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

Optimal. Leaf size=125 \[ -\frac {a d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^3}-\frac {a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {a \cosh (c+d x)}{b^2 (a+b x)} \]

[Out]

Chi(a*d/b+d*x)*cosh(-c+a*d/b)/b^2+a*cosh(d*x+c)/b^2/(b*x+a)-a*d*cosh(-c+a*d/b)*Shi(a*d/b+d*x)/b^3+a*d*Chi(a*d/
b+d*x)*sinh(-c+a*d/b)/b^3-Shi(a*d/b+d*x)*sinh(-c+a*d/b)/b^2

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Rubi [A]  time = 0.30, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 3297, 3303, 3298, 3301} \[ -\frac {a d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {a \cosh (c+d x)}{b^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(x*Cosh[c + d*x])/(a + b*x)^2,x]

[Out]

(a*Cosh[c + d*x])/(b^2*(a + b*x)) + (Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^2 - (a*d*CoshIntegral[(a
*d)/b + d*x]*Sinh[c - (a*d)/b])/b^3 - (a*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^3 + (Sinh[c - (a*d
)/b]*SinhIntegral[(a*d)/b + d*x])/b^2

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

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 {x \cosh (c+d x)}{(a+b x)^2} \, dx &=\int \left (-\frac {a \cosh (c+d x)}{b (a+b x)^2}+\frac {\cosh (c+d x)}{b (a+b x)}\right ) \, dx\\ &=\frac {\int \frac {\cosh (c+d x)}{a+b x} \, dx}{b}-\frac {a \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx}{b}\\ &=\frac {a \cosh (c+d x)}{b^2 (a+b x)}-\frac {(a d) \int \frac {\sinh (c+d x)}{a+b x} \, dx}{b^2}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b}\\ &=\frac {a \cosh (c+d x)}{b^2 (a+b x)}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^2}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^2}-\frac {\left (a d \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2}-\frac {\left (a d \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2}\\ &=\frac {a \cosh (c+d x)}{b^2 (a+b x)}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^2}-\frac {a d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^3}-\frac {a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^2}\\ \end {align*}

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Mathematica [A]  time = 0.44, size = 97, normalized size = 0.78 \[ \frac {\text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (b \cosh \left (c-\frac {a d}{b}\right )-a d \sinh \left (c-\frac {a d}{b}\right )\right )+\text {Shi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (b \sinh \left (c-\frac {a d}{b}\right )-a d \cosh \left (c-\frac {a d}{b}\right )\right )+\frac {a b \cosh (c+d x)}{a+b x}}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*b*Cosh[c + d*x])/(a + b*x) + CoshIntegral[d*(a/b + x)]*(b*Cosh[c - (a*d)/b] - a*d*Sinh[c - (a*d)/b]) + (-(
a*d*Cosh[c - (a*d)/b]) + b*Sinh[c - (a*d)/b])*SinhIntegral[d*(a/b + x)])/b^3

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fricas [A]  time = 0.38, size = 200, normalized size = 1.60 \[ \frac {2 \, a b \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d - a b + {\left (a b d - b^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{2} d + a b + {\left (a b d + b^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) + {\left ({\left (a^{2} d - a b + {\left (a b d - b^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{2} d + a b + {\left (a b d + b^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{4} x + a b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a*b*cosh(d*x + c) - ((a^2*d - a*b + (a*b*d - b^2)*x)*Ei((b*d*x + a*d)/b) - (a^2*d + a*b + (a*b*d + b^2)
*x)*Ei(-(b*d*x + a*d)/b))*cosh(-(b*c - a*d)/b) + ((a^2*d - a*b + (a*b*d - b^2)*x)*Ei((b*d*x + a*d)/b) + (a^2*d
+ a*b + (a*b*d + b^2)*x)*Ei(-(b*d*x + a*d)/b))*sinh(-(b*c - a*d)/b))/(b^4*x + a*b^3)

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giac [B]  time = 0.19, size = 994, normalized size = 7.95 \[ -\frac {{\left ({\left (b x + a\right )} a {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - a b c d^{2} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} + a^{2} d^{3} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - {\left (b x + a\right )} a {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} + a b c d^{2} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - a^{2} d^{3} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - {\left (b x + a\right )} b {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} + b^{2} c d {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - a b d^{2} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - {\left (b x + a\right )} b {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} + b^{2} c d {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - a b d^{2} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - a b d^{2} e^{\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}{b}\right )} - a b d^{2} e^{\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}{b}\right )}\right )} b}{2 \, {\left ({\left (b x + a\right )} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b^{5} c + a b^{4} d\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) -
b*c + a*d)/b)*e^((b*c - a*d)/b) - a*b*c*d^2*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*
e^((b*c - a*d)/b) + a^2*d^3*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^((b*c - a*d)/b
) - (b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*Ei(-((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b
*c + a*d)/b)*e^(-(b*c - a*d)/b) + a*b*c*d^2*Ei(-((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)
*e^(-(b*c - a*d)/b) - a^2*d^3*Ei(-((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^(-(b*c - a*
d)/b) - (b*x + a)*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) -
b*c + a*d)/b)*e^((b*c - a*d)/b) + b^2*c*d*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^
((b*c - a*d)/b) - a*b*d^2*Ei(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^((b*c - a*d)/b)
- (b*x + a)*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d*Ei(-((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c +
a*d)/b)*e^(-(b*c - a*d)/b) + b^2*c*d*Ei(-((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^(-(
b*c - a*d)/b) - a*b*d^2*Ei(-((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*e^(-(b*c - a*d)/b)
- a*b*d^2*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - a*b*d^2*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*
x + a) + d)/b))*b/(((b*x + a)*b^4*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b^5*c + a*b^4*d)*d)

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maple [A]  time = 0.10, size = 215, normalized size = 1.72 \[ \frac {d \,{\mathrm e}^{-d x -c} a}{2 b^{2} \left (b d x +d a \right )}-\frac {d \,{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right ) a}{2 b^{3}}-\frac {{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right )}{2 b^{2}}-\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right )}{2 b^{2}}+\frac {d \,{\mathrm e}^{d x +c} a}{2 b^{3} \left (\frac {a d}{b}+d x \right )}+\frac {d \,{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right ) a}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*d*exp(-d*x-c)/b^2/(b*d*x+a*d)*a-1/2*d/b^3*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a-1/2/b^2*exp((a*d-b*c)
/b)*Ei(1,d*x+c+(a*d-b*c)/b)-1/2/b^2*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)+1/2*d/b^3*exp(d*x+c)/(a*d/b+d*x
)*a+1/2*d/b^3*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a

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maxima [A]  time = 0.41, size = 178, normalized size = 1.42 \[ -\frac {1}{2} \, {\left (a {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b^{3}} - \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b^{3}}\right )} + \frac {\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}}{b d} + \frac {2 \, \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{2} d}\right )} d + {\left (\frac {a}{b^{3} x + a b^{2}} + \frac {\log \left (b x + a\right )}{b^{2}}\right )} \cosh \left (d x + c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a*(e^(-c + a*d/b)*exp_integral_e(1, (b*x + a)*d/b)/b^3 - e^(c - a*d/b)*exp_integral_e(1, -(b*x + a)*d/b)
/b^3) + (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) + 2*cosh(d*x + c)*log(b*x + a)/(b^2*d))*d + (a/(b^3*x + a*b^2) + log(b*x + a)/b^2)*cosh(d*x + c)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*cosh(c + d*x))/(a + b*x)^2,x)

[Out]

int((x*cosh(c + d*x))/(a + b*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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