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

Optimal. Leaf size=41 \[ a \cosh (c) \text{Chi}(d x)+a \sinh (c) \text{Shi}(d x)-\frac{b \cosh (c+d x)}{d^2}+\frac{b x \sinh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0980776, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {5287, 3303, 3298, 3301, 3296, 2638} \[ a \cosh (c) \text{Chi}(d x)+a \sinh (c) \text{Shi}(d x)-\frac{b \cosh (c+d x)}{d^2}+\frac{b x \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)*Cosh[c + d*x])/x,x]

[Out]

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

Rule 5287

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]

Rule 3303

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 3298

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 3301

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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.109761, size = 55, normalized size = 1.34 \[ a \cosh (c) \text{Chi}(d x)+a \sinh (c) \text{Shi}(d x)+\frac{b \cosh (d x) (d x \sinh (c)-\cosh (c))}{d^2}+\frac{b \sinh (d x) (d x \cosh (c)-\sinh (c))}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.035, size = 81, normalized size = 2. \begin{align*} -{\frac{a{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{b{{\rm e}^{-dx-c}}x}{2\,d}}-{\frac{b{{\rm e}^{-dx-c}}}{2\,{d}^{2}}}-{\frac{a{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}}+{\frac{b{{\rm e}^{dx+c}}x}{2\,d}}-{\frac{b{{\rm e}^{dx+c}}}{2\,{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.16856, size = 165, normalized size = 4.02 \begin{align*} -\frac{1}{4} \,{\left (b{\left (\frac{{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac{{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}\right )} + \frac{2 \, a \cosh \left (d x + c\right ) \log \left (x^{2}\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}{d}\right )} d + \frac{1}{2} \,{\left (b x^{2} + a \log \left (x^{2}\right )\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b*((d^2*x^2*e^c - 2*d*x*e^c + 2*e^c)*e^(d*x)/d^3 + (d^2*x^2 + 2*d*x + 2)*e^(-d*x - c)/d^3) + 2*a*cosh(d*
x + c)*log(x^2)/d - 2*(Ei(-d*x)*e^(-c) + Ei(d*x)*e^c)*a/d)*d + 1/2*(b*x^2 + a*log(x^2))*cosh(d*x + c)

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Fricas [A]  time = 2.06274, size = 188, normalized size = 4.59 \begin{align*} \frac{2 \, b d x \sinh \left (d x + c\right ) - 2 \, b \cosh \left (d x + c\right ) +{\left (a d^{2}{\rm Ei}\left (d x\right ) + a d^{2}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) +{\left (a d^{2}{\rm Ei}\left (d x\right ) - a d^{2}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.56531, size = 49, normalized size = 1.2 \begin{align*} a \sinh{\left (c \right )} \operatorname{Shi}{\left (d x \right )} + a \cosh{\left (c \right )} \operatorname{Chi}\left (d x\right ) + b \left (\begin{cases} \frac{x \sinh{\left (c + d x \right )}}{d} - \frac{\cosh{\left (c + d x \right )}}{d^{2}} & \text{for}\: d \neq 0 \\\frac{x^{2} \cosh{\left (c \right )}}{2} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*sinh(c)*Shi(d*x) + a*cosh(c)*Chi(d*x) + b*Piecewise((x*sinh(c + d*x)/d - cosh(c + d*x)/d**2, Ne(d, 0)), (x**
2*cosh(c)/2, True))

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Giac [A]  time = 1.17627, size = 103, normalized size = 2.51 \begin{align*} \frac{a d^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d^{2}{\rm Ei}\left (d x\right ) e^{c} + b d x e^{\left (d x + c\right )} - b d x e^{\left (-d x - c\right )} - b e^{\left (d x + c\right )} - b e^{\left (-d x - c\right )}}{2 \, d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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