Integrand size = 29, antiderivative size = 78 \[ \int \frac {e^{2 (a+b x)}}{g \cosh (d+b x)+f \sinh (d+b x)} \, dx=\frac {2 e^{2 a-d+b x}}{b (f+g)}-\frac {2 e^{2 a-2 d} \sqrt {f-g} \text {arctanh}\left (\frac {e^{d+b x} \sqrt {f+g}}{\sqrt {f-g}}\right )}{b (f+g)^{3/2}} \] Output:
2*exp(b*x+2*a-d)/b/(f+g)-2*exp(2*a-2*d)*(f-g)^(1/2)*arctanh(exp(b*x+d)*(f+ g)^(1/2)/(f-g)^(1/2))/b/(f+g)^(3/2)
Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.91 \[ \int \frac {e^{2 (a+b x)}}{g \cosh (d+b x)+f \sinh (d+b x)} \, dx=\frac {2 e^{2 a-2 d} \left (\frac {e^{d+b x}}{f+g}-\frac {\sqrt {f-g} \text {arctanh}\left (\frac {e^{d+b x} \sqrt {f+g}}{\sqrt {f-g}}\right )}{(f+g)^{3/2}}\right )}{b} \] Input:
Integrate[E^(2*(a + b*x))/(g*Cosh[d + b*x] + f*Sinh[d + b*x]),x]
Output:
(2*E^(2*a - 2*d)*(E^(d + b*x)/(f + g) - (Sqrt[f - g]*ArcTanh[(E^(d + b*x)* Sqrt[f + g])/Sqrt[f - g]])/(f + g)^(3/2)))/b
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.81, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2720, 27, 262, 221}
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.
\(\displaystyle \int \frac {e^{2 (a+b x)}}{f \sinh (b x+d)+g \cosh (b x+d)} \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int -\frac {2 e^{2 a+2 b x}}{f-g-e^{2 b x} (f+g)}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 e^{2 a} \int \frac {e^{2 b x}}{f-g-e^{2 b x} (f+g)}de^{b x}}{b}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {2 e^{2 a} \left (\frac {(f-g) \int \frac {1}{f-g-e^{2 b x} (f+g)}de^{b x}}{f+g}-\frac {e^{b x}}{f+g}\right )}{b}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 e^{2 a} \left (\frac {\sqrt {f-g} \text {arctanh}\left (\frac {e^{b x} \sqrt {f+g}}{\sqrt {f-g}}\right )}{(f+g)^{3/2}}-\frac {e^{b x}}{f+g}\right )}{b}\) |
Input:
Int[E^(2*(a + b*x))/(g*Cosh[d + b*x] + f*Sinh[d + b*x]),x]
Output:
(-2*E^(2*a)*(-(E^(b*x)/(f + g)) + (Sqrt[f - g]*ArcTanh[(E^(b*x)*Sqrt[f + g ])/Sqrt[f - g]])/(f + g)^(3/2)))/b
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Leaf count of result is larger than twice the leaf count of optimal. \(142\) vs. \(2(67)=134\).
Time = 3.03 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.83
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{b x +2 a -d}}{b \left (f +g \right )}+\frac {\ln \left ({\mathrm e}^{b x +a}-\frac {\sqrt {\left (f -g \right ) \left (f +g \right )}\, {\mathrm e}^{a -d}}{f +g}\right ) \sqrt {\left (f -g \right ) \left (f +g \right )}\, {\mathrm e}^{2 a -2 d}}{b \left (f +g \right )^{2}}-\frac {\ln \left ({\mathrm e}^{b x +a}+\frac {\sqrt {\left (f -g \right ) \left (f +g \right )}\, {\mathrm e}^{a -d}}{f +g}\right ) \sqrt {\left (f -g \right ) \left (f +g \right )}\, {\mathrm e}^{2 a -2 d}}{b \left (f +g \right )^{2}}\) | \(143\) |
Input:
int(exp(2*b*x+2*a)/(g*cosh(b*x+d)+f*sinh(b*x+d)),x,method=_RETURNVERBOSE)
Output:
2*exp(b*x+2*a-d)/b/(f+g)+ln(exp(b*x+a)-1/(f+g)*((f-g)*(f+g))^(1/2)*exp(a-d ))/b*((f-g)*(f+g))^(1/2)/(f+g)^2*exp(2*a-2*d)-ln(exp(b*x+a)+1/(f+g)*((f-g) *(f+g))^(1/2)*exp(a-d))/b*((f-g)*(f+g))^(1/2)/(f+g)^2*exp(2*a-2*d)
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (67) = 134\).
