3.102 \(\int \frac {a+a \cosh (e+f x)}{c+d x} \, dx\)

Optimal. Leaf size=64 \[ \frac {a \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d}+\frac {a \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d}+\frac {a \log (c+d x)}{d} \]

[Out]

a*Chi(c*f/d+f*x)*cosh(-e+c*f/d)/d+a*ln(d*x+c)/d-a*Shi(c*f/d+f*x)*sinh(-e+c*f/d)/d

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Rubi [A]  time = 0.14, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3317, 3303, 3298, 3301} \[ \frac {a \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d}+\frac {a \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d}+\frac {a \log (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cosh[e + f*x])/(c + d*x),x]

[Out]

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

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 54, normalized size = 0.84 \[ \frac {a \left (\text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \cosh \left (e-\frac {c f}{d}\right )+\sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+\log (c+d x)\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 111, normalized size = 1.73 \[ \frac {{\left (a {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + a {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) + 2 \, a \log \left (d x + c\right ) - {\left (a {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - a {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a*Ei((d*f*x + c*f)/d) + a*Ei(-(d*f*x + c*f)/d))*cosh(-(d*e - c*f)/d) + 2*a*log(d*x + c) - (a*Ei((d*f*x +
c*f)/d) - a*Ei(-(d*f*x + c*f)/d))*sinh(-(d*e - c*f)/d))/d

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giac [A]  time = 0.14, size = 69, normalized size = 1.08 \[ \frac {a {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (\frac {c f}{d} - e\right )} + a {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (-\frac {c f}{d} + e\right )} + 2 \, a \log \left (d x + c\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.14, size = 94, normalized size = 1.47 \[ \frac {a \ln \left (d x +c \right )}{d}-\frac {a \,{\mathrm e}^{\frac {c f -d e}{d}} \Ei \left (1, f x +e +\frac {c f -d e}{d}\right )}{2 d}-\frac {a \,{\mathrm e}^{-\frac {c f -d e}{d}} \Ei \left (1, -f x -e -\frac {c f -d e}{d}\right )}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*ln(d*x+c)/d-1/2*a/d*exp((c*f-d*e)/d)*Ei(1,f*x+e+(c*f-d*e)/d)-1/2*a/d*exp(-(c*f-d*e)/d)*Ei(1,-f*x-e-(c*f-d*e)
/d)

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maxima [A]  time = 0.45, size = 70, normalized size = 1.09 \[ -\frac {1}{2} \, a {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} + \frac {a \log \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \frac {\cosh {\left (e + f x \right )}}{c + d x}\, dx + \int \frac {1}{c + d x}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(Integral(cosh(e + f*x)/(c + d*x), x) + Integral(1/(c + d*x), x))

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