3.34 \(\int \frac{x^2 \cosh (c+d x)}{(a+b x)^3} \, dx\)

Optimal. Leaf size=241 \[ \frac{a^2 d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 b^5}+\frac{a^2 d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 b^5}-\frac{a^2 d \sinh (c+d x)}{2 b^4 (a+b x)}-\frac{a^2 \cosh (c+d x)}{2 b^3 (a+b x)^2}-\frac{2 a d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^3}+\frac{\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^3}-\frac{2 a d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{2 a \cosh (c+d x)}{b^3 (a+b x)} \]

[Out]

-(a^2*Cosh[c + d*x])/(2*b^3*(a + b*x)^2) + (2*a*Cosh[c + d*x])/(b^3*(a + b*x)) + (Cosh[c - (a*d)/b]*CoshIntegr
al[(a*d)/b + d*x])/b^3 + (a^2*d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*b^5) - (2*a*d*CoshIntegral
[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^4 - (a^2*d*Sinh[c + d*x])/(2*b^4*(a + b*x)) - (2*a*d*Cosh[c - (a*d)/b]*Si
nhIntegral[(a*d)/b + d*x])/b^4 + (Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^3 + (a^2*d^2*Sinh[c - (a*d)
/b]*SinhIntegral[(a*d)/b + d*x])/(2*b^5)

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Rubi [A]  time = 0.530363, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6742, 3297, 3303, 3298, 3301} \[ \frac{a^2 d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 b^5}+\frac{a^2 d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 b^5}-\frac{a^2 d \sinh (c+d x)}{2 b^4 (a+b x)}-\frac{a^2 \cosh (c+d x)}{2 b^3 (a+b x)^2}-\frac{2 a d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^3}+\frac{\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^3}-\frac{2 a d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{2 a \cosh (c+d x)}{b^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*Cosh[c + d*x])/(a + b*x)^3,x]

[Out]

-(a^2*Cosh[c + d*x])/(2*b^3*(a + b*x)^2) + (2*a*Cosh[c + d*x])/(b^3*(a + b*x)) + (Cosh[c - (a*d)/b]*CoshIntegr
al[(a*d)/b + d*x])/b^3 + (a^2*d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*b^5) - (2*a*d*CoshIntegral
[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^4 - (a^2*d*Sinh[c + d*x])/(2*b^4*(a + b*x)) - (2*a*d*Cosh[c - (a*d)/b]*Si
nhIntegral[(a*d)/b + d*x])/b^4 + (Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^3 + (a^2*d^2*Sinh[c - (a*d)
/b]*SinhIntegral[(a*d)/b + d*x])/(2*b^5)

Rule 6742

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

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)^3} \, dx &=\int \left (\frac{a^2 \cosh (c+d x)}{b^2 (a+b x)^3}-\frac{2 a \cosh (c+d x)}{b^2 (a+b x)^2}+\frac{\cosh (c+d x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{\int \frac{\cosh (c+d x)}{a+b x} \, dx}{b^2}-\frac{(2 a) \int \frac{\cosh (c+d x)}{(a+b x)^2} \, dx}{b^2}+\frac{a^2 \int \frac{\cosh (c+d x)}{(a+b x)^3} \, dx}{b^2}\\ &=-\frac{a^2 \cosh (c+d x)}{2 b^3 (a+b x)^2}+\frac{2 a \cosh (c+d x)}{b^3 (a+b x)}-\frac{(2 a d) \int \frac{\sinh (c+d x)}{a+b x} \, dx}{b^3}+\frac{\left (a^2 d\right ) \int \frac{\sinh (c+d x)}{(a+b x)^2} \, dx}{2 b^3}+\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^2}+\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^2}\\ &=-\frac{a^2 \cosh (c+d x)}{2 b^3 (a+b x)^2}+\frac{2 a \cosh (c+d x)}{b^3 (a+b x)}+\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^3}-\frac{a^2 d \sinh (c+d x)}{2 b^4 (a+b x)}+\frac{\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^2\right ) \int \frac{\cosh (c+d x)}{a+b x} \, dx}{2 b^4}-\frac{\left (2 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^3}-\frac{\left (2 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^3}\\ &=-\frac{a^2 \cosh (c+d x)}{2 b^3 (a+b x)^2}+\frac{2 a \cosh (c+d x)}{b^3 (a+b x)}+\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^3}-\frac{2 a d \text{Chi}\left (\frac{a d}{b}+d x\right ) \sinh \left (c-\frac{a d}{b}\right )}{b^4}-\frac{a^2 d \sinh (c+d x)}{2 b^4 (a+b x)}-\frac{2 a d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^4}+\frac{\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^2 \cosh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cosh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{2 b^4}+\frac{\left (a^2 d^2 \sinh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sinh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{2 b^4}\\ &=-\frac{a^2 \cosh (c+d x)}{2 b^3 (a+b x)^2}+\frac{2 a \cosh (c+d x)}{b^3 (a+b x)}+\frac{\cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{a^2 d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{2 b^5}-\frac{2 a d \text{Chi}\left (\frac{a d}{b}+d x\right ) \sinh \left (c-\frac{a d}{b}\right )}{b^4}-\frac{a^2 d \sinh (c+d x)}{2 b^4 (a+b x)}-\frac{2 a d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^4}+\frac{\sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^3}+\frac{a^2 d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{2 b^5}\\ \end{align*}

