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

Optimal. Leaf size=182 \[ -\frac{a^3 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^5}+\frac{3 a^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{3 a^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^4}-\frac{a^3 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^5}+\frac{a^3 \cosh (c+d x)}{b^4 (a+b x)}-\frac{2 a \sinh (c+d x)}{b^3 d}-\frac{\cosh (c+d x)}{b^2 d^2}+\frac{x \sinh (c+d x)}{b^2 d} \]

[Out]

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

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Rubi [A]  time = 0.422891, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ -\frac{a^3 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^5}+\frac{3 a^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{3 a^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^4}-\frac{a^3 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^5}+\frac{a^3 \cosh (c+d x)}{b^4 (a+b x)}-\frac{2 a \sinh (c+d x)}{b^3 d}-\frac{\cosh (c+d x)}{b^2 d^2}+\frac{x \sinh (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.914422, size = 156, normalized size = 0.86 \[ \frac{\frac{b \left (\left (a^3 d^2-a b^2-b^3 x\right ) \cosh (c+d x)+b d \left (-2 a^2-a b x+b^2 x^2\right ) \sinh (c+d x)\right )}{d^2 (a+b x)}+a^2 \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (3 b \cosh \left (c-\frac{a d}{b}\right )-a d \sinh \left (c-\frac{a d}{b}\right )\right )-a^2 \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \cosh \left (c-\frac{a d}{b}\right )-3 b \sinh \left (c-\frac{a d}{b}\right )\right )}{b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 325, normalized size = 1.8 \begin{align*}{\frac{d{{\rm e}^{-dx-c}}{a}^{3}}{2\,{b}^{4} \left ( bdx+da \right ) }}-{\frac{d{a}^{3}}{2\,{b}^{5}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{3\,{a}^{2}}{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}}x}{2\,d{b}^{2}}}+{\frac{{{\rm e}^{-dx-c}}a}{d{b}^{3}}}-{\frac{{{\rm e}^{-dx-c}}}{2\,{d}^{2}{b}^{2}}}+{\frac{{{\rm e}^{dx+c}}x}{2\,d{b}^{2}}}+{\frac{d{{\rm e}^{dx+c}}{a}^{3}}{2\,{b}^{5}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}+{\frac{d{a}^{3}}{2\,{b}^{5}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }-{\frac{a{{\rm e}^{dx+c}}}{d{b}^{3}}}-{\frac{{{\rm e}^{dx+c}}}{2\,{d}^{2}{b}^{2}}}-{\frac{3\,{a}^{2}}{2\,{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^3*cosh(d*x+c)/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.3432, size = 421, normalized size = 2.31 \begin{align*} -\frac{1}{4} \,{\left (2 \, 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^{5}} - \frac{e^{\left (c - \frac{a d}{b}\right )} E_{1}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{b^{5}}\right )} + \frac{6 \, 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} + \frac{e^{\left (c - \frac{a d}{b}\right )} E_{1}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{b^{3} d} - \frac{4 \, a{\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^{3}} + \frac{\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}}}{b^{2}} + \frac{12 \, a^{2} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{4} d}\right )} d + \frac{1}{2} \,{\left (\frac{2 \, a^{3}}{b^{5} x + a b^{4}} + \frac{6 \, a^{2} \log \left (b x + a\right )}{b^{4}} + \frac{b x^{2} - 4 \, a x}{b^{3}}\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.03256, size = 663, normalized size = 3.64 \begin{align*} \frac{2 \,{\left (a^{3} b d^{2} - b^{4} x - a b^{3}\right )} \cosh \left (d x + c\right ) -{\left ({\left (a^{4} d^{3} - 3 \, a^{3} b d^{2} +{\left (a^{3} b d^{3} - 3 \, a^{2} b^{2} d^{2}\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\left (a^{4} d^{3} + 3 \, a^{3} b d^{2} +{\left (a^{3} b d^{3} + 3 \, a^{2} b^{2} d^{2}\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^{4} d x^{2} - a b^{3} d x - 2 \, a^{2} b^{2} d\right )} \sinh \left (d x + c\right ) +{\left ({\left (a^{4} d^{3} - 3 \, a^{3} b d^{2} +{\left (a^{3} b d^{3} - 3 \, a^{2} b^{2} d^{2}\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\left (a^{4} d^{3} + 3 \, a^{3} b d^{2} +{\left (a^{3} b d^{3} + 3 \, a^{2} b^{2} d^{2}\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^{6} d^{2} x + a b^{5} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18621, size = 393, normalized size = 2.16 \begin{align*} -\frac{a^{3} b d x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a^{3} b d x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + a^{4} d{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - 3 \, a^{2} b^{2} x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a^{4} d{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - 3 \, a^{2} b^{2} x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - 3 \, a^{3} b{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - 3 \, a^{3} b{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - a^{3} b e^{\left (d x + c\right )} - a^{3} b e^{\left (-d x - c\right )}}{2 \,{\left (b^{6} x + a b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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