3.26 \(\int \frac{x^4 \cosh (c+d x)}{(a+b x)^2} \, dx\)

Optimal. Leaf size=231 \[ \frac{a^4 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^6}-\frac{4 a^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^5}-\frac{4 a^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^5}+\frac{a^4 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^6}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}-\frac{a^4 \cosh (c+d x)}{b^5 (a+b x)}+\frac{2 a \cosh (c+d x)}{b^3 d^2}-\frac{2 a x \sinh (c+d x)}{b^3 d}+\frac{2 \sinh (c+d x)}{b^2 d^3}-\frac{2 x \cosh (c+d x)}{b^2 d^2}+\frac{x^2 \sinh (c+d x)}{b^2 d} \]

[Out]

(2*a*Cosh[c + d*x])/(b^3*d^2) - (2*x*Cosh[c + d*x])/(b^2*d^2) - (a^4*Cosh[c + d*x])/(b^5*(a + b*x)) - (4*a^3*C
osh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^5 + (a^4*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^6
+ (2*Sinh[c + d*x])/(b^2*d^3) + (3*a^2*Sinh[c + d*x])/(b^4*d) - (2*a*x*Sinh[c + d*x])/(b^3*d) + (x^2*Sinh[c +
d*x])/(b^2*d) + (a^4*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^6 - (4*a^3*Sinh[c - (a*d)/b]*SinhInteg
ral[(a*d)/b + d*x])/b^5

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Rubi [A]  time = 0.536144, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {6742, 2637, 3296, 2638, 3297, 3303, 3298, 3301} \[ \frac{a^4 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^6}-\frac{4 a^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^5}-\frac{4 a^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^5}+\frac{a^4 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^6}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}-\frac{a^4 \cosh (c+d x)}{b^5 (a+b x)}+\frac{2 a \cosh (c+d x)}{b^3 d^2}-\frac{2 a x \sinh (c+d x)}{b^3 d}+\frac{2 \sinh (c+d x)}{b^2 d^3}-\frac{2 x \cosh (c+d x)}{b^2 d^2}+\frac{x^2 \sinh (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*Cosh[c + d*x])/(a + b*x)^2,x]

[Out]

(2*a*Cosh[c + d*x])/(b^3*d^2) - (2*x*Cosh[c + d*x])/(b^2*d^2) - (a^4*Cosh[c + d*x])/(b^5*(a + b*x)) - (4*a^3*C
osh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^5 + (a^4*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^6
+ (2*Sinh[c + d*x])/(b^2*d^3) + (3*a^2*Sinh[c + d*x])/(b^4*d) - (2*a*x*Sinh[c + d*x])/(b^3*d) + (x^2*Sinh[c +
d*x])/(b^2*d) + (a^4*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^6 - (4*a^3*Sinh[c - (a*d)/b]*SinhInteg
ral[(a*d)/b + d*x])/b^5

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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[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^4 \cosh (c+d x)}{(a+b x)^2} \, dx &=\int \left (\frac{3 a^2 \cosh (c+d x)}{b^4}-\frac{2 a x \cosh (c+d x)}{b^3}+\frac{x^2 \cosh (c+d x)}{b^2}+\frac{a^4 \cosh (c+d x)}{b^4 (a+b x)^2}-\frac{4 a^3 \cosh (c+d x)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{\left (3 a^2\right ) \int \cosh (c+d x) \, dx}{b^4}-\frac{\left (4 a^3\right ) \int \frac{\cosh (c+d x)}{a+b x} \, dx}{b^4}+\frac{a^4 \int \frac{\cosh (c+d x)}{(a+b x)^2} \, dx}{b^4}-\frac{(2 a) \int x \cosh (c+d x) \, dx}{b^3}+\frac{\int x^2 \cosh (c+d x) \, dx}{b^2}\\ &=-\frac{a^4 \cosh (c+d x)}{b^5 (a+b x)}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}-\frac{2 a x \sinh (c+d x)}{b^3 d}+\frac{x^2 \sinh (c+d x)}{b^2 d}+\frac{(2 a) \int \sinh (c+d x) \, dx}{b^3 d}-\frac{2 \int x \sinh (c+d x) \, dx}{b^2 d}+\frac{\left (a^4 d\right ) \int \frac{\sinh (c+d x)}{a+b x} \, dx}{b^5}-\frac{\left (4 a^3 \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^4}-\frac{\left (4 a^3 \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^4}\\ &=\frac{2 a \cosh (c+d x)}{b^3 d^2}-\frac{2 x \cosh (c+d x)}{b^2 d^2}-\frac{a^4 \cosh (c+d x)}{b^5 (a+b x)}-\frac{4 a^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^5}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}-\frac{2 a x \sinh (c+d x)}{b^3 d}+\frac{x^2 \sinh (c+d x)}{b^2 d}-\frac{4 a^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^5}+\frac{2 \int \cosh (c+d x) \, dx}{b^2 d^2}+\frac{\left (a^4 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^5}+\frac{\left (a^4 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^5}\\ &=\frac{2 a \cosh (c+d x)}{b^3 d^2}-\frac{2 x \cosh (c+d x)}{b^2 d^2}-\frac{a^4 \cosh (c+d x)}{b^5 (a+b x)}-\frac{4 a^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^5}+\frac{a^4 d \text{Chi}\left (\frac{a d}{b}+d x\right ) \sinh \left (c-\frac{a d}{b}\right )}{b^6}+\frac{2 \sinh (c+d x)}{b^2 d^3}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}-\frac{2 a x \sinh (c+d x)}{b^3 d}+\frac{x^2 \sinh (c+d x)}{b^2 d}+\frac{a^4 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^6}-\frac{4 a^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^5}\\ \end{align*}