Mathematica [A]  time = 0.928054, size = 153, normalized size = 0.63 \[ \frac{\text{Chi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (\left (a^2 d^2+2 b^2\right ) \cosh \left (c-\frac{a d}{b}\right )-4 a b d \sinh \left (c-\frac{a d}{b}\right )\right )+\text{Shi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (\left (a^2 d^2+2 b^2\right ) \sinh \left (c-\frac{a d}{b}\right )-4 a b d \cosh \left (c-\frac{a d}{b}\right )\right )-\frac{a b (a d (a+b x) \sinh (c+d x)-b (3 a+4 b x) \cosh (c+d x))}{(a+b x)^2}}{2 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*Cosh[c + d*x])/(a + b*x)^3,x]

[Out]

(CoshIntegral[d*(a/b + x)]*((2*b^2 + a^2*d^2)*Cosh[c - (a*d)/b] - 4*a*b*d*Sinh[c - (a*d)/b]) - (a*b*(-(b*(3*a
+ 4*b*x)*Cosh[c + d*x]) + a*d*(a + b*x)*Sinh[c + d*x]))/(a + b*x)^2 + (-4*a*b*d*Cosh[c - (a*d)/b] + (2*b^2 + a
^2*d^2)*Sinh[c - (a*d)/b])*SinhIntegral[d*(a/b + x)])/(2*b^5)

