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

Optimal. Leaf size=68 \[ -\frac{a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^2}-\frac{a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^2}+\frac{\sinh (c+d x)}{b d} \]

[Out]

-((a*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^2) + Sinh[c + d*x]/(b*d) - (a*Sinh[c - (a*d)/b]*SinhInte
gral[(a*d)/b + d*x])/b^2

________________________________________________________________________________________

Rubi [A]  time = 0.176147, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6742, 2637, 3303, 3298, 3301} \[ -\frac{a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^2}-\frac{a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^2}+\frac{\sinh (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

Int[(x*Cosh[c + d*x])/(a + b*x),x]

[Out]

-((a*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^2) + Sinh[c + d*x]/(b*d) - (a*Sinh[c - (a*d)/b]*SinhInte
gral[(a*d)/b + d*x])/b^2

Rule 6742

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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{x \cosh (c+d x)}{a+b x} \, dx &=\int \left (\frac{\cosh (c+d x)}{b}-\frac{a \cosh (c+d x)}{b (a+b x)}\right ) \, dx\\ &=\frac{\int \cosh (c+d x) \, dx}{b}-\frac{a \int \frac{\cosh (c+d x)}{a+b x} \, dx}{b}\\ &=\frac{\sinh (c+d x)}{b d}-\frac{\left (a \cosh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cosh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b}-\frac{\left (a \sinh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sinh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b}\\ &=-\frac{a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^2}+\frac{\sinh (c+d x)}{b d}-\frac{a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^2}\\ \end{align*}

Mathematica [A]  time = 0.135784, size = 64, normalized size = 0.94 \[ \frac{-a d \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )-a d \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+b \sinh (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.023, size = 114, normalized size = 1.7 \begin{align*} -{\frac{{{\rm e}^{-dx-c}}}{2\,bd}}+{\frac{a}{2\,{b}^{2}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{{{\rm e}^{dx+c}}}{2\,bd}}+{\frac{a}{2\,{b}^{2}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.33287, size = 211, normalized size = 3.1 \begin{align*} \frac{1}{2} \, d{\left (\frac{a{\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 )}}{b d} - \frac{\frac{{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac{{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}}{b} + \frac{2 \, a \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{2} d}\right )} +{\left (\frac{x}{b} - \frac{a \log \left (b x + a\right )}{b^{2}}\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.9967, size = 251, normalized size = 3.69 \begin{align*} -\frac{{\left (a d{\rm Ei}\left (\frac{b d x + a d}{b}\right ) + a d{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) - 2 \, b \sinh \left (d x + c\right ) -{\left (a d{\rm Ei}\left (\frac{b d x + a d}{b}\right ) - a d{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{2 \, b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.18626, size = 78, normalized size = 1.15 \begin{align*} -\frac{a{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} + a{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )}}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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