Integrand size = 15, antiderivative size = 35 \[ \int \text {sech}^2(c+b x) \sinh (a+b x) \, dx=-\frac {\cosh (a-c) \text {sech}(c+b x)}{b}+\frac {\arctan (\sinh (c+b x)) \sinh (a-c)}{b} \] Output:
-cosh(a-c)*sech(b*x+c)/b+arctan(sinh(b*x+c))*sinh(a-c)/b
Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(35)=70\).
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.37 \[ \int \text {sech}^2(c+b x) \sinh (a+b x) \, dx=-\frac {\cosh (a-c) \text {sech}(c+b x)}{b}+\frac {2 \arctan \left (\frac {(\cosh (c)-\sinh (c)) \left (\cosh \left (\frac {b x}{2}\right ) \sinh (c)+\cosh (c) \sinh \left (\frac {b x}{2}\right )\right )}{\cosh (c) \cosh \left (\frac {b x}{2}\right )-\cosh \left (\frac {b x}{2}\right ) \sinh (c)}\right ) \sinh (a-c)}{b} \] Input:
Integrate[Sech[c + b*x]^2*Sinh[a + b*x],x]
Output:
-((Cosh[a - c]*Sech[c + b*x])/b) + (2*ArcTan[((Cosh[c] - Sinh[c])*(Cosh[(b *x)/2]*Sinh[c] + Cosh[c]*Sinh[(b*x)/2]))/(Cosh[c]*Cosh[(b*x)/2] - Cosh[(b* x)/2]*Sinh[c])]*Sinh[a - c])/b
Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6158, 3042, 26, 3086, 24, 4257}
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.
\(\displaystyle \int \sinh (a+b x) \text {sech}^2(b x+c) \, dx\) |
\(\Big \downarrow \) 6158 |
\(\displaystyle \sinh (a-c) \int \text {sech}(c+b x)dx+\cosh (a-c) \int \text {sech}(c+b x) \tanh (c+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sinh (a-c) \int \csc \left (i c+i b x+\frac {\pi }{2}\right )dx+\cosh (a-c) \int -i \sec (i c+i b x) \tan (i c+i b x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \sinh (a-c) \int \csc \left (i c+i b x+\frac {\pi }{2}\right )dx-i \cosh (a-c) \int \sec (i c+i b x) \tan (i c+i b x)dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {\cosh (a-c) \int 1d\text {sech}(c+b x)}{b}+\sinh (a-c) \int \csc \left (i c+i b x+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\cosh (a-c) \text {sech}(b x+c)}{b}+\sinh (a-c) \int \csc \left (i c+i b x+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {\sinh (a-c) \arctan (\sinh (b x+c))}{b}-\frac {\cosh (a-c) \text {sech}(b x+c)}{b}\) |
Input:
Int[Sech[c + b*x]^2*Sinh[a + b*x],x]
Output:
-((Cosh[a - c]*Sech[c + b*x])/b) + (ArcTan[Sinh[c + b*x]]*Sinh[a - c])/b
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_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[Sech[w_]^(n_.)*Sinh[v_], x_Symbol] :> Simp[Cosh[v - w] Int[Tanh[w]*Se ch[w]^(n - 1), x], x] + Simp[Sinh[v - w] Int[Sech[w]^(n - 1), x], x] /; G tQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
Result contains complex when optimal does not.
Time = 1.05 (sec) , antiderivative size = 181, normalized size of antiderivative = 5.17
method | result | size |
risch | \(-\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 a}+{\mathrm e}^{2 c}\right )}{b \left ({\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )}+\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}+\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b}\) | \(181\) |
Input:
int(sech(b*x+c)^2*sinh(b*x+a),x,method=_RETURNVERBOSE)
Output:
-1/b*exp(b*x+a)*(exp(2*a)+exp(2*c))/(exp(2*b*x+2*a+2*c)+exp(2*a))+1/2*I*ln (exp(b*x+a)+I*exp(a-c))/b*exp(-a-c)*exp(a)^2-1/2*I*ln(exp(b*x+a)+I*exp(a-c ))/b*exp(-a-c)*exp(c)^2-1/2*I*ln(exp(b*x+a)-I*exp(a-c))/b*exp(-a-c)*exp(a) ^2+1/2*I*ln(exp(b*x+a)-I*exp(a-c))/b*exp(-a-c)*exp(c)^2
Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (35) = 70\).
