3.65 \(\int \frac{x^4 \cosh (c+d x)}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=449 \[ -\frac{a d \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^3}-\frac{a d \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^3}+\frac{3 \sqrt{-a} \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}-\frac{3 \sqrt{-a} \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^{5/2}}-\frac{3 \sqrt{-a} \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}-\frac{3 \sqrt{-a} \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^{5/2}}+\frac{a d \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^3}-\frac{a d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^3}-\frac{x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac{\sinh (c+d x)}{b^2 d}+\frac{x \cosh (c+d x)}{2 b^2} \]

[Out]

(x*Cosh[c + d*x])/(2*b^2) - (x^3*Cosh[c + d*x])/(2*b*(a + b*x^2)) + (3*Sqrt[-a]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]
*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) - (3*Sqrt[-a]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegr
al[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2)) - (a*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]
*d)/Sqrt[b]])/(4*b^3) - (a*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*b^3)
+ Sinh[c + d*x]/(b^2*d) + (a*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^3
) - (3*Sqrt[-a]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) - (a*d*Co
sh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^3) - (3*Sqrt[-a]*Sinh[c - (Sqrt[-a
]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2))

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Rubi [A]  time = 0.859464, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {5291, 5293, 2637, 5281, 3303, 3298, 3301, 5292, 3296} \[ -\frac{a d \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^3}-\frac{a d \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^3}+\frac{3 \sqrt{-a} \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}-\frac{3 \sqrt{-a} \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^{5/2}}-\frac{3 \sqrt{-a} \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}-\frac{3 \sqrt{-a} \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^{5/2}}+\frac{a d \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^3}-\frac{a d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^3}-\frac{x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac{\sinh (c+d x)}{b^2 d}+\frac{x \cosh (c+d x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*Cosh[c + d*x])/(a + b*x^2)^2,x]

[Out]

(x*Cosh[c + d*x])/(2*b^2) - (x^3*Cosh[c + d*x])/(2*b*(a + b*x^2)) + (3*Sqrt[-a]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]
*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) - (3*Sqrt[-a]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegr
al[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2)) - (a*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]
*d)/Sqrt[b]])/(4*b^3) - (a*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*b^3)
+ Sinh[c + d*x]/(b^2*d) + (a*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^3
) - (3*Sqrt[-a]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) - (a*d*Co
sh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^3) - (3*Sqrt[-a]*Sinh[c - (Sqrt[-a
]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2))

Rule 5291

Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m - n + 1)*(a + b
*x^n)^(p + 1)*Cosh[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*(a + b*x^n)^(
p + 1)*Cosh[c + d*x], x], x] - Dist[d/(b*n*(p + 1)), Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d*x], x],
x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 0] && RationalQ[m] && (GtQ[m - n + 1, 0] || GtQ[n, 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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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])

