\(\int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx\) [369]

Optimal result

Integrand size = 19, antiderivative size = 271 \[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=\frac {\text {Chi}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right ) \sinh \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right )}{\sqrt {d^2-4 c e}}-\frac {\text {Chi}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right ) \sinh \left (a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}\right )}{\sqrt {d^2-4 c e}}+\frac {\cosh \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Shi}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\cosh \left (a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Shi}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}} \]

[Out]

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)-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)

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6860, 3384, 3379, 3382} \[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=\frac {\sinh \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Chi}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\sinh \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \text {Chi}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}+\frac {\cosh \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Shi}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\cosh \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \text {Shi}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}} \]

[In]

Int[Sinh[a + b*x]/(c + d*x + e*x^2),x]

[Out]

(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)])/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)])/
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] - (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]

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 6860

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] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 e \sinh (a+b x)}{\sqrt {d^2-4 c e} \left (d-\sqrt {d^2-4 c e}+2 e x\right )}-\frac {2 e \sinh (a+b x)}{\sqrt {d^2-4 c e} \left (d+\sqrt {d^2-4 c e}+2 e x\right )}\right ) \, dx \\ & = \frac {(2 e) \int \frac {\sinh (a+b x)}{d-\sqrt {d^2-4 c e}+2 e x} \, dx}{\sqrt {d^2-4 c e}}-\frac {(2 e) \int \frac {\sinh (a+b x)}{d+\sqrt {d^2-4 c e}+2 e x} \, dx}{\sqrt {d^2-4 c e}} \\ & = \frac {\left (2 e \cosh \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right )\right ) \int \frac {\sinh \left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{d-\sqrt {d^2-4 c e}+2 e x} \, dx}{\sqrt {d^2-4 c e}}-\frac {\left (2 e \cosh \left (a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}\right )\right ) \int \frac {\sinh \left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{d+\sqrt {d^2-4 c e}+2 e x} \, dx}{\sqrt {d^2-4 c e}}+\frac {\left (2 e \sinh \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right )\right ) \int \frac {\cosh \left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{d-\sqrt {d^2-4 c e}+2 e x} \, dx}{\sqrt {d^2-4 c e}}-\frac {\left (2 e \sinh \left (a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}\right )\right ) \int \frac {\cosh \left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{d+\sqrt {d^2-4 c e}+2 e x} \, dx}{\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 {\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.81 \[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=\frac {e^{-a-\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}} \left (-e^{\frac {b d}{e}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (d-\sqrt {d^2-4 c e}+2 e x\right )}{2 e}\right )+e^{2 a+\frac {b \sqrt {d^2-4 c e}}{e}} \operatorname {ExpIntegralEi}\left (\frac {b \left (d-\sqrt {d^2-4 c e}+2 e x\right )}{2 e}\right )+e^{\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{e}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (d+\sqrt {d^2-4 c e}+2 e x\right )}{2 e}\right )-e^{2 a} \operatorname {ExpIntegralEi}\left (\frac {b \left (d+\sqrt {d^2-4 c e}+2 e x\right )}{2 e}\right )\right )}{2 \sqrt {d^2-4 c e}} \]

[In]

Integrate[Sinh[a + b*x]/(c + d*x + e*x^2),x]

[Out]

(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])

Maple [A] (verified)

Time = 1.88 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.39

method	result	size
risch	\(-\frac {b \,{\mathrm e}^{\frac {2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}} \operatorname {Ei}_{1}\left (\frac {-2 e \left (b x +a \right )+2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{2 \sqrt {-4 b^{2} c e +b^{2} d^{2}}}+\frac {b \,{\mathrm e}^{\frac {2 a e -b d -\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}} \operatorname {Ei}_{1}\left (-\frac {2 e \left (b x +a \right )-2 a e +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{2 \sqrt {-4 b^{2} c e +b^{2} d^{2}}}+\frac {b \,{\mathrm e}^{-\frac {2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}} \operatorname {Ei}_{1}\left (-\frac {-2 e \left (b x +a \right )+2 a e -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{2 \sqrt {-4 b^{2} c e +b^{2} d^{2}}}-\frac {b \,{\mathrm e}^{-\frac {2 a e -b d -\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}} \operatorname {Ei}_{1}\left (\frac {2 e \left (b x +a \right )-2 a e +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{2 \sqrt {-4 b^{2} c e +b^{2} d^{2}}}\)	\(376\)

[In]

int(sinh(b*x+a)/(e*x^2+d*x+c),x,method=_RETURNVERBOSE)

[Out]

-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)*ex
p(-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)

Fricas [B] (verification not implemented)

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 \[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=-\frac {{\left (e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 \, b e x + b d + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (-\frac {2 \, b e x + b d + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )\right )} \cosh \left (\frac {b d - 2 \, a e + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - {\left (e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 \, b e x + b d - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (-\frac {2 \, b e x + b d - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )\right )} \cosh \left (-\frac {b d - 2 \, a e - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - {\left (e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 \, b e x + b d + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (-\frac {2 \, b e x + b d + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )\right )} \sinh \left (\frac {b d - 2 \, a e + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - {\left (e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 \, b e x + b d - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (-\frac {2 \, b e x + b d - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )\right )} \sinh \left (-\frac {b d - 2 \, a e - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )}{2 \, {\left (b d^{2} - 4 \, b c e\right )}} \]

[In]

integrate(sinh(b*x+a)/(e*x^2+d*x+c),x, algorithm="fricas")

[Out]

-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*sqr
t((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))*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*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))*sin
h(-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)

Sympy [F]

\[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=\int \frac {\sinh {\left (a + b x \right )}}{c + d x + e x^{2}}\, dx \]

[In]

integrate(sinh(b*x+a)/(e*x**2+d*x+c),x)

[Out]

Integral(sinh(a + b*x)/(c + d*x + e*x**2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(sinh(b*x+a)/(e*x^2+d*x+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*c*e-d^2>0)', see `assume?` f
or more deta

Giac [F]

\[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=\int { \frac {\sinh \left (b x + a\right )}{e x^{2} + d x + c} \,d x } \]

[In]

integrate(sinh(b*x+a)/(e*x^2+d*x+c),x, algorithm="giac")

[Out]

integrate(sinh(b*x + a)/(e*x^2 + d*x + c), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh (a+b x)}{c+d x+e x^2} \, dx=\int \frac {\mathrm {sinh}\left (a+b\,x\right )}{e\,x^2+d\,x+c} \,d x \]

[In]

int(sinh(a + b*x)/(c + d*x + e*x^2),x)

[Out]

int(sinh(a + b*x)/(c + d*x + e*x^2), x)