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

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

[Out]

Chi(b*x+1/2*b*(d-(-4*c*e+d^2)^(1/2))/e)*cosh(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)*cosh(a-1/2*b*(d+(-4*c*e+d^2)^(1/2))/e)/(-4*c*e+d^2)^(1/2)+Shi(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)-Shi(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)

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Rubi [A]  time = 0.75, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {6728, 3303, 3298, 3301} \[ \frac {\cosh \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Chi}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\cosh \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \text {Chi}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}+\frac {\sinh \left (a-\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}\right ) \text {Shi}\left (\frac {b \left (d-\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}}-\frac {\sinh \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \text {Shi}\left (\frac {b \left (d+\sqrt {d^2-4 c e}\right )}{2 e}+b x\right )}{\sqrt {d^2-4 c e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[a - (b*(d - Sqrt[d^2 - 4*c*e]))/(2*e)]*CoshIntegral[(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)]*CoshIntegral[(b*(d + Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/
Sqrt[d^2 - 4*c*e] + (Sinh[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] - (Sinh[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 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]

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 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]

Rubi steps

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

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Mathematica [C]  time = 0.50, size = 248, normalized size = 0.92 \[ \frac {\cosh \left (a+\frac {b \left (\sqrt {d^2-4 c e}-d\right )}{2 e}\right ) \text {Ci}\left (\frac {i b \left (d+2 e x-\sqrt {d^2-4 c e}\right )}{2 e}\right )-\cosh \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \text {Ci}\left (\frac {i b \left (d+2 e x+\sqrt {d^2-4 c e}\right )}{2 e}\right )-\sinh \left (a-\frac {b \left (\sqrt {d^2-4 c e}+d\right )}{2 e}\right ) \text {Shi}\left (\frac {b \left (d+2 e x+\sqrt {d^2-4 c e}\right )}{2 e}\right )+i \sinh \left (a+\frac {b \left (\sqrt {d^2-4 c e}-d\right )}{2 e}\right ) \text {Si}\left (\frac {i b \left (\sqrt {d^2-4 c e}-d\right )}{2 e}-i b x\right )}{\sqrt {d^2-4 c e}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cosh[a + (b*(-d + Sqrt[d^2 - 4*c*e]))/(2*e)]*CosIntegral[((I/2)*b*(d - Sqrt[d^2 - 4*c*e] + 2*e*x))/e] - Cosh[
a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*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)]*SinhIntegral[(b*(d + Sqrt[d^2 - 4*c*e] + 2*e*x))/(2*e)] + I*Sinh[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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fricas [B]  time = 0.53, size = 671, normalized size = 2.48 \[ -\frac {{\left (e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 \, b e x + b d + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (-\frac {2 \, b e x + b d + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )\right )} \cosh \left (\frac {b d - 2 \, a e + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - {\left (e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 \, b e x + b d - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (-\frac {2 \, b e x + b d - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )\right )} \cosh \left (-\frac {b d - 2 \, a e - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - {\left (e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 \, b e x + b d + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (-\frac {2 \, b e x + b d + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )\right )} \sinh \left (\frac {b d - 2 \, a e + e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - {\left (e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (\frac {2 \, b e x + b d - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right ) - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}} {\rm Ei}\left (-\frac {2 \, b e x + b d - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )\right )} \sinh \left (-\frac {b d - 2 \, a e - e \sqrt {\frac {b^{2} d^{2} - 4 \, b^{2} c e}{e^{2}}}}{2 \, e}\right )}{2 \, {\left (b d^{2} - 4 \, b c e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (b x + a\right )}{e x^{2} + d x + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 376, normalized size = 1.39 \[ \frac {b \,{\mathrm e}^{-\frac {2 e a -b d -\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}} \Ei \left (1, \frac {2 e \left (b x +a \right )-2 e a +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{2 \sqrt {-4 b^{2} c e +b^{2} d^{2}}}-\frac {b \,{\mathrm e}^{-\frac {2 e a -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}} \Ei \left (1, -\frac {-2 e \left (b x +a \right )+2 e a -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{2 \sqrt {-4 b^{2} c e +b^{2} d^{2}}}-\frac {b \,{\mathrm e}^{\frac {2 e a -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}} \Ei \left (1, \frac {-2 e \left (b x +a \right )+2 e a -b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{2 \sqrt {-4 b^{2} c e +b^{2} d^{2}}}+\frac {b \,{\mathrm e}^{\frac {2 e a -b d -\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}} \Ei \left (1, -\frac {2 e \left (b x +a \right )-2 e a +b d +\sqrt {-4 b^{2} c e +b^{2} d^{2}}}{2 e}\right )}{2 \sqrt {-4 b^{2} c e +b^{2} d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(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/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)*e
xp(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/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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(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 details)Is 4*c*e-d^2 positive or negative?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {cosh}\left (a+b\,x\right )}{e\,x^2+d\,x+c} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\left (a + b x \right )}}{c + d x + e x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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