\(\int \frac {(a+b x)^2 \cosh (c+d x)}{x^3} \, dx\) [15]

Optimal result

Integrand size = 17, antiderivative size = 121 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^3} \, dx=-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{2 x}+2 a b d \cosh (c) \text {Shi}(d x)+b^2 \sinh (c) \text {Shi}(d x)+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3378, 3384, 3379, 3382} \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^3} \, dx=\frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x)-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d \sinh (c+d x)}{2 x}+2 a b d \sinh (c) \text {Chi}(d x)+2 a b d \cosh (c) \text {Shi}(d x)-\frac {2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+b^2 \sinh (c) \text {Shi}(d x) \]

[In]

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

[Out]

-1/2*(a^2*Cosh[c + d*x])/x^2 - (2*a*b*Cosh[c + d*x])/x + b^2*Cosh[c]*CoshIntegral[d*x] + (a^2*d^2*Cosh[c]*Cosh
Integral[d*x])/2 + 2*a*b*d*CoshIntegral[d*x]*Sinh[c] - (a^2*d*Sinh[c + d*x])/(2*x) + 2*a*b*d*Cosh[c]*SinhInteg
ral[d*x] + b^2*Sinh[c]*SinhIntegral[d*x] + (a^2*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^2 \cosh (c+d x)}{x^3}+\frac {2 a b \cosh (c+d x)}{x^2}+\frac {b^2 \cosh (c+d x)}{x}\right ) \, dx \\ & = a^2 \int \frac {\cosh (c+d x)}{x^3} \, dx+(2 a b) \int \frac {\cosh (c+d x)}{x^2} \, dx+b^2 \int \frac {\cosh (c+d x)}{x} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {2 a b \cosh (c+d x)}{x}+\frac {1}{2} \left (a^2 d\right ) \int \frac {\sinh (c+d x)}{x^2} \, dx+(2 a b d) \int \frac {\sinh (c+d x)}{x} \, dx+\left (b^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\left (b^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{2 x}+b^2 \sinh (c) \text {Shi}(d x)+\frac {1}{2} \left (a^2 d^2\right ) \int \frac {\cosh (c+d x)}{x} \, dx+(2 a b d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx+(2 a b d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{2 x}+2 a b d \cosh (c) \text {Shi}(d x)+b^2 \sinh (c) \text {Shi}(d x)+\frac {1}{2} \left (a^2 d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\frac {1}{2} \left (a^2 d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx \\ & = -\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{2 x}+2 a b d \cosh (c) \text {Shi}(d x)+b^2 \sinh (c) \text {Shi}(d x)+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^3} \, dx=\frac {1}{2} \left (\text {Chi}(d x) \left (\left (2 b^2+a^2 d^2\right ) \cosh (c)+4 a b d \sinh (c)\right )-\frac {a ((a+4 b x) \cosh (c+d x)+a d x \sinh (c+d x))}{x^2}+\left (4 a b d \cosh (c)+\left (2 b^2+a^2 d^2\right ) \sinh (c)\right ) \text {Shi}(d x)\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.54

method	result	size
risch	\(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a^{2} d^{2} x^{2}+{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a^{2} d^{2} x^{2}+4 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a b d \,x^{2}-4 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a b d \,x^{2}+2 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) b^{2} x^{2}+2 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b^{2} x^{2}-{\mathrm e}^{-d x -c} a^{2} d x +{\mathrm e}^{d x +c} a^{2} d x +4 \,{\mathrm e}^{-d x -c} a b x +4 \,{\mathrm e}^{d x +c} a b x +{\mathrm e}^{-d x -c} a^{2}+{\mathrm e}^{d x +c} a^{2}}{4 x^{2}}\)	\(186\)
meijerg	\(\frac {b^{2} \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +2 \ln \left (x \right )+2 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Chi}\left (d x \right )-2 \ln \left (d x \right )-2 \gamma }{\sqrt {\pi }}\right )}{2}+b^{2} \operatorname {Shi}\left (d x \right ) \sinh \left (c \right )+\frac {i d a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{2}+\frac {d b a \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Chi}\left (d x \right )-4 \ln \left (d x \right )-4 \gamma }{\sqrt {\pi }}\right )}{2}-\frac {a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d^{2} \left (\frac {4}{\sqrt {\pi }\, x^{2} d^{2}}-\frac {2 \left (2 \gamma -3+2 \ln \left (x \right )+2 \ln \left (i d \right )\right )}{\sqrt {\pi }}-\frac {4 \left (\frac {9 x^{2} d^{2}}{2}+3\right )}{3 \sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \cosh \left (d x \right )}{\sqrt {\pi }\, x^{2} d^{2}}+\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}-\frac {4 \left (\operatorname {Chi}\left (d x \right )-\ln \left (d x \right )-\gamma \right )}{\sqrt {\pi }}\right )}{8}+\frac {i a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d^{2} \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}+\frac {4 i \sinh \left (d x \right )}{x^{2} d^{2} \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{8}\)	\(343\)

[In]

int((b*x+a)^2*cosh(d*x+c)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^3} \, dx=-\frac {2 \, a^{2} d x \sinh \left (d x + c\right ) + 2 \, {\left (4 \, a b x + a^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{2} + 4 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{2} - 4 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left ({\left (a^{2} d^{2} + 4 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{2} - 4 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{4 \, x^{2}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^3} \, dx=\int \frac {\left (a + b x\right )^{2} \cosh {\left (c + d x \right )}}{x^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^3} \, dx=\frac {1}{4} \, {\left ({\left (d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + d e^{c} \Gamma \left (-1, -d x\right )\right )} a^{2} - 4 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - {\rm Ei}\left (d x\right ) e^{c}\right )} a b - \frac {4 \, b^{2} \cosh \left (d x + c\right ) \log \left (x\right )}{d} + \frac {2 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} b^{2}}{d}\right )} d + \frac {1}{2} \, {\left (2 \, b^{2} \log \left (x\right ) - \frac {4 \, a b x + a^{2}}{x^{2}}\right )} \cosh \left (d x + c\right ) \]

[In]

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

[Out]

1/4*((d*e^(-c)*gamma(-1, d*x) + d*e^c*gamma(-1, -d*x))*a^2 - 4*(Ei(-d*x)*e^(-c) - Ei(d*x)*e^c)*a*b - 4*b^2*cos
h(d*x + c)*log(x)/d + 2*(Ei(-d*x)*e^(-c) + Ei(d*x)*e^c)*b^2/d)*d + 1/2*(2*b^2*log(x) - (4*a*b*x + a^2)/x^2)*co
sh(d*x + c)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.50 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^3} \, dx=\frac {a^{2} d^{2} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{2} x^{2} {\rm Ei}\left (d x\right ) e^{c} - 4 \, a b d x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 4 \, a b d x^{2} {\rm Ei}\left (d x\right ) e^{c} + 2 \, b^{2} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 2 \, b^{2} x^{2} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d x e^{\left (d x + c\right )} + a^{2} d x e^{\left (-d x - c\right )} - 4 \, a b x e^{\left (d x + c\right )} - 4 \, a b x e^{\left (-d x - c\right )} - a^{2} e^{\left (d x + c\right )} - a^{2} e^{\left (-d x - c\right )}}{4 \, x^{2}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^3} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (a+b\,x\right )}^2}{x^3} \,d x \]

[In]

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

[Out]

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