Integrand size = 19, antiderivative size = 271 \[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=\frac {\text {Chi}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right ) \sinh \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right )}{\sqrt {d^2-4 c e}}-\frac {\text {Chi}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right ) \sinh \left (a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}\right )}{\sqrt {d^2-4 c e}}+\frac {\cosh \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Shi}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\cosh \left (a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Shi}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}} \]
cosh(a-1/2*b*(d-(-4*c*e+d^2)^(1/2))/e)*Shi(b*x+1/2*b*(d-(-4*c*e+d^2)^(1/2) )/e)/(-4*c*e+d^2)^(1/2)-cosh(a-1/2*b*(d+(-4*c*e+d^2)^(1/2))/e)*Shi(b*x+1/2 *b*(d+(-4*c*e+d^2)^(1/2))/e)/(-4*c*e+d^2)^(1/2)+Chi(b*x+1/2*b*(d-(-4*c*e+d ^2)^(1/2))/e)*sinh(a-1/2*b*(d-(-4*c*e+d^2)^(1/2))/e)/(-4*c*e+d^2)^(1/2)-Ch i(b*x+1/2*b*(d+(-4*c*e+d^2)^(1/2))/e)*sinh(a-1/2*b*(d+(-4*c*e+d^2)^(1/2))/ e)/(-4*c*e+d^2)^(1/2)
Time = 1.13 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.81 \[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=\frac {e^{-a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}} \left (-e^{\frac {b d}{e}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (d-\sqrt {d^2-4 c e}+2 e x\right )}{2 e}\right )+e^{2 a+\frac {b \sqrt {d^2-4 c e}}{e}} \operatorname {ExpIntegralEi}\left (\frac {b \left (d-\sqrt {d^2-4 c e}+2 e x\right )}{2 e}\right )+e^{\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{e}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (d+\sqrt {d^2-4 c e}+2 e x\right )}{2 e}\right )-e^{2 a} \operatorname {ExpIntegralEi}\left (\frac {b \left (d+\sqrt {d^2-4 c e}+2 e x\right )}{2 e}\right )\right )}{2 \sqrt {d^2-4 c e}} \]
(E^(-a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e))*(-(E^((b*d)/e)*ExpIntegralEi[- 1/2*(b*(d - Sqrt[d^2 - 4*c*e] + 2*e*x))/e]) + E^(2*a + (b*Sqrt[d^2 - 4*c*e ])/e)*ExpIntegralEi[(b*(d - Sqrt[d^2 - 4*c*e] + 2*e*x))/(2*e)] + E^((b*(d + Sqrt[d^2 - 4*c*e]))/e)*ExpIntegralEi[-1/2*(b*(d + Sqrt[d^2 - 4*c*e] + 2* e*x))/e] - E^(2*a)*ExpIntegralEi[(b*(d + Sqrt[d^2 - 4*c*e] + 2*e*x))/(2*e) ]))/(2*Sqrt[d^2 - 4*c*e])
Time = 1.01 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {7279, 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.
| \(\displaystyle \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {2 e \sinh (a+b x)}{\sqrt {d^2-4 c e} \left (-\sqrt {d^2-4 c e}+d+2 e x\right )}-\frac {2 e \sinh (a+b x)}{\sqrt {d^2-4 c e} \left (\sqrt {d^2-4 c e}+d+2 e x\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sinh \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Chi}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\sinh \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \text {Chi}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}+\frac {\cosh \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Shi}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\cosh \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \text {Shi}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}\) |
(CoshIntegral[(b*(d - Sqrt[d^2 - 4*c*e]))/(2*e) + b*x]*Sinh[a - (b*(d - Sq rt[d^2 - 4*c*e]))/(2*e)])/Sqrt[d^2 - 4*c*e] - (CoshIntegral[(b*(d + Sqrt[d ^2 - 4*c*e]))/(2*e) + b*x]*Sinh[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)])/Sq rt[d^2 - 4*c*e] + (Cosh[a - (b*(d - Sqrt[d^2 - 4*c*e]))/(2*e)]*SinhIntegra l[(b*(d - Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*e] - (Cosh[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*SinhIntegral[(b*(d + Sqrt[d^2 - 4*c*e]) )/(2*e) + b*x])/Sqrt[d^2 - 4*c*e]
3.4.69.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 1.88 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.39
| method | result | size |
| risch | \(-\frac {b \,{\mathrm e}^{\frac {2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}} \operatorname {Ei}_{1}\left (\frac {-2 e \left (b x +a \right )+2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{2 \sqrt {-4 b^{2} c e +b^{2} d^{2}}}+\frac {b \,{\mathrm e}^{\frac {2 a e -b d -\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}} \operatorname {Ei}_{1}\left (-\frac {2 e \left (b x +a \right )-2 a e +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{2 \sqrt {-4 b^{2} c e +b^{2} d^{2}}}+\frac {b \,{\mathrm e}^{-\frac {2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}} \operatorname {Ei}_{1}\left (-\frac {-2 e \left (b x +a \right )+2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{2 \sqrt {-4 b^{2} c e +b^{2} d^{2}}}-\frac {b \,{\mathrm e}^{-\frac {2 a e -b d -\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}} \operatorname {Ei}_{1}\left (\frac {2 e \left (b x +a \right )-2 a e +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{2 \sqrt {-4 b^{2} c e +b^{2} d^{2}}}\) | \(376\) |
-1/2*b/(-4*b^2*c*e+b^2*d^2)^(1/2)*exp(1/2/e*(2*a*e-b*d+(-4*b^2*c*e+b^2*d^2 )^(1/2)))*Ei(1,1/2*(-2*e*(b*x+a)+2*a*e-b*d+(-4*b^2*c*e+b^2*d^2)^(1/2))/e)+ 1/2*b/(-4*b^2*c*e+b^2*d^2)^(1/2)*exp(1/2/e*(2*a*e-b*d-(-4*b^2*c*e+b^2*d^2) ^(1/2)))*Ei(1,-1/2*(2*e*(b*x+a)-2*a*e+b*d+(-4*b^2*c*e+b^2*d^2)^(1/2))/e)+1 /2*b/(-4*b^2*c*e+b^2*d^2)^(1/2)*exp(-1/2/e*(2*a*e-b*d+(-4*b^2*c*e+b^2*d^2) ^(1/2)))*Ei(1,-1/2*(-2*e*(b*x+a)+2*a*e-b*d+(-4*b^2*c*e+b^2*d^2)^(1/2))/e)- 1/2*b/(-4*b^2*c*e+b^2*d^2)^(1/2)*exp(-1/2/e*(2*a*e-b*d-(-4*b^2*c*e+b^2*d^2 )^(1/2)))*Ei(1,1/2*(2*e*(b*x+a)-2*a*e+b*d+(-4*b^2*c*e+b^2*d^2)^(1/2))/e)
Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (231) = 462\).
