\(\int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx\) [8]

Optimal result

Integrand size = 15, antiderivative size = 132 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=-\frac {a \cosh (c+d x)}{3 x^3}-\frac {b \cosh (c+d x)}{2 x^2}-\frac {a d^2 \cosh (c+d x)}{6 x}+\frac {1}{2} b d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{6} a d^3 \text {Chi}(d x) \sinh (c)-\frac {a d \sinh (c+d x)}{6 x^2}-\frac {b d \sinh (c+d x)}{2 x}+\frac {1}{6} a d^3 \cosh (c) \text {Shi}(d x)+\frac {1}{2} b d^2 \sinh (c) \text {Shi}(d x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6874, 3378, 3384, 3379, 3382} \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=\frac {1}{6} a d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a d^3 \cosh (c) \text {Shi}(d x)-\frac {a d^2 \cosh (c+d x)}{6 x}-\frac {a \cosh (c+d x)}{3 x^3}-\frac {a d \sinh (c+d x)}{6 x^2}+\frac {1}{2} b d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} b d^2 \sinh (c) \text {Shi}(d x)-\frac {b \cosh (c+d x)}{2 x^2}-\frac {b d \sinh (c+d x)}{2 x} \]

[In]

Int[((a + b*x)*Cosh[c + d*x])/x^4,x]

[Out]

-1/3*(a*Cosh[c + d*x])/x^3 - (b*Cosh[c + d*x])/(2*x^2) - (a*d^2*Cosh[c + d*x])/(6*x) + (b*d^2*Cosh[c]*CoshInte
gral[d*x])/2 + (a*d^3*CoshIntegral[d*x]*Sinh[c])/6 - (a*d*Sinh[c + d*x])/(6*x^2) - (b*d*Sinh[c + d*x])/(2*x) +
(a*d^3*Cosh[c]*SinhIntegral[d*x])/6 + (b*d^2*Sinh[c]*SinhIntegral[d*x])/2

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 {a \cosh (c+d x)}{x^4}+\frac {b \cosh (c+d x)}{x^3}\right ) \, dx \\ & = a \int \frac {\cosh (c+d x)}{x^4} \, dx+b \int \frac {\cosh (c+d x)}{x^3} \, dx \\ & = -\frac {a \cosh (c+d x)}{3 x^3}-\frac {b \cosh (c+d x)}{2 x^2}+\frac {1}{3} (a d) \int \frac {\sinh (c+d x)}{x^3} \, dx+\frac {1}{2} (b d) \int \frac {\sinh (c+d x)}{x^2} \, dx \\ & = -\frac {a \cosh (c+d x)}{3 x^3}-\frac {b \cosh (c+d x)}{2 x^2}-\frac {a d \sinh (c+d x)}{6 x^2}-\frac {b d \sinh (c+d x)}{2 x}+\frac {1}{6} \left (a d^2\right ) \int \frac {\cosh (c+d x)}{x^2} \, dx+\frac {1}{2} \left (b d^2\right ) \int \frac {\cosh (c+d x)}{x} \, dx \\ & = -\frac {a \cosh (c+d x)}{3 x^3}-\frac {b \cosh (c+d x)}{2 x^2}-\frac {a d^2 \cosh (c+d x)}{6 x}-\frac {a d \sinh (c+d x)}{6 x^2}-\frac {b d \sinh (c+d x)}{2 x}+\frac {1}{6} \left (a d^3\right ) \int \frac {\sinh (c+d x)}{x} \, dx+\frac {1}{2} \left (b d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\frac {1}{2} \left (b d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx \\ & = -\frac {a \cosh (c+d x)}{3 x^3}-\frac {b \cosh (c+d x)}{2 x^2}-\frac {a d^2 \cosh (c+d x)}{6 x}+\frac {1}{2} b d^2 \cosh (c) \text {Chi}(d x)-\frac {a d \sinh (c+d x)}{6 x^2}-\frac {b d \sinh (c+d x)}{2 x}+\frac {1}{2} b d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{6} \left (a d^3 \cosh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx+\frac {1}{6} \left (a d^3 \sinh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx \\ & = -\frac {a \cosh (c+d x)}{3 x^3}-\frac {b \cosh (c+d x)}{2 x^2}-\frac {a d^2 \cosh (c+d x)}{6 x}+\frac {1}{2} b d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{6} a d^3 \text {Chi}(d x) \sinh (c)-\frac {a d \sinh (c+d x)}{6 x^2}-\frac {b d \sinh (c+d x)}{2 x}+\frac {1}{6} a d^3 \cosh (c) \text {Shi}(d x)+\frac {1}{2} b d^2 \sinh (c) \text {Shi}(d x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=-\frac {2 a \cosh (c+d x)+3 b x \cosh (c+d x)+a d^2 x^2 \cosh (c+d x)-d^2 x^3 \text {Chi}(d x) (3 b \cosh (c)+a d \sinh (c))+a d x \sinh (c+d x)+3 b d x^2 \sinh (c+d x)-d^2 x^3 (a d \cosh (c)+3 b \sinh (c)) \text {Shi}(d x)}{6 x^3} \]

