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

Optimal. Leaf size=264 \[ -\frac{a^3 d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 b^6}+\frac{3 a^2 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^5}-\frac{a^3 d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 b^6}+\frac{3 a^2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^5}+\frac{a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}+\frac{a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}-\frac{3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac{3 a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^4}-\frac{3 a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{\sinh (c+d x)}{b^3 d} \]

[Out]

(a^3*Cosh[c + d*x])/(2*b^4*(a + b*x)^2) - (3*a^2*Cosh[c + d*x])/(b^4*(a + b*x)) - (3*a*Cosh[c - (a*d)/b]*CoshI
ntegral[(a*d)/b + d*x])/b^4 - (a^3*d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*b^6) + (3*a^2*d*CoshI
ntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^5 + Sinh[c + d*x]/(b^3*d) + (a^3*d*Sinh[c + d*x])/(2*b^5*(a + b*x)
) + (3*a^2*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^5 - (3*a*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b
+ d*x])/b^4 - (a^3*d^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*b^6)

________________________________________________________________________________________

Rubi [A]  time = 0.618072, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 15, 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} \[ -\frac{a^3 d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 b^6}+\frac{3 a^2 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^5}-\frac{a^3 d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 b^6}+\frac{3 a^2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^5}+\frac{a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}+\frac{a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}-\frac{3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac{3 a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^4}-\frac{3 a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{\sinh (c+d x)}{b^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Cosh[c + d*x])/(2*b^4*(a + b*x)^2) - (3*a^2*Cosh[c + d*x])/(b^4*(a + b*x)) - (3*a*Cosh[c - (a*d)/b]*CoshI
ntegral[(a*d)/b + d*x])/b^4 - (a^3*d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*b^6) + (3*a^2*d*CoshI
ntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^5 + Sinh[c + d*x]/(b^3*d) + (a^3*d*Sinh[c + d*x])/(2*b^5*(a + b*x)
) + (3*a^2*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^5 - (3*a*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b
+ d*x])/b^4 - (a^3*d^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*b^6)

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^3 \cosh (c+d x)}{(a+b x)^3} \, dx &=\int \left (\frac{\cosh (c+d x)}{b^3}-\frac{a^3 \cosh (c+d x)}{b^3 (a+b x)^3}+\frac{3 a^2 \cosh (c+d x)}{b^3 (a+b x)^2}-\frac{3 a \cosh (c+d x)}{b^3 (a+b x)}\right ) \, dx\\ &=\frac{\int \cosh (c+d x) \, dx}{b^3}-\frac{(3 a) \int \frac{\cosh (c+d x)}{a+b x} \, dx}{b^3}+\frac{\left (3 a^2\right ) \int \frac{\cosh (c+d x)}{(a+b x)^2} \, dx}{b^3}-\frac{a^3 \int \frac{\cosh (c+d x)}{(a+b x)^3} \, dx}{b^3}\\ &=\frac{a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}-\frac{3 a^2 \cosh (c+d x)}{b^4 (a+b x)}+\frac{\sinh (c+d x)}{b^3 d}+\frac{\left (3 a^2 d\right ) \int \frac{\sinh (c+d x)}{a+b x} \, dx}{b^4}-\frac{\left (a^3 d\right ) \int \frac{\sinh (c+d x)}{(a+b x)^2} \, dx}{2 b^4}-\frac{\left (3 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^3}-\frac{\left (3 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^3}\\ &=\frac{a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}-\frac{3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac{3 a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^4}+\frac{\sinh (c+d x)}{b^3 d}+\frac{a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}-\frac{3 a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^4}-\frac{\left (a^3 d^2\right ) \int \frac{\cosh (c+d x)}{a+b x} \, dx}{2 b^5}+\frac{\left (3 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^4}+\frac{\left (3 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^4}\\ &=\frac{a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}-\frac{3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac{3 a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^4}+\frac{3 a^2 d \text{Chi}\left (\frac{a d}{b}+d x\right ) \sinh \left (c-\frac{a d}{b}\right )}{b^5}+\frac{\sinh (c+d x)}{b^3 d}+\frac{a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}+\frac{3 a^2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^5}-\frac{3 a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^4}-\frac{\left (a^3 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^5}-\frac{\left (a^3 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^5}\\ &=\frac{a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}-\frac{3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac{3 a \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^4}-\frac{a^3 d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{2 b^6}+\frac{3 a^2 d \text{Chi}\left (\frac{a d}{b}+d x\right ) \sinh \left (c-\frac{a d}{b}\right )}{b^5}+\frac{\sinh (c+d x)}{b^3 d}+\frac{a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}+\frac{3 a^2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^5}-\frac{3 a \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^4}-\frac{a^3 d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{2 b^6}\\ \end{align*}

