\(\int \frac {(a+b x) \cosh (c+d x)}{x^5} \, dx\) [9]

Optimal result

Integrand size = 15, antiderivative size = 166 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^5} \, dx=-\frac {a \cosh (c+d x)}{4 x^4}-\frac {b \cosh (c+d x)}{3 x^3}-\frac {a d^2 \cosh (c+d x)}{24 x^2}-\frac {b d^2 \cosh (c+d x)}{6 x}+\frac {1}{24} a d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{6} b d^3 \text {Chi}(d x) \sinh (c)-\frac {a d \sinh (c+d x)}{12 x^3}-\frac {b d \sinh (c+d x)}{6 x^2}-\frac {a d^3 \sinh (c+d x)}{24 x}+\frac {1}{6} b d^3 \cosh (c) \text {Shi}(d x)+\frac {1}{24} a d^4 \sinh (c) \text {Shi}(d x) \]

[Out]

1/24*a*d^4*Chi(d*x)*cosh(c)-1/4*a*cosh(d*x+c)/x^4-1/3*b*cosh(d*x+c)/x^3-1/24*a*d^2*cosh(d*x+c)/x^2-1/6*b*d^2*c
osh(d*x+c)/x+1/6*b*d^3*cosh(c)*Shi(d*x)+1/6*b*d^3*Chi(d*x)*sinh(c)+1/24*a*d^4*Shi(d*x)*sinh(c)-1/12*a*d*sinh(d
*x+c)/x^3-1/6*b*d*sinh(d*x+c)/x^2-1/24*a*d^3*sinh(d*x+c)/x

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6874, 3378, 3384, 3379, 3382} \[ \int \frac {(a+b x) \cosh (c+d x)}{x^5} \, dx=\frac {1}{24} a d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a d^4 \sinh (c) \text {Shi}(d x)-\frac {a d^3 \sinh (c+d x)}{24 x}-\frac {a d^2 \cosh (c+d x)}{24 x^2}-\frac {a \cosh (c+d x)}{4 x^4}-\frac {a d \sinh (c+d x)}{12 x^3}+\frac {1}{6} b d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} b d^3 \cosh (c) \text {Shi}(d x)-\frac {b d^2 \cosh (c+d x)}{6 x}-\frac {b \cosh (c+d x)}{3 x^3}-\frac {b d \sinh (c+d x)}{6 x^2} \]

[In]

Int[((a + b*x)*Cosh[c + d*x])/x^5,x]

[Out]

-1/4*(a*Cosh[c + d*x])/x^4 - (b*Cosh[c + d*x])/(3*x^3) - (a*d^2*Cosh[c + d*x])/(24*x^2) - (b*d^2*Cosh[c + d*x]
)/(6*x) + (a*d^4*Cosh[c]*CoshIntegral[d*x])/24 + (b*d^3*CoshIntegral[d*x]*Sinh[c])/6 - (a*d*Sinh[c + d*x])/(12
*x^3) - (b*d*Sinh[c + d*x])/(6*x^2) - (a*d^3*Sinh[c + d*x])/(24*x) + (b*d^3*Cosh[c]*SinhIntegral[d*x])/6 + (a*
d^4*Sinh[c]*SinhIntegral[d*x])/24