[In]

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

[Out]

-1/6*(2*a*Cosh[c + d*x] + 3*b*x*Cosh[c + d*x] + a*d^2*x^2*Cosh[c + d*x] - d^2*x^3*CoshIntegral[d*x]*(3*b*Cosh[
c] + a*d*Sinh[c]) + a*d*x*Sinh[c + d*x] + 3*b*d*x^2*Sinh[c + d*x] - d^2*x^3*(a*d*Cosh[c] + 3*b*Sinh[c])*SinhIn
tegral[d*x])/x^3

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.55

method	result	size
risch	\(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a \,d^{3} x^{3}-{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a \,d^{3} x^{3}+3 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) b \,d^{2} x^{3}+3 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b \,d^{2} x^{3}+{\mathrm e}^{d x +c} a \,d^{2} x^{2}+{\mathrm e}^{-d x -c} a \,d^{2} x^{2}+3 \,{\mathrm e}^{d x +c} b d \,x^{2}-3 \,{\mathrm e}^{-d x -c} b d \,x^{2}+{\mathrm e}^{d x +c} a d x -{\mathrm e}^{-d x -c} a d x +3 \,{\mathrm e}^{d x +c} b x +3 \,{\mathrm e}^{-d x -c} b x +2 a \,{\mathrm e}^{d x +c}+2 \,{\mathrm e}^{-d x -c} a}{12 x^{3}}\)	\(204\)
meijerg	\(-\frac {d^{2} b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{\sqrt {\pi }}-\frac {4 \left (\frac {9 x^{2} d^{2}}{2}+3\right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \cosh \left (d x \right )}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}-\frac {4 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{\sqrt {\pi }}\right )}{8}+\frac {i d^{2} b \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}+\frac {4 i \sinh \left (d x \right )}{x^{2} d^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{8}-\frac {i a \cosh \left (c \right ) \sqrt {\pi }\, d^{3} \left (-\frac {8 i \left (x^{2} d^{2}+2\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \sinh \left (d x \right )}{3 x^{2} d^{2} \sqrt {\pi }}+\frac {8 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{16}-\frac {a \sinh \left (c \right ) \sqrt {\pi }\, d^{3} \left (\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{3 \sqrt {\pi }}-\frac {8 \left (\frac {55 x^{2} d^{2}}{2}+45\right )}{45 \sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \cosh \left (d x \right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {16 \left (\frac {5 x^{2} d^{2}}{2}+5\right ) \sinh \left (d x \right )}{15 \sqrt {\pi }\, x^{3} d^{3}}-\frac {8 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{3 \sqrt {\pi }}\right )}{16}\)	\(359\)

[In]

int((b*x+a)*cosh(d*x+c)/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/12*(exp(c)*Ei(1,-d*x)*a*d^3*x^3-exp(-c)*Ei(1,d*x)*a*d^3*x^3+3*exp(c)*Ei(1,-d*x)*b*d^2*x^3+3*exp(-c)*Ei(1,d*
x)*b*d^2*x^3+exp(d*x+c)*a*d^2*x^2+exp(-d*x-c)*a*d^2*x^2+3*exp(d*x+c)*b*d*x^2-3*exp(-d*x-c)*b*d*x^2+exp(d*x+c)*
a*d*x-exp(-d*x-c)*a*d*x+3*exp(d*x+c)*b*x+3*exp(-d*x-c)*b*x+2*a*exp(d*x+c)+2*exp(-d*x-c)*a)/x^3