Mathematica [A]  time = 0.999999, size = 236, normalized size = 0.89 \[ -\frac{a d (a+b x)^2 \left (\text{Chi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (\left (a^2 d^2+6 b^2\right ) \cosh \left (c-\frac{a d}{b}\right )-6 a b d \sinh \left (c-\frac{a d}{b}\right )\right )+\text{Shi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (\left (a^2 d^2+6 b^2\right ) \sinh \left (c-\frac{a d}{b}\right )-6 a b d \cosh \left (c-\frac{a d}{b}\right )\right )\right )+b \cosh (d x) \left (a^2 b d \cosh (c) (5 a+6 b x)-\sinh (c) (a+b x) \left (a^3 d^2+2 a b^2+2 b^3 x\right )\right )-b \sinh (d x) \left (\cosh (c) (a+b x) \left (a^3 d^2+2 a b^2+2 b^3 x\right )-a^2 b d \sinh (c) (5 a+6 b x)\right )}{2 b^6 d (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*Cosh[d*x]*(a^2*b*d*(5*a + 6*b*x)*Cosh[c] - (a + b*x)*(2*a*b^2 + a^3*d^2 + 2*b^3*x)*Sinh[c]) - b*((a + b*x)
*(2*a*b^2 + a^3*d^2 + 2*b^3*x)*Cosh[c] - a^2*b*d*(5*a + 6*b*x)*Sinh[c])*Sinh[d*x] + a*d*(a + b*x)^2*(CoshInteg
ral[d*(a/b + x)]*((6*b^2 + a^2*d^2)*Cosh[c - (a*d)/b] - 6*a*b*d*Sinh[c - (a*d)/b]) + (-6*a*b*d*Cosh[c - (a*d)/
b] + (6*b^2 + a^2*d^2)*Sinh[c - (a*d)/b])*SinhIntegral[d*(a/b + x)]))/(2*b^6*d*(a + b*x)^2)

________________________________________________________________________________________

Maple [B]  time = 0.117, size = 571, normalized size = 2.2 \begin{align*} -{\frac{{{\rm e}^{-dx-c}}}{2\,d{b}^{3}}}+{\frac{{d}^{2}{a}^{3}}{4\,{b}^{6}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{3\,d{a}^{2}}{2\,{b}^{5}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{3\,a}{2\,{b}^{4}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{5\,{d}^{2}{{\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{3\,{d}^{2}{{\rm e}^{-dx-c}}{a}^{2}x}{2\,{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}x}{4\,{b}^{4} \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}^{4}}{4\,{b}^{5} \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}-{\frac{3\,d{a}^{2}}{2\,{b}^{5}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }+{\frac{3\,a}{2\,{b}^{4}}{{\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}^{3}}}+{\frac{{d}^{2}{{\rm e}^{dx+c}}{a}^{3}}{4\,{b}^{6}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}+{\frac{{d}^{2}{a}^{3}}{4\,{b}^{6}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }-{\frac{3\,d{{\rm e}^{dx+c}}{a}^{2}}{2\,{b}^{5}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}+{\frac{{d}^{2}{{\rm e}^{dx+c}}{a}^{3}}{4\,{b}^{6}} \left ({\frac{da}{b}}+dx \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cosh(d*x+c)/(b*x+a)^3,x)

[Out]

