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

Optimal. Leaf size=239 \[ -\frac{d \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 \sqrt{-a} b^{3/2}}+\frac{d \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 \sqrt{-a} b^{3/2}}-\frac{d \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 \sqrt{-a} b^{3/2}}-\frac{d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 \sqrt{-a} b^{3/2}}-\frac{\cosh (c+d x)}{2 b \left (a+b x^2\right )} \]

[Out]

-Cosh[c + d*x]/(2*b*(a + b*x^2)) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])
/(4*Sqrt[-a]*b^(3/2)) + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*Sqrt[-a
]*b^(3/2)) - (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*Sqrt[-a]*b^(3/2))
- (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*Sqrt[-a]*b^(3/2))

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Rubi [A]  time = 0.325431, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {5289, 5280, 3303, 3298, 3301} \[ -\frac{d \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 \sqrt{-a} b^{3/2}}+\frac{d \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 \sqrt{-a} b^{3/2}}-\frac{d \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 \sqrt{-a} b^{3/2}}-\frac{d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 \sqrt{-a} b^{3/2}}-\frac{\cosh (c+d x)}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cosh[c + d*x]/(2*b*(a + b*x^2)) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])
/(4*Sqrt[-a]*b^(3/2)) + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*Sqrt[-a
]*b^(3/2)) - (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*Sqrt[-a]*b^(3/2))
- (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*Sqrt[-a]*b^(3/2))

Rule 5289

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(e^m*(a + b*x
^n)^(p + 1)*Cosh[c + d*x])/(b*n*(p + 1)), x] - Dist[(d*e^m)/(b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*Sinh[c + d*
x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && EqQ[m - n + 1, 0] && LtQ[p, -1] && (IntegerQ[n
] || GtQ[e, 0])

Rule 5280

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sinh[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])

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)}{\left (a+b x^2\right )^2} \, dx &=-\frac{\cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac{d \int \frac{\sinh (c+d x)}{a+b x^2} \, dx}{2 b}\\ &=-\frac{\cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac{d \int \left (\frac{\sqrt{-a} \sinh (c+d x)}{2 a \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{-a} \sinh (c+d x)}{2 a \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{2 b}\\ &=-\frac{\cosh (c+d x)}{2 b \left (a+b x^2\right )}-\frac{d \int \frac{\sinh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{4 \sqrt{-a} b}-\frac{d \int \frac{\sinh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{4 \sqrt{-a} b}\\ &=-\frac{\cosh (c+d x)}{2 b \left (a+b x^2\right )}-\frac{\left (d \cosh \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}{4 \sqrt{-a} b}+\frac{\left (d \cosh \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}{4 \sqrt{-a} b}-\frac{\left (d \sinh \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}{4 \sqrt{-a} b}-\frac{\left (d \sinh \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}{4 \sqrt{-a} b}\\ &=-\frac{\cosh (c+d x)}{2 b \left (a+b x^2\right )}-\frac{d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 \sqrt{-a} b^{3/2}}+\frac{d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 \sqrt{-a} b^{3/2}}-\frac{d \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 \sqrt{-a} b^{3/2}}-\frac{d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 \sqrt{-a} b^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.513809, size = 239, normalized size = 1. \[ \frac{i \left (d \left (a+b x^2\right ) \sinh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (-\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )-d \left (a+b x^2\right ) \sinh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+i \left (d \left (a+b x^2\right ) \cosh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )+d \left (a+b x^2\right ) \cosh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )+2 \sqrt{a} \sqrt{b} \cosh (c+d x)\right )\right )}{4 \sqrt{a} b^{3/2} \left (a+b x^2\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((I/4)*(d*(a + b*x^2)*CosIntegral[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x]*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]] - d*(a + b*x
^2)*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x]*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]] + I*(2*Sqrt[a]*Sqrt[b]*Cosh[c + d
*x] + d*(a + b*x^2)*Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] + d*(a + b*x^2)*C
osh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])))/(Sqrt[a]*b^(3/2)*(a + b*x^2))

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Maple [A]  time = 0.044, size = 291, normalized size = 1.2 \begin{align*} -{\frac{{d}^{2}{{\rm e}^{-dx-c}}}{4\,b \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) }}-{\frac{d}{8\,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}{\sqrt{-ab}}}}+{\frac{d}{8\,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}{\sqrt{-ab}}}}-{\frac{{d}^{2}{{\rm e}^{dx+c}}}{4\,b \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) }}-{\frac{d}{8\,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}{\sqrt{-ab}}}}+{\frac{d}{8\,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}{\sqrt{-ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*d^2*exp(-d*x-c)/b/(b*d^2*x^2+a*d^2)-1/8*d/b/(-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)+1/8*d/b/(-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*d^2*exp(d*x+c)/b/(b*d^2*x^2+a*d^2)-1/8*d/b/(-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)+1/8*d/b/(-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)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.16685, size = 1393, normalized size = 5.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*a*cosh(d*x + c) + (((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*sinh(d*x + c)^2)*sqrt(-a*d^2/b)*Ei(d*x -
sqrt(-a*d^2/b)) - ((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*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^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*sinh(d*x + c)^2)*sqrt(-a*d^2
/b)*Ei(d*x + sqrt(-a*d^2/b)) - ((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*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^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*sinh(d*x + c)^2
)*sqrt(-a*d^2/b)*Ei(d*x - sqrt(-a*d^2/b)) + ((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*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)) + (((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*sin
h(d*x + c)^2)*sqrt(-a*d^2/b)*Ei(d*x + sqrt(-a*d^2/b)) + ((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*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*b^2*x^2 + a^2*b)*cosh(d*x + c)^
2 - (a*b^2*x^2 + a^2*b)*sinh(d*x + c)^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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