\(\int \frac {(a+b x^2) \cosh (c+d x)}{x^2} \, dx\) [45]

Optimal result

Integrand size = 17, antiderivative size = 42 \[ \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^2} \, dx=-\frac {a \cosh (c+d x)}{x}+a d \text {Chi}(d x) \sinh (c)+\frac {b \sinh (c+d x)}{d}+a d \cosh (c) \text {Shi}(d x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5395, 2717, 3378, 3384, 3379, 3382} \[ \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^2} \, dx=a d \sinh (c) \text {Chi}(d x)+a d \cosh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{x}+\frac {b \sinh (c+d x)}{d} \]

[In]

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

[Out]

-((a*Cosh[c + d*x])/x) + a*d*CoshIntegral[d*x]*Sinh[c] + (b*Sinh[c + d*x])/d + a*d*Cosh[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 5395

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^2} \, dx=-\frac {a \cosh (c+d x)}{x}+a d \text {Chi}(d x) \sinh (c)+\frac {b \sinh (c+d x)}{d}+a d \cosh (c) \text {Shi}(d x) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.98

method	result	size
risch	\(-\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) a \,d^{2} x -{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) a \,d^{2} x +d \,{\mathrm e}^{-d x -c} a +{\mathrm e}^{-d x -c} b x +a d \,{\mathrm e}^{d x +c}-{\mathrm e}^{d x +c} b x}{2 d x}\)	\(83\)
meijerg	\(\frac {b \cosh \left (c \right ) \sinh \left (d x \right )}{d}-\frac {b \sinh \left (c \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}+\frac {i a \cosh \left (c \right ) \sqrt {\pi }\, d \left (\frac {4 i \cosh \left (d x \right )}{d x \sqrt {\pi }}-\frac {4 i \operatorname {Shi}\left (d x \right )}{\sqrt {\pi }}\right )}{4}+\frac {a \sinh \left (c \right ) \sqrt {\pi }\, d \left (\frac {4 \gamma -4+4 \ln \left (x \right )+4 \ln \left (i d \right )}{\sqrt {\pi }}+\frac {4}{\sqrt {\pi }}-\frac {4 \sinh \left (d x \right )}{\sqrt {\pi }\, x d}+\frac {4 \,\operatorname {Chi}\left (d x \right )-4 \ln \left (d x \right )-4 \gamma }{\sqrt {\pi }}\right )}{4}\)	\(144\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.95 \[ \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^2} \, dx=-\frac {2 \, a d \cosh \left (d x + c\right ) - 2 \, b x \sinh \left (d x + c\right ) - {\left (a d^{2} x {\rm Ei}\left (d x\right ) - a d^{2} x {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left (a d^{2} x {\rm Ei}\left (d x\right ) + a d^{2} x {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d x} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate((b*x**2+a)*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 = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^2} \, dx=-\frac {1}{2} \, {\left (a {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a {\rm Ei}\left (d x\right ) e^{c} + \frac {{\left (d x e^{c} - e^{c}\right )} b e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} b e^{\left (-d x - c\right )}}{d^{2}}\right )} d + {\left (b x - \frac {a}{x}\right )} \cosh \left (d x + c\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a+b x^2\right ) \cosh (c+d x)}{x^2} \, dx=-\frac {a d^{2} x {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a d^{2} x {\rm Ei}\left (d x\right ) e^{c} + a d e^{\left (d x + c\right )} - b x e^{\left (d x + c\right )} + a d e^{\left (-d x - c\right )} + b x e^{\left (-d x - c\right )}}{2 \, d x} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[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)