Time = 0.10 (sec) , antiderivative size = 385, normalized size of antiderivative = 4.94 \[ \int \frac {e^{2 (a+b x)}}{g \cosh (d+b x)+f \sinh (d+b x)} \, dx=\left [\frac {\sqrt {\frac {f - g}{f + g}} {\left (\cosh \left (-2 \, a + 2 \, d\right ) - \sinh \left (-2 \, a + 2 \, d\right )\right )} \log \left (\frac {{\left (f + g\right )} \cosh \left (b x + d\right )^{2} + 2 \, {\left (f + g\right )} \cosh \left (b x + d\right ) \sinh \left (b x + d\right ) + {\left (f + g\right )} \sinh \left (b x + d\right )^{2} - 2 \, {\left ({\left (f + g\right )} \cosh \left (b x + d\right ) + {\left (f + g\right )} \sinh \left (b x + d\right )\right )} \sqrt {\frac {f - g}{f + g}} + f - g}{{\left (f + g\right )} \cosh \left (b x + d\right )^{2} + 2 \, {\left (f + g\right )} \cosh \left (b x + d\right ) \sinh \left (b x + d\right ) + {\left (f + g\right )} \sinh \left (b x + d\right )^{2} - f + g}\right ) + 2 \, \cosh \left (b x + d\right ) \cosh \left (-2 \, a + 2 \, d\right ) + 2 \, {\left (\cosh \left (-2 \, a + 2 \, d\right ) - \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right ) - 2 \, \cosh \left (b x + d\right ) \sinh \left (-2 \, a + 2 \, d\right )}{b f + b g}, -\frac {2 \, {\left (\sqrt {-\frac {f - g}{f + g}} {\left (\cosh \left (-2 \, a + 2 \, d\right ) - \sinh \left (-2 \, a + 2 \, d\right )\right )} \arctan \left (-\frac {{\left ({\left (f + g\right )} \cosh \left (b x + d\right ) + {\left (f + g\right )} \sinh \left (b x + d\right )\right )} \sqrt {-\frac {f - g}{f + g}}}{f - g}\right ) - \cosh \left (b x + d\right ) \cosh \left (-2 \, a + 2 \, d\right ) - {\left (\cosh \left (-2 \, a + 2 \, d\right ) - \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right ) + \cosh \left (b x + d\right ) \sinh \left (-2 \, a + 2 \, d\right )\right )}}{b f + b g}\right ] \] Input:
integrate(exp(2*b*x+2*a)/(g*cosh(b*x+d)+f*sinh(b*x+d)),x, algorithm="frica s")
Output:
[(sqrt((f - g)/(f + g))*(cosh(-2*a + 2*d) - sinh(-2*a + 2*d))*log(((f + g) *cosh(b*x + d)^2 + 2*(f + g)*cosh(b*x + d)*sinh(b*x + d) + (f + g)*sinh(b* x + d)^2 - 2*((f + g)*cosh(b*x + d) + (f + g)*sinh(b*x + d))*sqrt((f - g)/ (f + g)) + f - g)/((f + g)*cosh(b*x + d)^2 + 2*(f + g)*cosh(b*x + d)*sinh( b*x + d) + (f + g)*sinh(b*x + d)^2 - f + g)) + 2*cosh(b*x + d)*cosh(-2*a + 2*d) + 2*(cosh(-2*a + 2*d) - sinh(-2*a + 2*d))*sinh(b*x + d) - 2*cosh(b*x + d)*sinh(-2*a + 2*d))/(b*f + b*g), -2*(sqrt(-(f - g)/(f + g))*(cosh(-2*a + 2*d) - sinh(-2*a + 2*d))*arctan(-((f + g)*cosh(b*x + d) + (f + g)*sinh( b*x + d))*sqrt(-(f - g)/(f + g))/(f - g)) - cosh(b*x + d)*cosh(-2*a + 2*d) - (cosh(-2*a + 2*d) - sinh(-2*a + 2*d))*sinh(b*x + d) + cosh(b*x + d)*sin h(-2*a + 2*d))/(b*f + b*g)]
\[ \int \frac {e^{2 (a+b x)}}{g \cosh (d+b x)+f \sinh (d+b x)} \, dx=e^{2 a} \int \frac {e^{2 b x}}{f \sinh {\left (b x + d \right )} + g \cosh {\left (b x + d \right )}}\, dx \] Input:
integrate(exp(2*b*x+2*a)/(g*cosh(b*x+d)+f*sinh(b*x+d)),x)
Output:
exp(2*a)*Integral(exp(2*b*x)/(f*sinh(b*x + d) + g*cosh(b*x + d)), x)