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 5292

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

Rubi steps

\begin{align*} \int \frac{x^4 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx &=-\frac{x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac{3 \int \frac{x^2 \cosh (c+d x)}{a+b x^2} \, dx}{2 b}+\frac{d \int \frac{x^3 \sinh (c+d x)}{a+b x^2} \, dx}{2 b}\\ &=-\frac{x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac{3 \int \left (\frac{\cosh (c+d x)}{b}-\frac{a \cosh (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx}{2 b}+\frac{d \int \left (\frac{x \sinh (c+d x)}{b}-\frac{a x \sinh (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx}{2 b}\\ &=-\frac{x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac{3 \int \cosh (c+d x) \, dx}{2 b^2}-\frac{(3 a) \int \frac{\cosh (c+d x)}{a+b x^2} \, dx}{2 b^2}+\frac{d \int x \sinh (c+d x) \, dx}{2 b^2}-\frac{(a d) \int \frac{x \sinh (c+d x)}{a+b x^2} \, dx}{2 b^2}\\ &=\frac{x \cosh (c+d x)}{2 b^2}-\frac{x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac{3 \sinh (c+d x)}{2 b^2 d}-\frac{\int \cosh (c+d x) \, dx}{2 b^2}-\frac{(3 a) \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}{2 b^2}-\frac{(a d) \int \left (-\frac{\sinh (c+d x)}{2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sinh (c+d x)}{2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{2 b^2}\\ &=\frac{x \cosh (c+d x)}{2 b^2}-\frac{x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac{\sinh (c+d x)}{b^2 d}-\frac{\left (3 \sqrt{-a}\right ) \int \frac{\cosh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{4 b^2}-\frac{\left (3 \sqrt{-a}\right ) \int \frac{\cosh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{4 b^2}+\frac{(a d) \int \frac{\sinh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{4 b^{5/2}}-\frac{(a d) \int \frac{\sinh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{4 b^{5/2}}\\ &=\frac{x \cosh (c+d x)}{2 b^2}-\frac{x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac{\sinh (c+d x)}{b^2 d}-\frac{\left (3 \sqrt{-a} \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}{4 b^2}-\frac{\left (a 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 b^{5/2}}-\frac{\left (3 \sqrt{-a} \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}{4 b^2}-\frac{\left (a 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 b^{5/2}}-\frac{\left (3 \sqrt{-a} \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}{4 b^2}-\frac{\left (a 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 b^{5/2}}+\frac{\left (3 \sqrt{-a} \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}{4 b^2}+\frac{\left (a 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 b^{5/2}}\\ &=\frac{x \cosh (c+d x)}{2 b^2}-\frac{x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac{3 \sqrt{-a} \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}-\frac{3 \sqrt{-a} \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 b^{5/2}}-\frac{a d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^3}-\frac{a d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^3}+\frac{\sinh (c+d x)}{b^2 d}+\frac{a d \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^3}-\frac{3 \sqrt{-a} \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}-\frac{a d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 b^3}-\frac{3 \sqrt{-a} \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 b^{5/2}}\\ \end{align*}

Mathematica [C]  time = 1.50657, size = 621, normalized size = 1.38 \[ \frac{-\frac{3 \sqrt{a} \sinh (c) \left (\sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (-\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+\sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )-\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )+\text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )\right )\right )}{\sqrt{b}}-\frac{a d \sinh (c) \left (\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (-\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+\sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )+\text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )\right )\right )}{b}-\frac{3 i \sqrt{a} \cosh (c) \left (\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (-\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )-\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+\sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )-\text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )\right )\right )}{\sqrt{b}}+\frac{i a d \cosh (c) \left (\sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (-\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )-\sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )-\text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )\right )\right )}{b}+2 \cosh (d x) \left (\frac{a x \cosh (c)}{a+b x^2}+\frac{2 \sinh (c)}{d}\right )+2 \sinh (d x) \left (\frac{a x \sinh (c)}{a+b x^2}+\frac{2 \cosh (c)}{d}\right )}{4 b^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*Cosh[c + d*x])/(a + b*x^2)^2,x]

[Out]

(2*Cosh[d*x]*((a*x*Cosh[c])/(a + b*x^2) + (2*Sinh[c])/d) + 2*((2*Cosh[c])/d + (a*x*Sinh[c])/(a + b*x^2))*Sinh[
d*x] - ((3*I)*Sqrt[a]*Cosh[c]*(Cos[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] - Cos[(Sqr
t[a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] + Sin[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[(Sqrt[a]*d)/
Sqrt[b] - I*d*x] - SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])))/Sqrt[b] + (I*a*d*Cosh[c]*(CosIntegral[-((Sqrt[a
]*d)/Sqrt[b]) + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]] - CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x]*Sin[(Sqrt[a]*d)/Sqr
t[b]] + Cos[(Sqrt[a]*d)/Sqrt[b]]*(-SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] + SinIntegral[(Sqrt[a]*d)/Sqrt[b]
+ I*d*x])))/b - (3*Sqrt[a]*Sinh[c]*(CosIntegral[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]] + Cos
Integral[(Sqrt[a]*d)/Sqrt[b] + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]] - Cos[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[(Sqrt[a
]*d)/Sqrt[b] - I*d*x] + SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])))/Sqrt[b] - (a*d*Sinh[c]*(Cos[(Sqrt[a]*d)/Sq
rt[b]]*CosIntegral[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] + Cos[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[a]*d)/Sqrt[b]
+ I*d*x] + Sin[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] + SinIntegral[(Sqrt[a]*d)/Sqrt[b
] + I*d*x])))/b)/(4*b^2)