-1/2/d*exp(-d*x-c)/b^3+1/4*d^2/b^6*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^3+3/2*d/b^5*exp((a*d-b*c)/b)*Ei(
1,d*x+c+(a*d-b*c)/b)*a^2+3/2/b^4*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a-5/4*d^2*exp(-d*x-c)/b^4/(b^2*d^2*x
^2+2*a*b*d^2*x+a^2*d^2)*a^3-3/2*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*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*x-1/4*d^3*exp(-d*x-c)/b^5/(b^2*d^2*x^2+2*a*b*d^2*x+a^2*d^2)*a^4-3/
2*d/b^5*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^2+3/2/b^4*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a+1/
2/d/b^3*exp(d*x+c)+1/4*d^2/b^6*exp(d*x+c)/(1/b*d*a+d*x)*a^3+1/4*d^2/b^6*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c
)/b)*a^3-3/2*d/b^5*exp(d*x+c)/(1/b*d*a+d*x)*a^2+1/4*d^2/b^6*exp(d*x+c)/(1/b*d*a+d*x)^2*a^3

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3}{2} \, a^{2} d \int \frac{x e^{\left (d x + c\right )}}{b^{5} d^{2} x^{4} + 4 \, a b^{4} d^{2} x^{3} + 6 \, a^{2} b^{3} d^{2} x^{2} + 4 \, a^{3} b^{2} d^{2} x + a^{4} b d^{2}}\,{d x} - \frac{3}{2} \, a^{2} d \int \frac{x}{b^{5} d^{2} x^{4} e^{\left (d x + c\right )} + 4 \, a b^{4} d^{2} x^{3} e^{\left (d x + c\right )} + 6 \, a^{2} b^{3} d^{2} x^{2} e^{\left (d x + c\right )} + 4 \, a^{3} b^{2} d^{2} x e^{\left (d x + c\right )} + a^{4} b d^{2} e^{\left (d x + c\right )}}\,{d x} - 3 \, a b \int \frac{x e^{\left (d x + c\right )}}{b^{5} d^{2} x^{4} + 4 \, a b^{4} d^{2} x^{3} + 6 \, a^{2} b^{3} d^{2} x^{2} + 4 \, a^{3} b^{2} d^{2} x + a^{4} b d^{2}}\,{d x} - 3 \, a b \int \frac{x}{b^{5} d^{2} x^{4} e^{\left (d x + c\right )} + 4 \, a b^{4} d^{2} x^{3} e^{\left (d x + c\right )} + 6 \, a^{2} b^{3} d^{2} x^{2} e^{\left (d x + c\right )} + 4 \, a^{3} b^{2} d^{2} x e^{\left (d x + c\right )} + a^{4} b d^{2} e^{\left (d x + c\right )}}\,{d x} + \frac{{\left (b d x^{3} e^{\left (2 \, c\right )} - 3 \, a x e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} -{\left (b d x^{3} + 3 \, a x\right )} e^{\left (-d x\right )}}{2 \,{\left (b^{4} d^{2} x^{3} e^{c} + 3 \, a b^{3} d^{2} x^{2} e^{c} + 3 \, a^{2} b^{2} d^{2} x e^{c} + a^{3} b d^{2} e^{c}\right )}} - \frac{3 \, a^{2} 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^{2} d^{2}} - \frac{3 \, a^{2} 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^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a^2*d*integrate(x*e^(d*x + c)/(b^5*d^2*x^4 + 4*a*b^4*d^2*x^3 + 6*a^2*b^3*d^2*x^2 + 4*a^3*b^2*d^2*x + a^4*b
*d^2), x) - 3/2*a^2*d*integrate(x/(b^5*d^2*x^4*e^(d*x + c) + 4*a*b^4*d^2*x^3*e^(d*x + c) + 6*a^2*b^3*d^2*x^2*e
^(d*x + c) + 4*a^3*b^2*d^2*x*e^(d*x + c) + a^4*b*d^2*e^(d*x + c)), x) - 3*a*b*integrate(x*e^(d*x + c)/(b^5*d^2
*x^4 + 4*a*b^4*d^2*x^3 + 6*a^2*b^3*d^2*x^2 + 4*a^3*b^2*d^2*x + a^4*b*d^2), x) - 3*a*b*integrate(x/(b^5*d^2*x^4
*e^(d*x + c) + 4*a*b^4*d^2*x^3*e^(d*x + c) + 6*a^2*b^3*d^2*x^2*e^(d*x + c) + 4*a^3*b^2*d^2*x*e^(d*x + c) + a^4
*b*d^2*e^(d*x + c)), x) + 1/2*((b*d*x^3*e^(2*c) - 3*a*x*e^(2*c))*e^(d*x) - (b*d*x^3 + 3*a*x)*e^(-d*x))/(b^4*d^
2*x^3*e^c + 3*a*b^3*d^2*x^2*e^c + 3*a^2*b^2*d^2*x*e^c + a^3*b*d^2*e^c) - 3/2*a^2*e^(-c + a*d/b)*exp_integral_e
(4, (b*x + a)*d/b)/((b*x + a)^3*b^2*d^2) - 3/2*a^2*e^(c - a*d/b)*exp_integral_e(4, -(b*x + a)*d/b)/((b*x + a)^
3*b^2*d^2)

