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

Optimal. Leaf size=177 \[ \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b}+\frac{\cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b}-\frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b}+\frac{\sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.254037, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {5293, 3303, 3298, 3301} \[ \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b}+\frac{\cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b}-\frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b}+\frac{\sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(x*Cosh[c + d*x])/(a + b*x^2),x]

[Out]

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

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

Mathematica [C]  time = 0.174738, size = 171, normalized size = 0.97 \[ \frac{\cosh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (-\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+\cosh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+i \left (\sinh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )-\sinh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )\right )}{2 b} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(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] + I*(Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b]
- I*d*x] - Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x]))/(2*b)

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Maple [A]  time = 0.032, size = 200, normalized size = 1.1 \begin{align*} -{\frac{1}{4\,b}{{\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\,b}{{\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\,b}{{\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\,b}{{\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x e^{\left (d x + 2 \, c\right )} - x e^{\left (-d x\right )}}{2 \,{\left (b d x^{2} e^{c} + a d e^{c}\right )}} + \frac{1}{2} \, \int \frac{{\left (b x^{2} e^{c} - a e^{c}\right )} e^{\left (d x\right )}}{b^{2} d x^{4} + 2 \, a b d x^{2} + a^{2} d}\,{d x} - \frac{1}{2} \, \int \frac{{\left (b x^{2} - a\right )} e^{\left (-d x\right )}}{b^{2} d x^{4} e^{c} + 2 \, a b d x^{2} e^{c} + a^{2} d e^{c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.09171, size = 455, normalized size = 2.57 \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 ) +{\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 ) -{\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 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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