Time = 0.11 (sec) , antiderivative size = 405, normalized size of antiderivative = 11.57 \[ \int \text {sech}^2(c+b x) \sinh (a+b x) \, dx=\frac {2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} + {\left ({\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right )^{2} + {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} - 1\right )} \sinh \left (b x + c\right )^{2} + {\left (\cosh \left (b x + c\right )^{2} + 1\right )} \sinh \left (-a + c\right )^{2} + \cosh \left (-a + c\right )^{2} - 2 \, {\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2} - {\left (\cosh \left (-a + c\right )^{2} - 1\right )} \cosh \left (b x + c\right )\right )} \sinh \left (b x + c\right ) - 2 \, {\left (\cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) + \cosh \left (-a + c\right )\right )} \sinh \left (-a + c\right ) - 1\right )} \arctan \left (\cosh \left (b x + c\right ) + \sinh \left (b x + c\right )\right ) - {\left (\cosh \left (-a + c\right )^{2} + 1\right )} \cosh \left (b x + c\right ) - {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} + 1\right )} \sinh \left (b x + c\right )}{b \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right ) + {\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right )^{2} + b \cosh \left (-a + c\right ) + 2 \, {\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right ) - {\left (b \cosh \left (b x + c\right )^{2} + b\right )} \sinh \left (-a + c\right )} \] Input:
integrate(sech(b*x+c)^2*sinh(b*x+a),x, algorithm="fricas")
Output:
(2*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 + ((cosh(-a + c)^2 - 1)*cosh(b*x + c)^2 + (cosh(-a + c)^2 - 2*cosh(-a + c) *sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c)^2 + (cosh(b*x + c)^2 + 1 )*sinh(-a + c)^2 + cosh(-a + c)^2 - 2*(2*cosh(b*x + c)*cosh(-a + c)*sinh(- a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 - 1)*cosh(b*x + c) )*sinh(b*x + c) - 2*(cosh(b*x + c)^2*cosh(-a + c) + cosh(-a + c))*sinh(-a + c) - 1)*arctan(cosh(b*x + c) + sinh(b*x + c)) - (cosh(-a + c)^2 + 1)*cos h(b*x + c) - (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^ 2 + 1)*sinh(b*x + c))/(b*cosh(b*x + c)^2*cosh(-a + c) + (b*cosh(-a + c) - b*sinh(-a + c))*sinh(b*x + c)^2 + b*cosh(-a + c) + 2*(b*cosh(b*x + c)*cosh (-a + c) - b*cosh(b*x + c)*sinh(-a + c))*sinh(b*x + c) - (b*cosh(b*x + c)^ 2 + b)*sinh(-a + c))
\[ \int \text {sech}^2(c+b x) \sinh (a+b x) \, dx=\int \sinh {\left (a + b x \right )} \operatorname {sech}^{2}{\left (b x + c \right )}\, dx \] Input:
integrate(sech(b*x+c)**2*sinh(b*x+a),x)
Output:
Integral(sinh(a + b*x)*sech(b*x + c)**2, x)
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.00 \[ \int \text {sech}^2(c+b x) \sinh (a+b x) \, dx=-\frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (-b x - c\right )}\right ) e^{\left (-a - c\right )}}{b} - \frac {{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-b x - a\right )}}{b {\left (e^{\left (-2 \, b x\right )} + e^{\left (2 \, c\right )}\right )}} \] Input:
integrate(sech(b*x+c)^2*sinh(b*x+a),x, algorithm="maxima")
Output:
-(e^(2*a) - e^(2*c))*arctan(e^(-b*x - c))*e^(-a - c)/b - (e^(2*a) + e^(2*c ))*e^(-b*x - a)/(b*(e^(-2*b*x) + e^(2*c)))
Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.00 \[ \int \text {sech}^2(c+b x) \sinh (a+b x) \, dx=\frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (b x + c\right )}\right ) e^{\left (-a - c\right )}}{b} - \frac {{\left (e^{\left (b x + 2 \, a\right )} + e^{\left (b x + 2 \, c\right )}\right )} e^{\left (-a\right )}}{b {\left (e^{\left (2 \, b x + 2 \, c\right )} + 1\right )}} \] Input:
integrate(sech(b*x+c)^2*sinh(b*x+a),x, algorithm="giac")
Output:
(e^(2*a) - e^(2*c))*arctan(e^(b*x + c))*e^(-a - c)/b - (e^(b*x + 2*a) + e^ (b*x + 2*c))*e^(-a)/(b*(e^(2*b*x + 2*c) + 1))
Time = 1.43 (sec) , antiderivative size = 150, normalized size of antiderivative = 4.29 \[ \int \text {sech}^2(c+b x) \sinh (a+b x) \, dx=-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{b\,x}\,\left (\sqrt {b^2}-{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\,\sqrt {b^2}\right )}{b\,\sqrt {{\mathrm {e}}^{-2\,a}\,{\mathrm {e}}^{2\,c}\,\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,c}-2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}+1\right )}}\right )\,\sqrt {{\mathrm {e}}^{2\,c-2\,a}\,\left ({\mathrm {e}}^{4\,a-4\,c}-2\,{\mathrm {e}}^{2\,a-2\,c}+1\right )}}{\sqrt {b^2}}-\frac {{\mathrm {e}}^{a+b\,x}\,\left ({\mathrm {e}}^{2\,a-2\,c}+1\right )}{b\,\left ({\mathrm {e}}^{2\,a-2\,c}+{\mathrm {e}}^{2\,a+2\,b\,x}\right )} \] Input:
int(sinh(a + b*x)/cosh(c + b*x)^2,x)
Output:
- (atan((exp(-a)*exp(2*c)*exp(b*x)*((b^2)^(1/2) - exp(2*a)*exp(-2*c)*(b^2) ^(1/2)))/(b*(exp(-2*a)*exp(2*c)*(exp(4*a)*exp(-4*c) - 2*exp(2*a)*exp(-2*c) + 1))^(1/2)))*(exp(2*c - 2*a)*(exp(4*a - 4*c) - 2*exp(2*a - 2*c) + 1))^(1 /2))/(b^2)^(1/2) - (exp(a + b*x)*(exp(2*a - 2*c) + 1))/(b*(exp(2*a - 2*c) + exp(2*a + 2*b*x)))
Time = 0.24 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.43 \[ \int \text {sech}^2(c+b x) \sinh (a+b x) \, dx=\frac {e^{2 b x +2 a +2 c} \mathit {atan} \left (e^{b x +c}\right )-e^{2 b x +4 c} \mathit {atan} \left (e^{b x +c}\right )+e^{2 a} \mathit {atan} \left (e^{b x +c}\right )-e^{2 c} \mathit {atan} \left (e^{b x +c}\right )-e^{b x +2 a +c}-e^{b x +3 c}}{e^{a +c} b \left (e^{2 b x +2 c}+1\right )} \] Input:
int(sech(b*x+c)^2*sinh(b*x+a),x)
Output:
(e**(2*a + 2*b*x + 2*c)*atan(e**(b*x + c)) - e**(2*b*x + 4*c)*atan(e**(b*x + c)) + e**(2*a)*atan(e**(b*x + c)) - e**(2*c)*atan(e**(b*x + c)) - e**(2 *a + b*x + c) - e**(b*x + 3*c))/(e**(a + c)*b*(e**(2*b*x + 2*c) + 1))