\(\int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx\) [33]

Optimal result

Integrand size = 17, antiderivative size = 264 \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=\frac {a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}-\frac {3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac {3 a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {a^3 d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{2 b^6}+\frac {3 a^2 d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^5}+\frac {\sinh (c+d x)}{b^3 d}+\frac {a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}+\frac {3 a^2 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {3 a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {a^3 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{2 b^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {6874, 2717, 3378, 3384, 3379, 3382} \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {a^3 d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{2 b^6}-\frac {a^3 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{2 b^6}+\frac {a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}+\frac {a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}+\frac {3 a^2 d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {3 a^2 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac {3 a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {3 a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {\sinh (c+d x)}{b^3 d} \]

[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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3378

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 3379

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 3382

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]

Rule 3384

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.63 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {b \cosh (d x) \left (a^2 b d (5 a+6 b x) \cosh (c)-(a+b x) \left (2 a b^2+a^3 d^2+2 b^3 x\right ) \sinh (c)\right )-b \left ((a+b x) \left (2 a b^2+a^3 d^2+2 b^3 x\right ) \cosh (c)-a^2 b d (5 a+6 b x) \sinh (c)\right ) \sinh (d x)+a d (a+b x)^2 \left (\text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (\left (6 b^2+a^2 d^2\right ) \cosh \left (c-\frac {a d}{b}\right )-6 a b d \sinh \left (c-\frac {a d}{b}\right )\right )+\left (-6 a b d \cosh \left (c-\frac {a d}{b}\right )+\left (6 b^2+a^2 d^2\right ) \sinh \left (c-\frac {a d}{b}\right )\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )\right )}{2 b^6 d (a+b x)^2} \]

[In]

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

[Out]

-1/2*(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*(CoshI
ntegral[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)]))/(b^6*d*(a + b*x)^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1054\) vs. \(2(262)=524\).

Time = 0.36 (sec) , antiderivative size = 1055, normalized size of antiderivative = 4.00

method	result	size
risch	\(\text {Expression too large to display}\)	\(1055\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (262) = 524\).

Time = 0.26 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.14 \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {2 \, {\left (6 \, a^{2} b^{3} d x + 5 \, a^{3} b^{2} d\right )} \cosh \left (d x + c\right ) + {\left ({\left (a^{5} d^{3} - 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} - 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} - 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{5} d^{3} + 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} + 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} + 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (a^{4} b d^{2} + 2 \, b^{5} x^{2} + 2 \, a^{2} b^{3} + {\left (a^{3} b^{2} d^{2} + 4 \, a b^{4}\right )} x\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{5} d^{3} - 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} - 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} - 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{5} d^{3} + 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} + 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} + 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (b^{8} d x^{2} + 2 \, a b^{7} d x + a^{2} b^{6} d\right )}} \]

[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]

\[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=\int \frac {x^{3} \cosh {\left (c + d x \right )}}{\left (a + b x\right )^{3}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=\int { \frac {x^{3} \cosh \left (d x + c\right )}{{\left (b x + a\right )}^{3}} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 879 vs. \(2 (262) = 524\).

Time = 0.27 (sec) , antiderivative size = 879, normalized size of antiderivative = 3.33 \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {a^{3} b^{2} d^{3} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{3} b^{2} d^{3} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 2 \, a^{4} b d^{3} x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 6 \, a^{2} b^{3} d^{2} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, a^{4} b d^{3} x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 6 \, a^{2} b^{3} d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + a^{5} d^{3} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 12 \, a^{3} b^{2} d^{2} x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 6 \, a b^{4} d x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{5} d^{3} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 12 \, a^{3} b^{2} d^{2} x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 6 \, a b^{4} d x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a^{3} b^{2} d^{2} x e^{\left (d x + c\right )} + a^{3} b^{2} d^{2} x e^{\left (-d x - c\right )} - 6 \, a^{4} b d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 12 \, a^{2} b^{3} d x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 6 \, a^{4} b d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 12 \, a^{2} b^{3} 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^{2} e^{\left (d x + c\right )} + 6 \, a^{2} b^{3} d x e^{\left (d x + c\right )} - 2 \, b^{5} x^{2} e^{\left (d x + c\right )} + a^{4} b d^{2} e^{\left (-d x - c\right )} + 6 \, a^{2} b^{3} d x e^{\left (-d x - c\right )} + 2 \, b^{5} x^{2} e^{\left (-d x - c\right )} + 6 \, a^{3} b^{2} d {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 6 \, a^{3} b^{2} d {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 5 \, a^{3} b^{2} d e^{\left (d x + c\right )} - 4 \, a b^{4} x e^{\left (d x + c\right )} + 5 \, a^{3} b^{2} d e^{\left (-d x - c\right )} + 4 \, a b^{4} x e^{\left (-d x - c\right )} - 2 \, a^{2} b^{3} e^{\left (d x + c\right )} + 2 \, a^{2} b^{3} e^{\left (-d x - c\right )}}{4 \, {\left (b^{8} d x^{2} + 2 \, a b^{7} d x + a^{2} b^{6} d\right )}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=\int \frac {x^3\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+b\,x\right )}^3} \,d x \]

[In]

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

[Out]

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