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

Optimal. Leaf size=147 \[ \frac{a^2 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{a^2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^4}-\frac{a^2 \cosh (c+d x)}{b^3 (a+b x)}-\frac{2 a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^3}-\frac{2 a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^3}+\frac{\sinh (c+d x)}{b^2 d} \]

[Out]

-((a^2*Cosh[c + d*x])/(b^3*(a + b*x))) - (2*a*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^3 + (a^2*d*Cosh
Integral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^4 + Sinh[c + d*x]/(b^2*d) + (a^2*d*Cosh[c - (a*d)/b]*SinhIntegral
[(a*d)/b + d*x])/b^4 - (2*a*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^3

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Rubi [A]  time = 0.374579, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {6742, 2637, 3297, 3303, 3298, 3301} \[ \frac{a^2 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{a^2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^4}-\frac{a^2 \cosh (c+d x)}{b^3 (a+b x)}-\frac{2 a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^3}-\frac{2 a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^3}+\frac{\sinh (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*Cosh[c + d*x])/(b^3*(a + b*x))) - (2*a*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^3 + (a^2*d*Cosh
Integral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^4 + Sinh[c + d*x]/(b^2*d) + (a^2*d*Cosh[c - (a*d)/b]*SinhIntegral
[(a*d)/b + d*x])/b^4 - (2*a*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^3

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 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 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^2 \cosh (c+d x)}{(a+b x)^2} \, dx &=\int \left (\frac{\cosh (c+d x)}{b^2}+\frac{a^2 \cosh (c+d x)}{b^2 (a+b x)^2}-\frac{2 a \cosh (c+d x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{\int \cosh (c+d x) \, dx}{b^2}-\frac{(2 a) \int \frac{\cosh (c+d x)}{a+b x} \, dx}{b^2}+\frac{a^2 \int \frac{\cosh (c+d x)}{(a+b x)^2} \, dx}{b^2}\\ &=-\frac{a^2 \cosh (c+d x)}{b^3 (a+b x)}+\frac{\sinh (c+d x)}{b^2 d}+\frac{\left (a^2 d\right ) \int \frac{\sinh (c+d x)}{a+b x} \, dx}{b^3}-\frac{\left (2 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^2}-\frac{\left (2 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^2}\\ &=-\frac{a^2 \cosh (c+d x)}{b^3 (a+b x)}-\frac{2 a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{\sinh (c+d x)}{b^2 d}-\frac{2 a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{\left (a^2 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^3}+\frac{\left (a^2 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^3}\\ &=-\frac{a^2 \cosh (c+d x)}{b^3 (a+b x)}-\frac{2 a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{a^2 d \text{Chi}\left (\frac{a d}{b}+d x\right ) \sinh \left (c-\frac{a d}{b}\right )}{b^4}+\frac{\sinh (c+d x)}{b^2 d}+\frac{a^2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^4}-\frac{2 a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^3}\\ \end{align*}

Mathematica [A]  time = 0.692046, size = 115, normalized size = 0.78 \[ \frac{b \left (\frac{b \sinh (c+d x)}{d}-\frac{a^2 \cosh (c+d x)}{a+b x}\right )+a \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \sinh \left (c-\frac{a d}{b}\right )-2 b \cosh \left (c-\frac{a d}{b}\right )\right )+a \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \cosh \left (c-\frac{a d}{b}\right )-2 b \sinh \left (c-\frac{a d}{b}\right )\right )}{b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*CoshIntegral[d*(a/b + x)]*(-2*b*Cosh[c - (a*d)/b] + a*d*Sinh[c - (a*d)/b]) + b*(-((a^2*Cosh[c + d*x])/(a +
b*x)) + (b*Sinh[c + d*x])/d) + a*(a*d*Cosh[c - (a*d)/b] - 2*b*Sinh[c - (a*d)/b])*SinhIntegral[d*(a/b + x)])/b^
4

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Maple [A]  time = 0.039, size = 254, normalized size = 1.7 \begin{align*} -{\frac{{{\rm e}^{-dx-c}}}{2\,d{b}^{2}}}-{\frac{d{{\rm e}^{-dx-c}}{a}^{2}}{2\,{b}^{3} \left ( bdx+da \right ) }}+{\frac{d{a}^{2}}{2\,{b}^{4}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{a}{{b}^{3}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{{{\rm e}^{dx+c}}}{2\,d{b}^{2}}}-{\frac{d{{\rm e}^{dx+c}}{a}^{2}}{2\,{b}^{4}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}-{\frac{d{a}^{2}}{2\,{b}^{4}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }+{\frac{a}{{b}^{3}}{{\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^2*cosh(d*x+c)/(b*x+a)^2,x)

