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

Optimal. Leaf size=73 \[ -\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a}+\frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.26, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6742, 3303, 3298, 3301} \[ -\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a}+\frac {\cosh (c) \text {Chi}(d x)}{a}+\frac {\sinh (c) \text {Shi}(d x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(x*(a + b*x)),x]

[Out]

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

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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 63, normalized size = 0.86 \[ \frac {-\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )-\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )+\cosh (c) \text {Chi}(d x)+\sinh (c) \text {Shi}(d x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.64, size = 123, normalized size = 1.68 \[ \frac {{\left ({\rm Ei}\left (d x\right ) + {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - {\left ({\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) + {\left ({\rm Ei}\left (d x\right ) - {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c) + {\left ({\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.14, size = 75, normalized size = 1.03 \[ \frac {{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + {\rm Ei}\left (d x\right ) e^{c} - {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.08, size = 108, normalized size = 1.48 \[ \frac {{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right )}{2 a}-\frac {{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2 a}+\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right )}{2 a}-\frac {{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.41, size = 155, normalized size = 2.12 \[ \frac {1}{2} \, d {\left (\frac {b {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{a d} + \frac {2 \, \cosh \left (d x + c\right ) \log \left (b x + a\right )}{a d} - \frac {2 \, \cosh \left (d x + c\right ) \log \relax (x)}{a d} + \frac {{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}}{a d}\right )} - {\left (\frac {\log \left (b x + a\right )}{a} - \frac {\log \relax (x)}{a}\right )} \cosh \left (d x + c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x\,\left (a+b\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)/(x*(a + b*x)),x)

[Out]

int(cosh(c + d*x)/(x*(a + b*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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