Time = 0.29 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.48 \[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=-\frac {{\left (e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 \, b e x + b d + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (-\frac {2 \, b e x + b d + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )\right )} \cosh \left (\frac {b d - 2 \, a e + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - {\left (e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 \, b e x + b d - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (-\frac {2 \, b e x + b d - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )\right )} \cosh \left (-\frac {b d - 2 \, a e - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - {\left (e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 \, b e x + b d + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (-\frac {2 \, b e x + b d + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )\right )} \sinh \left (\frac {b d - 2 \, a e + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - {\left (e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 \, b e x + b d - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (-\frac {2 \, b e x + b d - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )\right )} \sinh \left (-\frac {b d - 2 \, a e - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )}{2 \, {\left (b d^{2} - 4 \, b c e\right )}} \]
-1/2*((e*sqrt((b^2*d^2 - 4*b^2*c*e)/e^2)*Ei(1/2*(2*b*e*x + b*d + e*sqrt((b ^2*d^2 - 4*b^2*c*e)/e^2))/e) - e*sqrt((b^2*d^2 - 4*b^2*c*e)/e^2)*Ei(-1/2*( 2*b*e*x + b*d + e*sqrt((b^2*d^2 - 4*b^2*c*e)/e^2))/e))*cosh(1/2*(b*d - 2*a *e + e*sqrt((b^2*d^2 - 4*b^2*c*e)/e^2))/e) - (e*sqrt((b^2*d^2 - 4*b^2*c*e) /e^2)*Ei(1/2*(2*b*e*x + b*d - e*sqrt((b^2*d^2 - 4*b^2*c*e)/e^2))/e) - e*sq rt((b^2*d^2 - 4*b^2*c*e)/e^2)*Ei(-1/2*(2*b*e*x + b*d - e*sqrt((b^2*d^2 - 4 *b^2*c*e)/e^2))/e))*cosh(-1/2*(b*d - 2*a*e - e*sqrt((b^2*d^2 - 4*b^2*c*e)/ e^2))/e) - (e*sqrt((b^2*d^2 - 4*b^2*c*e)/e^2)*Ei(1/2*(2*b*e*x + b*d + e*sq rt((b^2*d^2 - 4*b^2*c*e)/e^2))/e) + e*sqrt((b^2*d^2 - 4*b^2*c*e)/e^2)*Ei(- 1/2*(2*b*e*x + b*d + e*sqrt((b^2*d^2 - 4*b^2*c*e)/e^2))/e))*sinh(1/2*(b*d - 2*a*e + e*sqrt((b^2*d^2 - 4*b^2*c*e)/e^2))/e) - (e*sqrt((b^2*d^2 - 4*b^2 *c*e)/e^2)*Ei(1/2*(2*b*e*x + b*d - e*sqrt((b^2*d^2 - 4*b^2*c*e)/e^2))/e) + e*sqrt((b^2*d^2 - 4*b^2*c*e)/e^2)*Ei(-1/2*(2*b*e*x + b*d - e*sqrt((b^2*d^ 2 - 4*b^2*c*e)/e^2))/e))*sinh(-1/2*(b*d - 2*a*e - e*sqrt((b^2*d^2 - 4*b^2* c*e)/e^2))/e))/(b*d^2 - 4*b*c*e)
\[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=\int \frac {\sinh {\left (a + b x \right )}}{c + d x + e x^{2}}\, dx \]
Exception generated. \[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*c*e-d^2>0)', see `assume?` for more deta
\[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=\int { \frac {\sinh \left (b x + a\right )}{e x^{2} + d x + c} \,d x } \]
Timed out. \[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=\int \frac {\mathrm {sinh}\left (a+b\,x\right )}{e\,x^2+d\,x+c} \,d x \]