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^5}+\frac {b \cosh (c+d x)}{x^4}\right ) \, dx \\ & = a \int \frac {\cosh (c+d x)}{x^5} \, dx+b \int \frac {\cosh (c+d x)}{x^4} \, dx \\ & = -\frac {a \cosh (c+d x)}{4 x^4}-\frac {b \cosh (c+d x)}{3 x^3}+\frac {1}{4} (a d) \int \frac {\sinh (c+d x)}{x^4} \, dx+\frac {1}{3} (b d) \int \frac {\sinh (c+d x)}{x^3} \, dx \\ & = -\frac {a \cosh (c+d x)}{4 x^4}-\frac {b \cosh (c+d x)}{3 x^3}-\frac {a d \sinh (c+d x)}{12 x^3}-\frac {b d \sinh (c+d x)}{6 x^2}+\frac {1}{12} \left (a d^2\right ) \int \frac {\cosh (c+d x)}{x^3} \, dx+\frac {1}{6} \left (b d^2\right ) \int \frac {\cosh (c+d x)}{x^2} \, dx \\ & = -\frac {a \cosh (c+d x)}{4 x^4}-\frac {b \cosh (c+d x)}{3 x^3}-\frac {a d^2 \cosh (c+d x)}{24 x^2}-\frac {b d^2 \cosh (c+d x)}{6 x}-\frac {a d \sinh (c+d x)}{12 x^3}-\frac {b d \sinh (c+d x)}{6 x^2}+\frac {1}{24} \left (a d^3\right ) \int \frac {\sinh (c+d x)}{x^2} \, dx+\frac {1}{6} \left (b d^3\right ) \int \frac {\sinh (c+d x)}{x} \, dx \\ & = -\frac {a \cosh (c+d x)}{4 x^4}-\frac {b \cosh (c+d x)}{3 x^3}-\frac {a d^2 \cosh (c+d x)}{24 x^2}-\frac {b d^2 \cosh (c+d x)}{6 x}-\frac {a d \sinh (c+d x)}{12 x^3}-\frac {b d \sinh (c+d x)}{6 x^2}-\frac {a d^3 \sinh (c+d x)}{24 x}+\frac {1}{24} \left (a d^4\right ) \int \frac {\cosh (c+d x)}{x} \, dx+\frac {1}{6} \left (b d^3 \cosh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx+\frac {1}{6} \left (b d^3 \sinh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx \\ & = -\frac {a \cosh (c+d x)}{4 x^4}-\frac {b \cosh (c+d x)}{3 x^3}-\frac {a d^2 \cosh (c+d x)}{24 x^2}-\frac {b d^2 \cosh (c+d x)}{6 x}+\frac {1}{6} b d^3 \text {Chi}(d x) \sinh (c)-\frac {a d \sinh (c+d x)}{12 x^3}-\frac {b d \sinh (c+d x)}{6 x^2}-\frac {a d^3 \sinh (c+d x)}{24 x}+\frac {1}{6} b d^3 \cosh (c) \text {Shi}(d x)+\frac {1}{24} \left (a d^4 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\frac {1}{24} \left (a d^4 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx \\ & = -\frac {a \cosh (c+d x)}{4 x^4}-\frac {b \cosh (c+d x)}{3 x^3}-\frac {a d^2 \cosh (c+d x)}{24 x^2}-\frac {b d^2 \cosh (c+d x)}{6 x}+\frac {1}{24} a d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{6} b d^3 \text {Chi}(d x) \sinh (c)-\frac {a d \sinh (c+d x)}{12 x^3}-\frac {b d \sinh (c+d x)}{6 x^2}-\frac {a d^3 \sinh (c+d x)}{24 x}+\frac {1}{6} b d^3 \cosh (c) \text {Shi}(d x)+\frac {1}{24} a d^4 \sinh (c) \text {Shi}(d x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^5} \, dx=-\frac {6 a \cosh (c+d x)+8 b x \cosh (c+d x)+a d^2 x^2 \cosh (c+d x)+4 b d^2 x^3 \cosh (c+d x)-d^3 x^4 \text {Chi}(d x) (a d \cosh (c)+4 b \sinh (c))+2 a d x \sinh (c+d x)+4 b d x^2 \sinh (c+d x)+a d^3 x^3 \sinh (c+d x)-d^3 x^4 (4 b \cosh (c)+a d \sinh (c)) \text {Shi}(d x)}{24 x^4} \]

[In]

Integrate[((a + b*x)*Cosh[c + d*x])/x^5,x]

[Out]

-1/24*(6*a*Cosh[c + d*x] + 8*b*x*Cosh[c + d*x] + a*d^2*x^2*Cosh[c + d*x] + 4*b*d^2*x^3*Cosh[c + d*x] - d^3*x^4
*CoshIntegral[d*x]*(a*d*Cosh[c] + 4*b*Sinh[c]) + 2*a*d*x*Sinh[c + d*x] + 4*b*d*x^2*Sinh[c + d*x] + a*d^3*x^3*S
inh[c + d*x] - d^3*x^4*(4*b*Cosh[c] + a*d*Sinh[c])*SinhIntegral[d*x])/x^4

