,b}} \]