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

Optimal. Leaf size=71 \[ \frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {\cosh (c+d x)}{b (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3297, 3303, 3298, 3301} \[ \frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {\cosh (c+d x)}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(a + b*x)^2,x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx &=-\frac {\cosh (c+d x)}{b (a+b x)}+\frac {d \int \frac {\sinh (c+d x)}{a+b x} \, dx}{b}\\ &=-\frac {\cosh (c+d x)}{b (a+b x)}+\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}{b}+\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}{b}\\ &=-\frac {\cosh (c+d x)}{b (a+b x)}+\frac {d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^2}+\frac {d \cosh \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 = 0.21, size = 65, normalized size = 0.92 \[ \frac {d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )+d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )-\frac {b \cosh (c+d x)}{a+b x}}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.57, size = 149, normalized size = 2.10 \[ -\frac {2 \, b \cosh \left (d x + c\right ) - {\left ({\left (b d x + a d\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (b d x + a d\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) + {\left ({\left (b d x + a d\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (b d x + a d\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{3} x + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.17, size = 615, normalized size = 8.66 \[ \frac {{\left ({\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - b c d^{2} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} + a d^{3} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - b d^{2} e^{\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}{b}\right )}\right )} b^{2}}{2 \, {\left ({\left (b x + a\right )} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b^{5} c + a b^{4} d\right )} d} - \frac {{\left ({\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - b c d^{2} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} + a d^{3} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} + b d^{2} e^{\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}{b}\right )}\right )} b^{2}}{2 \, {\left ({\left (b x + a\right )} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b^{5} c + a b^{4} d\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((b*x + 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) - 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*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*d
^2*e^((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b))*b^2/(((b*x + a)*b^4*(b*c/(b*x + a) - a*d/(b*x + a) + d
) - b^5*c + a*b^4*d)*d) - 1/2*((b*x + 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) - 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*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*d^2*e^(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b))*b^2/(((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 = 0.08, size = 132, normalized size = 1.86 \[ -\frac {d \,{\mathrm e}^{-d x -c}}{2 b \left (b d x +d a \right )}+\frac {d \,{\mathrm e}^{\frac {d a -c b}{b}} \Ei \left (1, d x +c +\frac {d a -c b}{b}\right )}{2 b^{2}}-\frac {d \,{\mathrm e}^{d x +c}}{2 b^{2} \left (\frac {a d}{b}+d x \right )}-\frac {d \,{\mathrm e}^{-\frac {d a -c b}{b}} \Ei \left (1, -d x -c -\frac {d a -c b}{b}\right )}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 81, normalized size = 1.14 \[ \frac {d {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} - \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{2 \, b} - \frac {\cosh \left (d x + c\right )}{{\left (b x + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)/(a + b*x)^2,x)

[Out]

int(cosh(c + d*x)/(a + b*x)^2, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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