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.62

method	result	size
risch	\(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a \,d^{4} x^{4}+{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a \,d^{4} x^{4}+4 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) b \,d^{3} x^{4}-4 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b \,d^{3} x^{4}+{\mathrm e}^{d x +c} a \,d^{3} x^{3}-{\mathrm e}^{-d x -c} a \,d^{3} x^{3}+4 \,{\mathrm e}^{d x +c} b \,d^{2} x^{3}+4 \,{\mathrm e}^{-d x -c} b \,d^{2} x^{3}+{\mathrm e}^{d x +c} a \,d^{2} x^{2}+{\mathrm e}^{-d x -c} a \,d^{2} x^{2}+4 \,{\mathrm e}^{d x +c} b d \,x^{2}-4 \,{\mathrm e}^{-d x -c} b d \,x^{2}+2 \,{\mathrm e}^{d x +c} a d x -2 \,{\mathrm e}^{-d x -c} a d x +8 \,{\mathrm e}^{d x +c} b x +8 \,{\mathrm e}^{-d x -c} b x +6 a \,{\mathrm e}^{d x +c}+6 \,{\mathrm e}^{-d x -c} a}{48 x^{4}}\)	\(269\)
meijerg	\(-\frac {i d^{3} b \cosh \left (c \right ) \sqrt {\pi }\, \left (-\frac {8 i \left (x^{2} d^{2}+2\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \sinh \left (d x \right )}{3 x^{2} d^{2} \sqrt {\pi }}+\frac {8 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{16}-\frac {d^{3} b \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{3 \sqrt {\pi }}-\frac {8 \left (\frac {55 x^{2} d^{2}}{2}+45\right )}{45 \sqrt {\pi }\, x^{2} d^{2}}+\frac {8 \cosh \left (d x \right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {16 \left (\frac {5 x^{2} d^{2}}{2}+5\right ) \sinh \left (d x \right )}{15 \sqrt {\pi }\, x^{3} d^{3}}-\frac {8 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{3 \sqrt {\pi }}\right )}{16}+\frac {a \cosh \left (c \right ) \sqrt {\pi }\, d^{4} \left (-\frac {8}{\sqrt {\pi }\, x^{4} d^{4}}-\frac {8}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {\frac {4 \gamma }{3}-\frac {25}{9}+\frac {4 \ln \left (x \right )}{3}+\frac {4 \ln \left (i d \right )}{3}}{\sqrt {\pi }}+\frac {\frac {25}{9} d^{4} x^{4}+8 x^{2} d^{2}+8}{\sqrt {\pi }\, x^{4} d^{4}}-\frac {8 \left (\frac {15 x^{2} d^{2}}{2}+45\right ) \cosh \left (d x \right )}{45 \sqrt {\pi }\, x^{4} d^{4}}-\frac {8 \left (\frac {15 x^{2} d^{2}}{2}+15\right ) \sinh \left (d x \right )}{45 \sqrt {\pi }\, x^{3} d^{3}}+\frac {\frac {4 \,\operatorname {Chi}\left (d x \right )}{3}-\frac {4 \ln \left (d x \right )}{3}-\frac {4 \gamma }{3}}{\sqrt {\pi }}\right )}{32}-\frac {i a \sinh \left (c \right ) \sqrt {\pi }\, d^{4} \left (-\frac {8 i \left (\frac {x^{2} d^{2}}{2}+1\right ) \cosh \left (d x \right )}{3 d^{3} x^{3} \sqrt {\pi }}-\frac {8 i \left (\frac {x^{2} d^{2}}{2}+3\right ) \sinh \left (d x \right )}{3 d^{4} x^{4} \sqrt {\pi }}+\frac {4 i \operatorname {Shi}\left (d x \right )}{3 \sqrt {\pi }}\right )}{32}\)	\(418\)

[In]

int((b*x+a)*cosh(d*x+c)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/48*(exp(c)*Ei(1,-d*x)*a*d^4*x^4+exp(-c)*Ei(1,d*x)*a*d^4*x^4+4*exp(c)*Ei(1,-d*x)*b*d^3*x^4-4*exp(-c)*Ei(1,d*
x)*b*d^3*x^4+exp(d*x+c)*a*d^3*x^3-exp(-d*x-c)*a*d^3*x^3+4*exp(d*x+c)*b*d^2*x^3+4*exp(-d*x-c)*b*d^2*x^3+exp(d*x
+c)*a*d^2*x^2+exp(-d*x-c)*a*d^2*x^2+4*exp(d*x+c)*b*d*x^2-4*exp(-d*x-c)*b*d*x^2+2*exp(d*x+c)*a*d*x-2*exp(-d*x-c
)*a*d*x+8*exp(d*x+c)*b*x+8*exp(-d*x-c)*b*x+6*a*exp(d*x+c)+6*exp(-d*x-c)*a)/x^4

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^5} \, dx=-\frac {2 \, {\left (4 \, b d^{2} x^{3} + a d^{2} x^{2} + 8 \, b x + 6 \, a\right )} \cosh \left (d x + c\right ) - {\left ({\left (a d^{4} + 4 \, b d^{3}\right )} x^{4} {\rm Ei}\left (d x\right ) + {\left (a d^{4} - 4 \, b d^{3}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (a d^{3} x^{3} + 4 \, b d x^{2} + 2 \, a d x\right )} \sinh \left (d x + c\right ) - {\left ({\left (a d^{4} + 4 \, b d^{3}\right )} x^{4} {\rm Ei}\left (d x\right ) - {\left (a d^{4} - 4 \, b d^{3}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{48 \, x^{4}} \]

