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

Optimal. Leaf size=249 \[ \frac{\sqrt{b} \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{3/2}}-\frac{\sqrt{b} \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{3/2}}-\frac{\sqrt{b} \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{3/2}}-\frac{\sqrt{b} \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{3/2}}+\frac{d \sinh (c) \text{Chi}(d x)}{a}+\frac{d \cosh (c) \text{Shi}(d x)}{a}-\frac{\cosh (c+d x)}{a x} \]

[Out]

-(Cosh[c + d*x]/(a*x)) + (Sqrt[b]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*
(-a)^(3/2)) - (Sqrt[b]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*(-a)^(3/2))
+ (d*CoshIntegral[d*x]*Sinh[c])/a + (d*Cosh[c]*SinhIntegral[d*x])/a - (Sqrt[b]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]
*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*(-a)^(3/2)) - (Sqrt[b]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegr
al[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*(-a)^(3/2))

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Rubi [A]  time = 0.495823, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {5293, 3297, 3303, 3298, 3301, 5281} \[ \frac{\sqrt{b} \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{3/2}}-\frac{\sqrt{b} \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{3/2}}-\frac{\sqrt{b} \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{3/2}}-\frac{\sqrt{b} \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{3/2}}+\frac{d \sinh (c) \text{Chi}(d x)}{a}+\frac{d \cosh (c) \text{Shi}(d x)}{a}-\frac{\cosh (c+d x)}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cosh[c + d*x]/(a*x)) + (Sqrt[b]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*
(-a)^(3/2)) - (Sqrt[b]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*(-a)^(3/2))
+ (d*CoshIntegral[d*x]*Sinh[c])/a + (d*Cosh[c]*SinhIntegral[d*x])/a - (Sqrt[b]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]
*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*(-a)^(3/2)) - (Sqrt[b]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegr
al[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*(-a)^(3/2))

Rule 5293

Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cosh[c
+ d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (Eq
Q[n, 2] || EqQ[p, -1])

Rule 3297

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 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 5281

Int[Cosh[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cosh[c + d*x], (a
+ b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])

Rubi steps

\begin{align*} \int \frac{\cosh (c+d x)}{x^2 \left (a+b x^2\right )} \, dx &=\int \left (\frac{\cosh (c+d x)}{a x^2}-\frac{b \cosh (c+d x)}{a \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\cosh (c+d x)}{x^2} \, dx}{a}-\frac{b \int \frac{\cosh (c+d x)}{a+b x^2} \, dx}{a}\\ &=-\frac{\cosh (c+d x)}{a x}-\frac{b \int \left (\frac{\sqrt{-a} \cosh (c+d x)}{2 a \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{-a} \cosh (c+d x)}{2 a \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{a}+\frac{d \int \frac{\sinh (c+d x)}{x} \, dx}{a}\\ &=-\frac{\cosh (c+d x)}{a x}-\frac{b \int \frac{\cosh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 (-a)^{3/2}}-\frac{b \int \frac{\cosh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 (-a)^{3/2}}+\frac{(d \cosh (c)) \int \frac{\sinh (d x)}{x} \, dx}{a}+\frac{(d \sinh (c)) \int \frac{\cosh (d x)}{x} \, dx}{a}\\ &=-\frac{\cosh (c+d x)}{a x}+\frac{d \text{Chi}(d x) \sinh (c)}{a}+\frac{d \cosh (c) \text{Shi}(d x)}{a}-\frac{\left (b \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 (-a)^{3/2}}-\frac{\left (b \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 (-a)^{3/2}}-\frac{\left (b \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 (-a)^{3/2}}+\frac{\left (b \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 (-a)^{3/2}}\\ &=-\frac{\cosh (c+d x)}{a x}+\frac{\sqrt{b} \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{3/2}}-\frac{\sqrt{b} \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{3/2}}+\frac{d \text{Chi}(d x) \sinh (c)}{a}+\frac{d \cosh (c) \text{Shi}(d x)}{a}-\frac{\sqrt{b} \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{3/2}}-\frac{\sqrt{b} \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.357135, size = 243, normalized size = 0.98 \[ \frac{-i \sqrt{b} x \cosh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (-\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+i \sqrt{b} x \cosh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+\sqrt{b} x \sinh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )+\sqrt{b} x \sinh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )+2 \sqrt{a} d x \sinh (c) \text{Chi}(d x)+2 \sqrt{a} d x \cosh (c) \text{Shi}(d x)-2 \sqrt{a} \cosh (c+d x)}{2 a^{3/2} x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*Sqrt[a]*Cosh[c + d*x] - I*Sqrt[b]*x*Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[-((Sqrt[a]*d)/Sqrt[b]) + I
*d*x] + I*Sqrt[b]*x*Cosh[c + (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] + 2*Sqrt[a]*d*x*C
oshIntegral[d*x]*Sinh[c] + 2*Sqrt[a]*d*x*Cosh[c]*SinhIntegral[d*x] + Sqrt[b]*x*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]
*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] + Sqrt[b]*x*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/
Sqrt[b] + I*d*x])/(2*a^(3/2)*x)