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Maple [B]  time = 0.047, size = 527, normalized size = 2.2 \begin{align*}{\frac{{d}^{3}{{\rm e}^{-dx-c}}{a}^{2}x}{4\,{b}^{3} \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}+{\frac{{d}^{3}{{\rm e}^{-dx-c}}{a}^{3}}{4\,{b}^{4} \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}+{\frac{{d}^{2}{{\rm e}^{-dx-c}}ax}{{b}^{2} \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}+{\frac{3\,{d}^{2}{{\rm e}^{-dx-c}}{a}^{2}}{4\,{b}^{3} \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}-{\frac{{a}^{2}{d}^{2}}{4\,{b}^{5}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{da}{{b}^{4}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{1}{2\,{b}^{3}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{1}{2\,{b}^{3}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }-{\frac{{a}^{2}{d}^{2}}{4\,{b}^{5}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }-{\frac{{d}^{2}{{\rm e}^{dx+c}}{a}^{2}}{4\,{b}^{5}} \left ({\frac{da}{b}}+dx \right ) ^{-2}}-{\frac{{d}^{2}{{\rm e}^{dx+c}}{a}^{2}}{4\,{b}^{5}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}+{\frac{d{{\rm e}^{dx+c}}a}{{b}^{4}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}+{\frac{da}{{b}^{4}}{{\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)^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{3}{2} \, a d \int \frac{x e^{\left (d x + c\right )}}{b^{4} d^{2} x^{4} + 4 \, a b^{3} d^{2} x^{3} + 6 \, a^{2} b^{2} d^{2} x^{2} + 4 \, a^{3} b d^{2} x + a^{4} d^{2}}\,{d x} + \frac{3}{2} \, a d \int \frac{x}{b^{4} d^{2} x^{4} e^{\left (d x + c\right )} + 4 \, a b^{3} d^{2} x^{3} e^{\left (d x + c\right )} + 6 \, a^{2} b^{2} d^{2} x^{2} e^{\left (d x + c\right )} + 4 \, a^{3} b d^{2} x e^{\left (d x + c\right )} + a^{4} d^{2} e^{\left (d x + c\right )}}\,{d x} + b \int \frac{x e^{\left (d x + c\right )}}{b^{4} d^{2} x^{4} + 4 \, a b^{3} d^{2} x^{3} + 6 \, a^{2} b^{2} d^{2} x^{2} + 4 \, a^{3} b d^{2} x + a^{4} d^{2}}\,{d x} + b \int \frac{x}{b^{4} d^{2} x^{4} e^{\left (d x + c\right )} + 4 \, a b^{3} d^{2} x^{3} e^{\left (d x + c\right )} + 6 \, a^{2} b^{2} d^{2} x^{2} e^{\left (d x + c\right )} + 4 \, a^{3} b d^{2} x e^{\left (d x + c\right )} + a^{4} d^{2} e^{\left (d x + c\right )}}\,{d x} + \frac{{\left (d x^{2} e^{\left (2 \, c\right )} + x e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} -{\left (d x^{2} - x\right )} e^{\left (-d x\right )}}{2 \,{\left (b^{3} d^{2} x^{3} e^{c} + 3 \, a b^{2} d^{2} x^{2} e^{c} + 3 \, a^{2} b d^{2} x e^{c} + a^{3} d^{2} e^{c}\right )}} + \frac{a e^{\left (-c + \frac{a d}{b}\right )} E_{4}\left (\frac{{\left (b x + a\right )} d}{b}\right )}{2 \,{\left (b x + a\right )}^{3} b d^{2}} + \frac{a e^{\left (c - \frac{a d}{b}\right )} E_{4}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{2 \,{\left (b x + a\right )}^{3} b d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(d*x+c)/(b*x+a)^3,x, algorithm="maxima")

[Out]

-3/2*a*d*integrate(x*e^(d*x + c)/(b^4*d^2*x^4 + 4*a*b^3*d^2*x^3 + 6*a^2*b^2*d^2*x^2 + 4*a^3*b*d^2*x + a^4*d^2)
, x) + 3/2*a*d*integrate(x/(b^4*d^2*x^4*e^(d*x + c) + 4*a*b^3*d^2*x^3*e^(d*x + c) + 6*a^2*b^2*d^2*x^2*e^(d*x +
c) + 4*a^3*b*d^2*x*e^(d*x + c) + a^4*d^2*e^(d*x + c)), x) + b*integrate(x*e^(d*x + c)/(b^4*d^2*x^4 + 4*a*b^3*
d^2*x^3 + 6*a^2*b^2*d^2*x^2 + 4*a^3*b*d^2*x + a^4*d^2), x) + b*integrate(x/(b^4*d^2*x^4*e^(d*x + c) + 4*a*b^3*
d^2*x^3*e^(d*x + c) + 6*a^2*b^2*d^2*x^2*e^(d*x + c) + 4*a^3*b*d^2*x*e^(d*x + c) + a^4*d^2*e^(d*x + c)), x) + 1
/2*((d*x^2*e^(2*c) + x*e^(2*c))*e^(d*x) - (d*x^2 - x)*e^(-d*x))/(b^3*d^2*x^3*e^c + 3*a*b^2*d^2*x^2*e^c + 3*a^2
*b*d^2*x*e^c + a^3*d^2*e^c) + 1/2*a*e^(-c + a*d/b)*exp_integral_e(4, (b*x + a)*d/b)/((b*x + a)^3*b*d^2) + 1/2*
a*e^(c - a*d/b)*exp_integral_e(4, -(b*x + a)*d/b)/((b*x + a)^3*b*d^2)