Exception generated. \[ \int \frac {e^{2 (a+b x)}}{g \cosh (d+b x)+f \sinh (d+b x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(exp(2*b*x+2*a)/(g*cosh(b*x+d)+f*sinh(b*x+d)),x, algorithm="maxim a")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*f^2-4*g^2>0)', see `assume?` f or more de
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.21 \[ \int \frac {e^{2 (a+b x)}}{g \cosh (d+b x)+f \sinh (d+b x)} \, dx=\frac {2 \, {\left (\frac {{\left (f - g\right )} \arctan \left (\frac {f e^{\left (b x + d\right )} + g e^{\left (b x + d\right )}}{\sqrt {-f^{2} + g^{2}}}\right )}{\sqrt {-f^{2} + g^{2}} {\left (f e^{\left (2 \, d\right )} + g e^{\left (2 \, d\right )}\right )}} + \frac {e^{\left (b x + d\right )}}{f e^{\left (2 \, d\right )} + g e^{\left (2 \, d\right )}}\right )} e^{\left (2 \, a\right )}}{b} \] Input:
integrate(exp(2*b*x+2*a)/(g*cosh(b*x+d)+f*sinh(b*x+d)),x, algorithm="giac" )
Output:
2*((f - g)*arctan((f*e^(b*x + d) + g*e^(b*x + d))/sqrt(-f^2 + g^2))/(sqrt( -f^2 + g^2)*(f*e^(2*d) + g*e^(2*d))) + e^(b*x + d)/(f*e^(2*d) + g*e^(2*d)) )*e^(2*a)/b
Time = 3.00 (sec) , antiderivative size = 245, normalized size of antiderivative = 3.14 \[ \int \frac {e^{2 (a+b x)}}{g \cosh (d+b x)+f \sinh (d+b x)} \, dx=\frac {2\,{\mathrm {e}}^{2\,a-d+b\,x}}{b\,\left (f+g\right )}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^d\,\left (f\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,d}\,\sqrt {b^2\,f^3+3\,b^2\,f^2\,g+3\,b^2\,f\,g^2+b^2\,g^3}-g\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,d}\,\sqrt {b^2\,f^3+3\,b^2\,f^2\,g+3\,b^2\,f\,g^2+b^2\,g^3}\right )}{b\,f^2\,\sqrt {g\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,d}-f\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,d}}-b\,g^2\,\sqrt {g\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,d}-f\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,d}}}\right )\,\sqrt {g\,{\mathrm {e}}^{4\,a-4\,d}-f\,{\mathrm {e}}^{4\,a-4\,d}}}{\sqrt {b^2\,f^3+3\,b^2\,f^2\,g+3\,b^2\,f\,g^2+b^2\,g^3}} \] Input:
int(exp(2*a + 2*b*x)/(g*cosh(d + b*x) + f*sinh(d + b*x)),x)
Output:
(2*exp(2*a - d + b*x))/(b*(f + g)) - (2*atan((exp(b*x)*exp(d)*(f*exp(2*a)* exp(-2*d)*(b^2*f^3 + b^2*g^3 + 3*b^2*f*g^2 + 3*b^2*f^2*g)^(1/2) - g*exp(2* a)*exp(-2*d)*(b^2*f^3 + b^2*g^3 + 3*b^2*f*g^2 + 3*b^2*f^2*g)^(1/2)))/(b*f^ 2*(g*exp(4*a)*exp(-4*d) - f*exp(4*a)*exp(-4*d))^(1/2) - b*g^2*(g*exp(4*a)* exp(-4*d) - f*exp(4*a)*exp(-4*d))^(1/2)))*(g*exp(4*a - 4*d) - f*exp(4*a - 4*d))^(1/2))/(b^2*f^3 + b^2*g^3 + 3*b^2*f*g^2 + 3*b^2*f^2*g)^(1/2)
Time = 0.24 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.21 \[ \int \frac {e^{2 (a+b x)}}{g \cosh (d+b x)+f \sinh (d+b x)} \, dx=\frac {2 e^{2 a} \left (-\sqrt {-f^{2}+g^{2}}\, \mathit {atan} \left (\frac {e^{b x +d} f +e^{b x +d} g}{\sqrt {-f^{2}+g^{2}}}\right )+e^{b x +d} f +e^{b x +d} g \right )}{e^{2 d} b \left (f^{2}+2 f g +g^{2}\right )} \] Input:
int(exp(2*b*x+2*a)/(g*cosh(b*x+d)+f*sinh(b*x+d)),x)
Output:
(2*e**(2*a)*( - sqrt( - f**2 + g**2)*atan((e**(b*x + d)*f + e**(b*x + d)*g )/sqrt( - f**2 + g**2)) + e**(b*x + d)*f + e**(b*x + d)*g))/(e**(2*d)*b*(f **2 + 2*f*g + g**2))