3.5 \(\int \frac {\cosh (a+b x)}{c+d x} \, dx\)

Optimal. Leaf size=51 \[ \frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d} \]

[Out]

Chi(b*c/d+b*x)*cosh(a-b*c/d)/d+Shi(b*c/d+b*x)*sinh(a-b*c/d)/d

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Rubi [A]  time = 0.10, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3303, 3298, 3301} \[ \frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]/(c + d*x),x]

[Out]

(Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x])/d + (Sinh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d

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]

Rubi steps

\begin {align*} \int \frac {\cosh (a+b x)}{c+d x} \, dx &=\cosh \left (a-\frac {b c}{d}\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx+\sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx\\ &=\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 49, normalized size = 0.96 \[ \frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )+\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]/(c + d*x),x]

[Out]

(Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x] + Sinh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d

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fricas [A]  time = 0.48, size = 94, normalized size = 1.84 \[ \frac {{\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) + {\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

1/2*((Ei((b*d*x + b*c)/d) + Ei(-(b*d*x + b*c)/d))*cosh(-(b*c - a*d)/d) + (Ei((b*d*x + b*c)/d) - Ei(-(b*d*x + b
*c)/d))*sinh(-(b*c - a*d)/d))/d

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giac [A]  time = 0.12, size = 56, normalized size = 1.10 \[ \frac {{\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.08, size = 82, normalized size = 1.61 \[ -\frac {{\mathrm e}^{-\frac {d a -c b}{d}} \Ei \left (1, b x +a -\frac {d a -c b}{d}\right )}{2 d}-\frac {{\mathrm e}^{\frac {d a -c b}{d}} \Ei \left (1, -b x -a -\frac {-d a +c b}{d}\right )}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.55, size = 57, normalized size = 1.12 \[ -\frac {e^{\left (-a + \frac {b c}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{2 \, d} - \frac {e^{\left (a - \frac {b c}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

-1/2*e^(-a + b*c/d)*exp_integral_e(1, (d*x + c)*b/d)/d - 1/2*e^(a - b*c/d)*exp_integral_e(1, -(d*x + c)*b/d)/d

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {cosh}\left (a+b\,x\right )}{c+d\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)/(c + d*x),x)

[Out]

int(cosh(a + b*x)/(c + d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(a + b*x)/(c + d*x), x)

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