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Maple [A]  time = 0.319, size = 532, normalized size = 1.2 \begin{align*}{\frac{{d}^{2}{{\rm e}^{-dx-c}}ax}{4\,{b}^{2} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) }}-{\frac{3\,a}{8\,{b}^{2}}{{\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{3\,a}{8\,{b}^{2}}{{\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{{{\rm e}^{-dx-c}}}{2\,d{b}^{2}}}-{\frac{da}{8\,{b}^{3}}{{\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{da}{8\,{b}^{3}}{{\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}^{dx+c}}}{2\,d{b}^{2}}}+{\frac{{d}^{2}{{\rm e}^{dx+c}}ax}{4\,{b}^{2} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) }}+{\frac{3\,a}{8\,{b}^{2}}{{\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{3\,a}{8\,{b}^{2}}{{\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{da}{8\,{b}^{3}}{{\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{da}{8\,{b}^{3}}{{\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^4*cosh(d*x+c)/(b*x^2+a)^2,x)

[Out]

1/4*d^2*exp(-d*x-c)*a/b^2/(b*d^2*x^2+a*d^2)*x-3/8/b^2*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)+3/8/b^2*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*exp(-d*x-c)/b^2-1/8*d/b^3*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
/8*d/b^3*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/d/b^2*exp(d*x+c)+1/4*d^2*e
xp(d*x+c)*a/b^2/(b*d^2*x^2+a*d^2)*x+3/8/b^2*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)-3/8/b^2*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/
8*d/b^3*a*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)+1/8*d/b^3*a*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^4*cosh(d*x+c)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.31252, size = 2488, normalized size = 5.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*cosh(d*x+c)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/8*(4*a*b*d*x*cosh(d*x + c) - (((a*b*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x +
c)^2 - 3*((b^2*x^2 + a*b)*cosh(d*x + c)^2 - (b^2*x^2 + a*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*
d^2/b)) - ((a*b*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x + c)^2 + 3*((b^2*x^2 + a
*b)*cosh(d*x + c)^2 - (b^2*x^2 + a*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(-d*x + sqrt(-a*d^2/b)))*cosh(c + sqr
t(-a*d^2/b)) - (((a*b*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x + c)^2 + 3*((b^2*x
^2 + a*b)*cosh(d*x + c)^2 - (b^2*x^2 + a*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) - ((a*b*
d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x + c)^2 - 3*((b^2*x^2 + a*b)*cosh(d*x + c
)^2 - (b^2*x^2 + a*b)*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)) +
8*(b^2*x^2 + a*b)*sinh(d*x + c) - (((a*b*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x
+ c)^2 - 3*((b^2*x^2 + a*b)*cosh(d*x + c)^2 - (b^2*x^2 + a*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(
-a*d^2/b)) + ((a*b*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x + c)^2 + 3*((b^2*x^2
+ a*b)*cosh(d*x + c)^2 - (b^2*x^2 + a*b)*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*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x + c)^2 + 3*((b^
2*x^2 + a*b)*cosh(d*x + c)^2 - (b^2*x^2 + a*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) + ((a
*b*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x + c)^2 - 3*((b^2*x^2 + a*b)*cosh(d*x
+ c)^2 - (b^2*x^2 + a*b)*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^4*d*x^2 + a*b^3*d)*cosh(d*x + c)^2 - (b^4*d*x^2 + a*b^3*d)*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**4*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^{4} \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^4*cosh(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

integrate(x^4*cosh(d*x + c)/(b*x^2 + a)^2, x)