Mathematica [A]  time = 1.20078, size = 173, normalized size = 0.75 \[ \frac{\frac{b^2 \left (3 a^2 d^2-2 a b d^2 x+b^2 \left (d^2 x^2+2\right )\right ) \sinh (c+d x)}{d^3}-\frac{b \left (-2 a^2 b^2+a^4 d^2+2 b^4 x^2\right ) \cosh (c+d x)}{d^2 (a+b x)}+a^3 \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \sinh \left (c-\frac{a d}{b}\right )-4 b \cosh \left (c-\frac{a d}{b}\right )\right )+a^3 \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \cosh \left (c-\frac{a d}{b}\right )-4 b \sinh \left (c-\frac{a d}{b}\right )\right )}{b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*Cosh[c + d*x])/(a + b*x)^2,x]

[Out]

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

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Maple [A]  time = 0.122, size = 431, normalized size = 1.9 \begin{align*} -{\frac{{{\rm e}^{-dx-c}}}{{d}^{3}{b}^{2}}}+{\frac{d{a}^{4}}{2\,{b}^{6}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+2\,{\frac{{a}^{3}}{{b}^{5}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{{{\rm e}^{-dx-c}}x}{{b}^{2}{d}^{2}}}+{\frac{{{\rm e}^{-dx-c}}a}{{d}^{2}{b}^{3}}}-{\frac{{{\rm e}^{-dx-c}}{x}^{2}}{2\,d{b}^{2}}}-{\frac{3\,{{\rm e}^{-dx-c}}{a}^{2}}{2\,d{b}^{4}}}-{\frac{d{{\rm e}^{-dx-c}}{a}^{4}}{2\,{b}^{5} \left ( bdx+da \right ) }}+{\frac{{{\rm e}^{-dx-c}}ax}{d{b}^{3}}}-{\frac{a{{\rm e}^{dx+c}}x}{d{b}^{3}}}+{\frac{3\,{{\rm e}^{dx+c}}{a}^{2}}{2\,d{b}^{4}}}+{\frac{a{{\rm e}^{dx+c}}}{{d}^{2}{b}^{3}}}+{\frac{{{\rm e}^{dx+c}}}{{d}^{3}{b}^{2}}}-{\frac{d{{\rm e}^{dx+c}}{a}^{4}}{2\,{b}^{6}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}-{\frac{d{a}^{4}}{2\,{b}^{6}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }+2\,{\frac{{a}^{3}}{{b}^{5}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }+{\frac{{{\rm e}^{dx+c}}{x}^{2}}{2\,d{b}^{2}}}-{\frac{{{\rm e}^{dx+c}}x}{{b}^{2}{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*cosh(d*x+c)/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.51568, size = 548, normalized size = 2.37 \begin{align*} \frac{1}{6} \,{\left (3 \, a^{4}{\left (\frac{e^{\left (-c + \frac{a d}{b}\right )} E_{1}\left (\frac{{\left (b x + a\right )} d}{b}\right )}{b^{6}} - \frac{e^{\left (c - \frac{a d}{b}\right )} E_{1}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{b^{6}}\right )} + \frac{12 \, a^{3}{\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^{4} d} - \frac{9 \, a^{2}{\left (\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}}\right )}}{b^{4}} + \frac{3 \, a{\left (\frac{{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac{{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}\right )}}{b^{3}} - \frac{\frac{{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} e^{\left (d x\right )}}{d^{4}} + \frac{{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} e^{\left (-d x - c\right )}}{d^{4}}}{b^{2}} + \frac{24 \, a^{3} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{5} d}\right )} d - \frac{1}{3} \,{\left (\frac{3 \, a^{4}}{b^{6} x + a b^{5}} + \frac{12 \, a^{3} \log \left (b x + a\right )}{b^{5}} - \frac{b^{2} x^{3} - 3 \, a b x^{2} + 9 \, a^{2} x}{b^{4}}\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*cosh(d*x+c)/(b*x+a)^2,x, algorithm="maxima")