________________________________________________________________________________________

Fricas [B]  time = 2.50905, size = 1142, normalized size = 4.33 \begin{align*} -\frac{2 \,{\left (6 \, a^{2} b^{3} d x + 5 \, a^{3} b^{2} d\right )} \cosh \left (d x + c\right ) +{\left ({\left (a^{5} d^{3} - 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d +{\left (a^{3} b^{2} d^{3} - 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \,{\left (a^{4} b d^{3} - 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\left (a^{5} d^{3} + 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d +{\left (a^{3} b^{2} d^{3} + 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \,{\left (a^{4} b d^{3} + 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} 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 (a^{4} b d^{2} + 2 \, b^{5} x^{2} + 2 \, a^{2} b^{3} +{\left (a^{3} b^{2} d^{2} + 4 \, a b^{4}\right )} x\right )} \sinh \left (d x + c\right ) -{\left ({\left (a^{5} d^{3} - 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d +{\left (a^{3} b^{2} d^{3} - 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \,{\left (a^{4} b d^{3} - 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\left (a^{5} d^{3} + 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d +{\left (a^{3} b^{2} d^{3} + 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \,{\left (a^{4} b d^{3} + 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} 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 )}{4 \,{\left (b^{8} d x^{2} + 2 \, a b^{7} d x + a^{2} b^{6} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \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**3*cosh(d*x+c)/(b*x+a)**3,x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.16071, size = 1019, normalized size = 3.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(a^3*b^2*d^2*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + a^3*b^2*d^2*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b)
+ 2*a^4*b*d^2*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) - 6*a^2*b^3*d*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 2*a^4*
b*d^2*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 6*a^2*b^3*d*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + a^5*d^2*Ei
((b*d*x + a*d)/b)*e^(c - a*d/b) - 12*a^3*b^2*d*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 6*a*b^4*x^2*Ei((b*d*x + a
*d)/b)*e^(c - a*d/b) + a^5*d^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 12*a^3*b^2*d*x*Ei(-(b*d*x + a*d)/b)*e^(-c
+ a*d/b) + 6*a*b^4*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - a^3*b^2*d*x*e^(d*x + c) + a^3*b^2*d*x*e^(-d*x -
c) - 6*a^4*b*d*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 12*a^2*b^3*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 6*a^4*b*d*
Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 12*a^2*b^3*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - a^4*b*d*e^(d*x + c) +
6*a^2*b^3*x*e^(d*x + c) + a^4*b*d*e^(-d*x - c) + 6*a^2*b^3*x*e^(-d*x - c) + 6*a^3*b^2*Ei((b*d*x + a*d)/b)*e^(
c - a*d/b) + 6*a^3*b^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 5*a^3*b^2*e^(d*x + c) + 5*a^3*b^2*e^(-d*x - c))/(
b^8*x^2 + 2*a*b^7*x + a^2*b^6)