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

Optimal. Leaf size=197 \[ -\frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a}-\frac{\cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a}+\frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a}-\frac{\sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a}+\frac{\cosh (c) \text{Chi}(d x)}{a}+\frac{\sinh (c) \text{Shi}(d x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.372347, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {5293, 3303, 3298, 3301} \[ -\frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a}-\frac{\cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a}+\frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a}-\frac{\sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a}+\frac{\cosh (c) \text{Chi}(d x)}{a}+\frac{\sinh (c) \text{Shi}(d x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(x*(a + b*x^2)),x]

[Out]

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

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 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]

Rubi steps

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

Mathematica [C]  time = 0.310288, size = 187, normalized size = 0.95 \[ -\frac{\cosh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (-\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+\cosh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+i \sinh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )-i \sinh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )-2 \cosh (c) \text{Chi}(d x)-2 \sinh (c) \text{Shi}(d x)}{2 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(-2*Cosh[c]*CoshIntegral[d*x] + Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] +
Cosh[c + (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] - 2*Sinh[c]*SinhIntegral[d*x] + I*Si
nh[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] - I*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*Sin
Integral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])/(2*a)

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Maple [A]  time = 0.043, size = 227, normalized size = 1.2 \begin{align*}{\frac{1}{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}{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{{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,a}}+{\frac{1}{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}{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{{{\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/(b*x^2+a),x)

[Out]

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

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Maxima [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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.20263, size = 545, normalized size = 2.77 \begin{align*} -\frac{{\left ({\rm Ei}\left (d x - \sqrt{-\frac{a d^{2}}{b}}\right ) +{\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 ({\rm Ei}\left (d x\right ) +{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) +{\left ({\rm Ei}\left (d x + \sqrt{-\frac{a d^{2}}{b}}\right ) +{\rm Ei}\left (-d x - \sqrt{-\frac{a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt{-\frac{a d^{2}}{b}}\right ) +{\left ({\rm Ei}\left (d x - \sqrt{-\frac{a d^{2}}{b}}\right ) -{\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 ({\rm Ei}\left (d x\right ) -{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) -{\left ({\rm Ei}\left (d x + \sqrt{-\frac{a d^{2}}{b}}\right ) -{\rm Ei}\left (-d x - \sqrt{-\frac{a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt{-\frac{a d^{2}}{b}}\right )}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Integral(cosh(c + d*x)/(x*(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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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