[Out]

-1/2/d*exp(-d*x-c)/b^2-1/2*d*exp(-d*x-c)/b^3/(b*d*x+a*d)*a^2+1/2*d/b^4*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b
)*a^2+1/b^3*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a+1/2/d/b^2*exp(d*x+c)-1/2*d/b^4*exp(d*x+c)/(1/b*d*a+d*x)
*a^2-1/2*d/b^4*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^2+1/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 = 1.34762, size = 319, normalized size = 2.17 \begin{align*} \frac{1}{2} \,{\left (a^{2}{\left (\frac{e^{\left (-c + \frac{a d}{b}\right )} E_{1}\left (\frac{{\left (b x + a\right )} d}{b}\right )}{b^{4}} - \frac{e^{\left (c - \frac{a d}{b}\right )} E_{1}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{b^{4}}\right )} + \frac{2 \, 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^{2} 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^{2}} + \frac{4 \, a \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{3} d}\right )} d -{\left (\frac{a^{2}}{b^{4} x + a b^{3}} - \frac{x}{b^{2}} + \frac{2 \, a \log \left (b x + a\right )}{b^{3}}\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^2*(e^(-c + a*d/b)*exp_integral_e(1, (b*x + a)*d/b)/b^4 - e^(c - a*d/b)*exp_integral_e(1, -(b*x + a)*d/b
)/b^4) + 2*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^2*d) - ((d*x*e^c - e^c)*e^(d*x)/d^2 + (d*x + 1)*e^(-d*x - c)/d^2)/b^2 + 4*a*cosh(d*x + c)*log(b*x +
a)/(b^3*d))*d - (a^2/(b^4*x + a*b^3) - x/b^2 + 2*a*log(b*x + a)/b^3)*cosh(d*x + c)

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Fricas [A]  time = 1.99239, size = 570, normalized size = 3.88 \begin{align*} -\frac{2 \, a^{2} b d \cosh \left (d x + c\right ) -{\left ({\left (a^{3} d^{2} - 2 \, a^{2} b d +{\left (a^{2} b d^{2} - 2 \, a b^{2} d\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\left (a^{3} d^{2} + 2 \, a^{2} b d +{\left (a^{2} b d^{2} + 2 \, a b^{2} d\right )} x\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) - 2 \,{\left (b^{3} x + a b^{2}\right )} \sinh \left (d x + c\right ) +{\left ({\left (a^{3} d^{2} - 2 \, a^{2} b d +{\left (a^{2} b d^{2} - 2 \, a b^{2} d\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\left (a^{3} d^{2} + 2 \, a^{2} b d +{\left (a^{2} b d^{2} + 2 \, a b^{2} d\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^{5} d x + a b^{4} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a^2*b*d*cosh(d*x + c) - ((a^3*d^2 - 2*a^2*b*d + (a^2*b*d^2 - 2*a*b^2*d)*x)*Ei((b*d*x + a*d)/b) - (a^3*
d^2 + 2*a^2*b*d + (a^2*b*d^2 + 2*a*b^2*d)*x)*Ei(-(b*d*x + a*d)/b))*cosh(-(b*c - a*d)/b) - 2*(b^3*x + a*b^2)*si
nh(d*x + c) + ((a^3*d^2 - 2*a^2*b*d + (a^2*b*d^2 - 2*a*b^2*d)*x)*Ei((b*d*x + a*d)/b) + (a^3*d^2 + 2*a^2*b*d +
(a^2*b*d^2 + 2*a*b^2*d)*x)*Ei(-(b*d*x + a*d)/b))*sinh(-(b*c - a*d)/b))/(b^5*d*x + a*b^4*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \cosh{\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22872, size = 387, normalized size = 2.63 \begin{align*} \frac{a^{2} b d x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a^{2} b d x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + a^{3} d{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - 2 \, a b^{2} x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a^{3} d{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - 2 \, a b^{2} x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - 2 \, a^{2} b{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - 2 \, a^{2} b{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - a^{2} b e^{\left (d x + c\right )} - a^{2} b e^{\left (-d x - c\right )}}{2 \,{\left (b^{5} x + a b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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