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=-\frac {2 \, {\left (a d^{2} x^{2} + 3 \, b x + 2 \, a\right )} \cosh \left (d x + c\right ) - {\left ({\left (a d^{3} + 3 \, b d^{2}\right )} x^{3} {\rm Ei}\left (d x\right ) - {\left (a d^{3} - 3 \, b d^{2}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (3 \, b d x^{2} + a d x\right )} \sinh \left (d x + c\right ) - {\left ({\left (a d^{3} + 3 \, b d^{2}\right )} x^{3} {\rm Ei}\left (d x\right ) + {\left (a d^{3} - 3 \, b d^{2}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{12 \, x^{3}} \]

[In]

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

[Out]

-1/12*(2*(a*d^2*x^2 + 3*b*x + 2*a)*cosh(d*x + c) - ((a*d^3 + 3*b*d^2)*x^3*Ei(d*x) - (a*d^3 - 3*b*d^2)*x^3*Ei(-
d*x))*cosh(c) + 2*(3*b*d*x^2 + a*d*x)*sinh(d*x + c) - ((a*d^3 + 3*b*d^2)*x^3*Ei(d*x) + (a*d^3 - 3*b*d^2)*x^3*E
i(-d*x))*sinh(c))/x^3

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.59 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=\frac {1}{12} \, {\left (2 \, a d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - 2 \, a d^{2} e^{c} \Gamma \left (-2, -d x\right ) + 3 \, b d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + 3 \, b d e^{c} \Gamma \left (-1, -d x\right )\right )} d - \frac {{\left (3 \, b x + 2 \, a\right )} \cosh \left (d x + c\right )}{6 \, x^{3}} \]

[In]

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

[Out]

1/12*(2*a*d^2*e^(-c)*gamma(-2, d*x) - 2*a*d^2*e^c*gamma(-2, -d*x) + 3*b*d*e^(-c)*gamma(-1, d*x) + 3*b*d*e^c*ga
mma(-1, -d*x))*d - 1/6*(3*b*x + 2*a)*cosh(d*x + c)/x^3

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.51 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=-\frac {a d^{3} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a d^{3} x^{3} {\rm Ei}\left (d x\right ) e^{c} - 3 \, b d^{2} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 3 \, b d^{2} x^{3} {\rm Ei}\left (d x\right ) e^{c} + a d^{2} x^{2} e^{\left (d x + c\right )} + a d^{2} x^{2} e^{\left (-d x - c\right )} + 3 \, b d x^{2} e^{\left (d x + c\right )} - 3 \, b d x^{2} e^{\left (-d x - c\right )} + a d x e^{\left (d x + c\right )} - a d x e^{\left (-d x - c\right )} + 3 \, b x e^{\left (d x + c\right )} + 3 \, b x e^{\left (-d x - c\right )} + 2 \, a e^{\left (d x + c\right )} + 2 \, a e^{\left (-d x - c\right )}}{12 \, x^{3}} \]

[In]

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

[Out]

-1/12*(a*d^3*x^3*Ei(-d*x)*e^(-c) - a*d^3*x^3*Ei(d*x)*e^c - 3*b*d^2*x^3*Ei(-d*x)*e^(-c) - 3*b*d^2*x^3*Ei(d*x)*e
^c + a*d^2*x^2*e^(d*x + c) + a*d^2*x^2*e^(-d*x - c) + 3*b*d*x^2*e^(d*x + c) - 3*b*d*x^2*e^(-d*x - c) + a*d*x*e
^(d*x + c) - a*d*x*e^(-d*x - c) + 3*b*x*e^(d*x + c) + 3*b*x*e^(-d*x - c) + 2*a*e^(d*x + c) + 2*a*e^(-d*x - c))
/x^3

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x) \cosh (c+d x)}{x^4} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (a+b\,x\right )}{x^4} \,d x \]

[In]

int((cosh(c + d*x)*(a + b*x))/x^4,x)

[Out]

int((cosh(c + d*x)*(a + b*x))/x^4, x)