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Maple [A]  time = 0.055, size = 288, normalized size = 1.2 \begin{align*} -{\frac{{{\rm e}^{-dx-c}}}{2\,ax}}+{\frac{b}{4\,a}{{\rm e}^{-{\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{b} \left ( d\sqrt{-ab}- \left ( dx+c \right ) b+cb \right ) } \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{b}{4\,a}{{\rm e}^{{\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) }}}{\it Ei} \left ( 1,{\frac{1}{b} \left ( d\sqrt{-ab}+ \left ( dx+c \right ) b-cb \right ) } \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{d{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,a}}-{\frac{{{\rm e}^{dx+c}}}{2\,ax}}+{\frac{b}{4\,a}{{\rm e}^{{\frac{1}{b} \left ( d\sqrt{-ab}+cb \right ) }}}{\it Ei} \left ( 1,{\frac{1}{b} \left ( d\sqrt{-ab}- \left ( dx+c \right ) b+cb \right ) } \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{b}{4\,a}{{\rm e}^{-{\frac{1}{b} \left ( d\sqrt{-ab}-cb \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{b} \left ( d\sqrt{-ab}+ \left ( dx+c \right ) b-cb \right ) } \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{d{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.49935, size = 1323, normalized size = 5.31 \begin{align*} -\frac{4 \, a d \cosh \left (d x + c\right ) -{\left ({\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt{-\frac{a d^{2}}{b}}{\rm Ei}\left (d x - \sqrt{-\frac{a d^{2}}{b}}\right ) +{\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt{-\frac{a d^{2}}{b}}{\rm Ei}\left (-d x + \sqrt{-\frac{a d^{2}}{b}}\right )\right )} \cosh \left (c + \sqrt{-\frac{a d^{2}}{b}}\right ) - 2 \,{\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 ({\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt{-\frac{a d^{2}}{b}}{\rm Ei}\left (d x + \sqrt{-\frac{a d^{2}}{b}}\right ) +{\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt{-\frac{a d^{2}}{b}}{\rm Ei}\left (-d x - \sqrt{-\frac{a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt{-\frac{a d^{2}}{b}}\right ) -{\left ({\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt{-\frac{a d^{2}}{b}}{\rm Ei}\left (d x - \sqrt{-\frac{a d^{2}}{b}}\right ) -{\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt{-\frac{a d^{2}}{b}}{\rm Ei}\left (-d x + \sqrt{-\frac{a d^{2}}{b}}\right )\right )} \sinh \left (c + \sqrt{-\frac{a d^{2}}{b}}\right ) - 2 \,{\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 ) -{\left ({\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt{-\frac{a d^{2}}{b}}{\rm Ei}\left (d x + \sqrt{-\frac{a d^{2}}{b}}\right ) -{\left (b x \cosh \left (d x + c\right )^{2} - b x \sinh \left (d x + c\right )^{2}\right )} \sqrt{-\frac{a d^{2}}{b}}{\rm Ei}\left (-d x - \sqrt{-\frac{a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt{-\frac{a d^{2}}{b}}\right )}{4 \,{\left (a^{2} d x \cosh \left (d x + c\right )^{2} - a^{2} d x \sinh \left (d x + c\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (c + d x \right )}}{x^{2} \left (a + b x^{2}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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