[In]

integrate((b*x+a)*cosh(d*x+c)/x^5,x, algorithm="fricas")

[Out]

-1/48*(2*(4*b*d^2*x^3 + a*d^2*x^2 + 8*b*x + 6*a)*cosh(d*x + c) - ((a*d^4 + 4*b*d^3)*x^4*Ei(d*x) + (a*d^4 - 4*b
*d^3)*x^4*Ei(-d*x))*cosh(c) + 2*(a*d^3*x^3 + 4*b*d*x^2 + 2*a*d*x)*sinh(d*x + c) - ((a*d^4 + 4*b*d^3)*x^4*Ei(d*
x) - (a*d^4 - 4*b*d^3)*x^4*Ei(-d*x))*sinh(c))/x^4

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x) \cosh (c+d x)}{x^5} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)*cosh(d*x+c)/x**5,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.49 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^5} \, dx=\frac {1}{24} \, {\left (3 \, a d^{3} e^{\left (-c\right )} \Gamma \left (-3, d x\right ) + 3 \, a d^{3} e^{c} \Gamma \left (-3, -d x\right ) + 4 \, b d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - 4 \, b d^{2} e^{c} \Gamma \left (-2, -d x\right )\right )} d - \frac {{\left (4 \, b x + 3 \, a\right )} \cosh \left (d x + c\right )}{12 \, x^{4}} \]

[In]

integrate((b*x+a)*cosh(d*x+c)/x^5,x, algorithm="maxima")

[Out]

1/24*(3*a*d^3*e^(-c)*gamma(-3, d*x) + 3*a*d^3*e^c*gamma(-3, -d*x) + 4*b*d^2*e^(-c)*gamma(-2, d*x) - 4*b*d^2*e^
c*gamma(-2, -d*x))*d - 1/12*(4*b*x + 3*a)*cosh(d*x + c)/x^4

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b x) \cosh (c+d x)}{x^5} \, dx=\frac {a d^{4} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d^{4} x^{4} {\rm Ei}\left (d x\right ) e^{c} - 4 \, b d^{3} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 4 \, b d^{3} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a d^{3} x^{3} e^{\left (d x + c\right )} + a d^{3} x^{3} e^{\left (-d x - c\right )} - 4 \, b d^{2} x^{3} e^{\left (d x + c\right )} - 4 \, b d^{2} x^{3} e^{\left (-d x - c\right )} - a d^{2} x^{2} e^{\left (d x + c\right )} - a d^{2} x^{2} e^{\left (-d x - c\right )} - 4 \, b d x^{2} e^{\left (d x + c\right )} + 4 \, b d x^{2} e^{\left (-d x - c\right )} - 2 \, a d x e^{\left (d x + c\right )} + 2 \, a d x e^{\left (-d x - c\right )} - 8 \, b x e^{\left (d x + c\right )} - 8 \, b x e^{\left (-d x - c\right )} - 6 \, a e^{\left (d x + c\right )} - 6 \, a e^{\left (-d x - c\right )}}{48 \, x^{4}} \]

[In]

integrate((b*x+a)*cosh(d*x+c)/x^5,x, algorithm="giac")

[Out]

1/48*(a*d^4*x^4*Ei(-d*x)*e^(-c) + a*d^4*x^4*Ei(d*x)*e^c - 4*b*d^3*x^4*Ei(-d*x)*e^(-c) + 4*b*d^3*x^4*Ei(d*x)*e^
c - a*d^3*x^3*e^(d*x + c) + a*d^3*x^3*e^(-d*x - c) - 4*b*d^2*x^3*e^(d*x + c) - 4*b*d^2*x^3*e^(-d*x - c) - a*d^
2*x^2*e^(d*x + c) - a*d^2*x^2*e^(-d*x - c) - 4*b*d*x^2*e^(d*x + c) + 4*b*d*x^2*e^(-d*x - c) - 2*a*d*x*e^(d*x +
c) + 2*a*d*x*e^(-d*x - c) - 8*b*x*e^(d*x + c) - 8*b*x*e^(-d*x - c) - 6*a*e^(d*x + c) - 6*a*e^(-d*x - c))/x^4

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x) \cosh (c+d x)}{x^5} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (a+b\,x\right )}{x^5} \,d x \]

[In]

int((cosh(c + d*x)*(a + b*x))/x^5,x)

[Out]

int((cosh(c + d*x)*(a + b*x))/x^5, x)