\(\int \frac {(a+b x)^2 \cosh (c+d x)}{x^2} \, dx\) [14]

Optimal result

Integrand size = 17, antiderivative size = 70 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^2} \, dx=-\frac {a^2 \cosh (c+d x)}{x}+2 a b \cosh (c) \text {Chi}(d x)+a^2 d \text {Chi}(d x) \sinh (c)+\frac {b^2 \sinh (c+d x)}{d}+a^2 d \cosh (c) \text {Shi}(d x)+2 a b \sinh (c) \text {Shi}(d x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {6874, 2717, 3378, 3384, 3379, 3382} \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^2} \, dx=a^2 d \sinh (c) \text {Chi}(d x)+a^2 d \cosh (c) \text {Shi}(d x)-\frac {a^2 \cosh (c+d x)}{x}+2 a b \cosh (c) \text {Chi}(d x)+2 a b \sinh (c) \text {Shi}(d x)+\frac {b^2 \sinh (c+d x)}{d} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^2} \, dx=-\frac {a^2 \cosh (c+d x)}{x}+a \text {Chi}(d x) (2 b \cosh (c)+a d \sinh (c))+\frac {b^2 \sinh (c+d x)}{d}+a (a d \cosh (c)+2 b \sinh (c)) \text {Shi}(d x) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.77

method	result	size
risch	\(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a^{2} d^{2} x -{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a^{2} d^{2} x +2 \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a b d x +2 \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a b d x +{\mathrm e}^{-d x -c} a^{2} d +{\mathrm e}^{-d x -c} b^{2} x +{\mathrm e}^{d x +c} a^{2} d -{\mathrm e}^{d x +c} b^{2} x}{2 d x}\)	\(124\)
meijerg	\(\frac {b^{2} \cosh \left (c \right ) \sinh \left (d x \right )}{d}-\frac {b^{2} \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}+a b \cosh \left (c \right ) \sqrt {\pi }\, \left (\frac {2 \gamma +2 \ln \left (x \right )+2 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Chi}\left (d x \right )-2 \ln \left (d x \right )-2 \gamma }{\sqrt {\pi }}\right )+2 a b \,\operatorname {Shi}\left (d x \right ) \sinh \left (c \right )+\frac {i a^{2} \cosh \left (c \right ) \sqrt {\pi }\, d \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{4}+\frac {a^{2} \sinh \left (c \right ) \sqrt {\pi }\, d \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Chi}\left (d x \right )-4 \ln \left (d x \right )-4 \gamma }{\sqrt {\pi }}\right )}{4}\)	\(209\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.74 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^2} \, dx=-\frac {2 \, a^{2} d \cosh \left (d x + c\right ) - 2 \, b^{2} x \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{2} + 2 \, a b d\right )} x {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{2} - 2 \, a b d\right )} x {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left ({\left (a^{2} d^{2} + 2 \, a b d\right )} x {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{2} - 2 \, a b d\right )} x {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d x} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^2} \, dx=-\frac {1}{2} \, {\left ({\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - {\rm Ei}\left (d x\right ) e^{c}\right )} a^{2} + b^{2} {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )} + \frac {4 \, a b \cosh \left (d x + c\right ) \log \left (x\right )}{d} - \frac {2 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a b}{d}\right )} d + {\left (b^{2} x + 2 \, a b \log \left (x\right ) - \frac {a^{2}}{x}\right )} \cosh \left (d x + c\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.70 \[ \int \frac {(a+b x)^2 \cosh (c+d x)}{x^2} \, dx=-\frac {a^{2} d^{2} x {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d^{2} x {\rm Ei}\left (d x\right ) e^{c} - 2 \, a b d x {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 2 \, a b d x {\rm Ei}\left (d x\right ) e^{c} + a^{2} d e^{\left (d x + c\right )} - b^{2} x e^{\left (d x + c\right )} + a^{2} d e^{\left (-d x - c\right )} + b^{2} x e^{\left (-d x - c\right )}}{2 \, d x} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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