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

Optimal. Leaf size=271 \[ \frac{\sinh \left (a-\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}\right ) \text{Chi}\left (\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}-\frac{\sinh \left (a-\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}\right ) \text{Chi}\left (\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}+\frac{\cosh \left (a-\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}\right ) \text{Shi}\left (\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}-\frac{\cosh \left (a-\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}\right ) \text{Shi}\left (\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}} \]

[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]

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Rubi [A]  time = 0.809164, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {6728, 3303, 3298, 3301} \[ \frac{\sinh \left (a-\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}\right ) \text{Chi}\left (\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}-\frac{\sinh \left (a-\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}\right ) \text{Chi}\left (\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}+\frac{\cosh \left (a-\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}\right ) \text{Shi}\left (\frac{b \left (d-\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}}-\frac{\cosh \left (a-\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}\right ) \text{Shi}\left (\frac{b \left (d+\sqrt{d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt{d^2-4 c e}} \]

Antiderivative was successfully verified.

[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 6728

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]

Rule 3303

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 3298

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 3301

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]

Rubi steps

\begin{align*} \int \frac{\sinh (a+b x)}{c+d x+e x^2} \, dx &=\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 [C]  time = 0.514847, size = 248, normalized size = 0.92 \[ \frac{\sinh \left (a+\frac{b \left (\sqrt{d^2-4 c e}-d\right )}{2 e}\right ) \text{CosIntegral}\left (\frac{i b \left (-\sqrt{d^2-4 c e}+d+2 e x\right )}{2 e}\right )-\sinh \left (a-\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}\right ) \text{CosIntegral}\left (\frac{i b \left (\sqrt{d^2-4 c e}+d+2 e x\right )}{2 e}\right )-\cosh \left (a-\frac{b \left (\sqrt{d^2-4 c e}+d\right )}{2 e}\right ) \text{Shi}\left (\frac{b \left (d+2 e x+\sqrt{d^2-4 c e}\right )}{2 e}\right )+i \cosh \left (a+\frac{b \left (\sqrt{d^2-4 c e}-d\right )}{2 e}\right ) \text{Si}\left (\frac{i b \left (\sqrt{d^2-4 c e}-d\right )}{2 e}-i b x\right )}{\sqrt{d^2-4 c e}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(CosIntegral[((I/2)*b*(d - Sqrt[d^2 - 4*c*e] + 2*e*x))/e]*Sinh[a + (b*(-d + Sqrt[d^2 - 4*c*e]))/(2*e)] - CosIn
tegral[((I/2)*b*(d + Sqrt[d^2 - 4*c*e] + 2*e*x))/e]*Sinh[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)] - Cosh[a - (b*
(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*SinhIntegral[(b*(d + Sqrt[d^2 - 4*c*e] + 2*e*x))/(2*e)] + I*Cosh[a + (b*(-d +
Sqrt[d^2 - 4*c*e]))/(2*e)]*SinIntegral[((I/2)*b*(-d + Sqrt[d^2 - 4*c*e]))/e - I*b*x])/Sqrt[d^2 - 4*c*e]

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Maple [A]  time = 0.039, size = 370, normalized size = 1.4 \begin{align*}{\frac{b}{2}{{\rm e}^{-{\frac{1}{2\,e} \left ( 2\,ea-bd+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{2\,e} \left ( -2\,e \left ( bx+a \right ) +2\,ea-bd+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}}}}}-{\frac{b}{2}{{\rm e}^{{\frac{1}{2\,e} \left ( -2\,ea+bd+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) }}}{\it Ei} \left ( 1,{\frac{1}{2\,e} \left ( 2\,e \left ( bx+a \right ) -2\,ea+bd+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}}}}}+{\frac{b}{2}{{\rm e}^{-{\frac{1}{2\,e} \left ( -2\,ea+bd+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{2\,e} \left ( 2\,e \left ( bx+a \right ) -2\,ea+bd+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}}}}}-{\frac{b}{2}{{\rm e}^{{\frac{1}{2\,e} \left ( 2\,ea-bd+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) }}}{\it Ei} \left ( 1,{\frac{1}{2\,e} \left ( -2\,e \left ( bx+a \right ) +2\,ea-bd+\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{-4\,{b}^{2}ce+{b}^{2}{d}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b/(-4*b^2*c*e+b^2*d^2)^(1/2)*exp(-1/2/e*(2*e*a-b*d+(-4*b^2*c*e+b^2*d^2)^(1/2)))*Ei(1,-1/2*(-2*e*(b*x+a)+2*
e*a-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*(-2*e*a+b*d+(-4*b^2*c*e+b^2*d^
2)^(1/2))/e)*Ei(1,1/2*(2*e*(b*x+a)-2*e*a+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)*e
xp(-1/2*(-2*e*a+b*d+(-4*b^2*c*e+b^2*d^2)^(1/2))/e)*Ei(1,-1/2*(2*e*(b*x+a)-2*e*a+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*e*a-b*d+(-4*b^2*c*e+b^2*d^2)^(1/2)))*Ei(1,1/2*(-2*e*(b*x+a)
+2*e*a-b*d+(-4*b^2*c*e+b^2*d^2)^(1/2))/e)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.22673, size = 1445, normalized size = 5.33 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\left (a + b x \right )}}{c + d x + e x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (b x + a\right )}{e x^{2} + d x + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)