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Fricas [A]  time = 2.47278, size = 973, normalized size = 4.04 \begin{align*} \frac{2 \,{\left (4 \, a b^{3} x + 3 \, a^{2} b^{2}\right )} \cosh \left (d x + c\right ) +{\left ({\left (a^{4} d^{2} - 4 \, a^{3} b d + 2 \, a^{2} b^{2} +{\left (a^{2} b^{2} d^{2} - 4 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \,{\left (a^{3} b d^{2} - 4 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\left (a^{4} d^{2} + 4 \, a^{3} b d + 2 \, a^{2} b^{2} +{\left (a^{2} b^{2} d^{2} + 4 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \,{\left (a^{3} b d^{2} + 4 \, a^{2} b^{2} d + 2 \, a b^{3}\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 (a^{2} b^{2} d x + a^{3} b d\right )} \sinh \left (d x + c\right ) -{\left ({\left (a^{4} d^{2} - 4 \, a^{3} b d + 2 \, a^{2} b^{2} +{\left (a^{2} b^{2} d^{2} - 4 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \,{\left (a^{3} b d^{2} - 4 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\left (a^{4} d^{2} + 4 \, a^{3} b d + 2 \, a^{2} b^{2} +{\left (a^{2} b^{2} d^{2} + 4 \, a b^{3} d + 2 \, b^{4}\right )} x^{2} + 2 \,{\left (a^{3} b d^{2} + 4 \, a^{2} b^{2} d + 2 \, a b^{3}\right )} x\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{4 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(d*x+c)/(b*x+a)^3,x, algorithm="fricas")

[Out]

1/4*(2*(4*a*b^3*x + 3*a^2*b^2)*cosh(d*x + c) + ((a^4*d^2 - 4*a^3*b*d + 2*a^2*b^2 + (a^2*b^2*d^2 - 4*a*b^3*d +
2*b^4)*x^2 + 2*(a^3*b*d^2 - 4*a^2*b^2*d + 2*a*b^3)*x)*Ei((b*d*x + a*d)/b) + (a^4*d^2 + 4*a^3*b*d + 2*a^2*b^2 +
(a^2*b^2*d^2 + 4*a*b^3*d + 2*b^4)*x^2 + 2*(a^3*b*d^2 + 4*a^2*b^2*d + 2*a*b^3)*x)*Ei(-(b*d*x + a*d)/b))*cosh(-
(b*c - a*d)/b) - 2*(a^2*b^2*d*x + a^3*b*d)*sinh(d*x + c) - ((a^4*d^2 - 4*a^3*b*d + 2*a^2*b^2 + (a^2*b^2*d^2 -
4*a*b^3*d + 2*b^4)*x^2 + 2*(a^3*b*d^2 - 4*a^2*b^2*d + 2*a*b^3)*x)*Ei((b*d*x + a*d)/b) - (a^4*d^2 + 4*a^3*b*d +
2*a^2*b^2 + (a^2*b^2*d^2 + 4*a*b^3*d + 2*b^4)*x^2 + 2*(a^3*b*d^2 + 4*a^2*b^2*d + 2*a*b^3)*x)*Ei(-(b*d*x + a*d
)/b))*sinh(-(b*c - a*d)/b))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)

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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 )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cosh(d*x+c)/(b*x+a)**3,x)

[Out]

Integral(x**2*cosh(c + d*x)/(a + b*x)**3, x)

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Giac [B]  time = 1.18058, size = 1000, normalized size = 4.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(d*x+c)/(b*x+a)^3,x, algorithm="giac")

[Out]

1/4*(a^2*b^2*d^2*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + a^2*b^2*d^2*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) +
2*a^3*b*d^2*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) - 4*a*b^3*d*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 2*a^3*b*d
^2*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 4*a*b^3*d*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + a^4*d^2*Ei((b*d
*x + a*d)/b)*e^(c - a*d/b) - 8*a^2*b^2*d*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 2*b^4*x^2*Ei((b*d*x + a*d)/b)*e
^(c - a*d/b) + a^4*d^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 8*a^2*b^2*d*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b)
+ 2*b^4*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - a^2*b^2*d*x*e^(d*x + c) + a^2*b^2*d*x*e^(-d*x - c) - 4*a^3*
b*d*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 4*a*b^3*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 4*a^3*b*d*Ei(-(b*d*x + a
*d)/b)*e^(-c + a*d/b) + 4*a*b^3*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - a^3*b*d*e^(d*x + c) + 4*a*b^3*x*e^(d*x
+ c) + a^3*b*d*e^(-d*x - c) + 4*a*b^3*x*e^(-d*x - c) + 2*a^2*b^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 2*a^2*b^
2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 3*a^2*b^2*e^(d*x + c) + 3*a^2*b^2*e^(-d*x - c))/(b^7*x^2 + 2*a*b^6*x +
a^2*b^5)