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

Optimal. Leaf size=150 \[ -\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^2}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}-\frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a b}-\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a b}+\frac {\cosh (c+d x)}{a (a+b x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.40, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6742, 3303, 3298, 3301, 3297} \[ -\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^2}+\frac {\cosh (c) \text {Chi}(d x)}{a^2}+\frac {\sinh (c) \text {Shi}(d x)}{a^2}-\frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a b}-\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a b}+\frac {\cosh (c+d x)}{a (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(x*(a + b*x)^2),x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 1.21, size = 139, normalized size = 0.93 \[ \frac {-\frac {\text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \sinh \left (c-\frac {a d}{b}\right )+b \cosh \left (c-\frac {a d}{b}\right )\right )}{b}-\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )-\frac {a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{b}+\frac {a \sinh (c) \sinh (d x)}{a+b x}+\frac {a \cosh (c) \cosh (d x)}{a+b x}+\cosh (c) \text {Chi}(d x)+\sinh (c) \text {Shi}(d x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 270, normalized size = 1.80 \[ \frac {2 \, a b \cosh \left (d x + c\right ) + {\left ({\left (b^{2} x + a b\right )} {\rm Ei}\left (d x\right ) + {\left (b^{2} x + a b\right )} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - {\left ({\left (a^{2} d + a b + {\left (a b d + b^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{2} d - a b + {\left (a b d - b^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) + {\left ({\left (b^{2} x + a b\right )} {\rm Ei}\left (d x\right ) - {\left (b^{2} x + a b\right )} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c) + {\left ({\left (a^{2} d + a b + {\left (a b d + b^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{2} d - a b + {\left (a b d - b^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (a^{2} b^{2} x + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.20, size = 1329, normalized size = 8.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

maple [A]  time = 0.10, size = 254, normalized size = 1.69 \[ \frac {{\mathrm e}^{-d x -c} d}{2 a \left (\left (d x +c \right ) b +d a -c b \right )}-\frac {{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right ) d}{2 a b}+\frac {{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right )}{2 a^{2}}-\frac {{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2 a^{2}}+\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right )}{2 a^{2}}+\frac {d \,{\mathrm e}^{d x +c}}{2 a b \left (\frac {a d}{b}+d x \right )}+\frac {d \,{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right )}{2 a b}-\frac {{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 227, normalized size = 1.51 \[ -\frac {1}{2} \, d {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{a b} - \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{a b} - \frac {b {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{a^{2} d} - \frac {2 \, \cosh \left (d x + c\right ) \log \left (b x + a\right )}{a^{2} d} + \frac {2 \, \cosh \left (d x + c\right ) \log \relax (x)}{a^{2} d} - \frac {{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}}{a^{2} d}\right )} + {\left (\frac {1}{a b x + a^{2}} - \frac {\log \left (b x + a\right )}{a^{2}} + \frac {\log \relax (x)}{a^{2}}\right )} \cosh \left (d x + c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)/(x*(a + b*x)^2),x)

[Out]

int(cosh(c + d*x)/(x*(a + b*x)^2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________