[Out]

1/6*(3*a^4*(e^(-c + a*d/b)*exp_integral_e(1, (b*x + a)*d/b)/b^6 - e^(c - a*d/b)*exp_integral_e(1, -(b*x + a)*d
/b)/b^6) + 12*a^3*(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^4*d) - 9*a^2*((d*x*e^c - e^c)*e^(d*x)/d^2 + (d*x + 1)*e^(-d*x - c)/d^2)/b^4 + 3*a*((d^2*x^2*e^c
- 2*d*x*e^c + 2*e^c)*e^(d*x)/d^3 + (d^2*x^2 + 2*d*x + 2)*e^(-d*x - c)/d^3)/b^3 - ((d^3*x^3*e^c - 3*d^2*x^2*e^
c + 6*d*x*e^c - 6*e^c)*e^(d*x)/d^4 + (d^3*x^3 + 3*d^2*x^2 + 6*d*x + 6)*e^(-d*x - c)/d^4)/b^2 + 24*a^3*cosh(d*x
+ c)*log(b*x + a)/(b^5*d))*d - 1/3*(3*a^4/(b^6*x + a*b^5) + 12*a^3*log(b*x + a)/b^5 - (b^2*x^3 - 3*a*b*x^2 +
9*a^2*x)/b^4)*cosh(d*x + c)

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Fricas [A]  time = 2.07989, size = 740, normalized size = 3.2 \begin{align*} -\frac{2 \,{\left (a^{4} b d^{3} + 2 \, b^{5} d x^{2} - 2 \, a^{2} b^{3} d\right )} \cosh \left (d x + c\right ) -{\left ({\left (a^{5} d^{4} - 4 \, a^{4} b d^{3} +{\left (a^{4} b d^{4} - 4 \, a^{3} b^{2} d^{3}\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\left (a^{5} d^{4} + 4 \, a^{4} b d^{3} +{\left (a^{4} b d^{4} + 4 \, a^{3} b^{2} d^{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 (b^{5} d^{2} x^{3} - a b^{4} d^{2} x^{2} + 3 \, a^{3} b^{2} d^{2} + 2 \, a b^{4} +{\left (a^{2} b^{3} d^{2} + 2 \, b^{5}\right )} x\right )} \sinh \left (d x + c\right ) +{\left ({\left (a^{5} d^{4} - 4 \, a^{4} b d^{3} +{\left (a^{4} b d^{4} - 4 \, a^{3} b^{2} d^{3}\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\left (a^{5} d^{4} + 4 \, a^{4} b d^{3} +{\left (a^{4} b d^{4} + 4 \, a^{3} b^{2} d^{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 )}{2 \,{\left (b^{7} d^{3} x + a b^{6} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*cosh(d*x+c)/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \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**4*cosh(d*x+c)/(b*x+a)**2,x)

[Out]

Integral(x**4*cosh(c + d*x)/(a + b*x)**2, x)

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Giac [A]  time = 1.18744, size = 393, normalized size = 1.7 \begin{align*} \frac{a^{4} b d x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a^{4} b d x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + a^{5} d{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - 4 \, a^{3} b^{2} x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a^{5} d{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - 4 \, a^{3} b^{2} x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - 4 \, a^{4} b{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - 4 \, a^{4} b{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - a^{4} b e^{\left (d x + c\right )} - a^{4} b e^{\left (-d x - c\right )}}{2 \,{\left (b^{7} x + a b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*cosh(d*x+c)/(b*x+a)^2,x, algorithm="giac")

[Out]

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