\(\int \frac {x^2 \cosh (c+d x)}{(a+b x^2)^3} \, dx\) [73]

Optimal result

Integrand size = 19, antiderivative size = 746 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}+\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^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 )}{16 a b^2}-\frac {d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}+\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2}+\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}} \]

[Out]

-1/4*x*cosh(d*x+c)/b/(b*x^2+a)^2+1/16*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c-d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^
(3/2)-1/16*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c+d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(3/2)-1/16*d*cosh(c+d*(-a)
^(1/2)/b^(1/2))*Shi(d*x-d*(-a)^(1/2)/b^(1/2))/a/b^2-1/16*d*cosh(c-d*(-a)^(1/2)/b^(1/2))*Shi(d*x+d*(-a)^(1/2)/b
^(1/2))/a/b^2-1/8*d*sinh(d*x+c)/b^2/(b*x^2+a)-1/16*d*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2)
)/a/b^2+1/16*Shi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/(-a)^(3/2)/b^(3/2)-1/16*d*Chi(-d*x+d*(
-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2))/a/b^2-1/16*Shi(d*x-d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b
^(1/2))/(-a)^(3/2)/b^(3/2)-1/16*d^2*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c-d*(-a)^(1/2)/b^(1/2))/b^(5/2)/(-a)^(1
/2)+1/16*d^2*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c+d*(-a)^(1/2)/b^(1/2))/b^(5/2)/(-a)^(1/2)-1/16*d^2*Shi(d*x+d
*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/b^(5/2)/(-a)^(1/2)+1/16*d^2*Shi(d*x-d*(-a)^(1/2)/b^(1/2))*si
nh(c+d*(-a)^(1/2)/b^(1/2))/b^(5/2)/(-a)^(1/2)-1/16*cosh(d*x+c)/a/b^(3/2)/((-a)^(1/2)-x*b^(1/2))+1/16*cosh(d*x+
c)/a/b^(3/2)/((-a)^(1/2)+x*b^(1/2))

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 746, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5399, 5389, 3378, 3384, 3379, 3382, 5396} \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {d^2 \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}-\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d^2 \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}+\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^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 )}{16 a b^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 )}{16 a b^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 )}{16 a b^2}-\frac {d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2} \]

[In]

Int[(x^2*Cosh[c + d*x])/(a + b*x^2)^3,x]

[Out]

-1/16*Cosh[c + d*x]/(a*b^(3/2)*(Sqrt[-a] - Sqrt[b]*x)) + Cosh[c + d*x]/(16*a*b^(3/2)*(Sqrt[-a] + Sqrt[b]*x)) -
(x*Cosh[c + d*x])/(4*b*(a + b*x^2)^2) - (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d
*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(
16*Sqrt[-a]*b^(5/2)) + (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2
)*b^(3/2)) - (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*Sqrt[-a]*b^(5/2
)) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a*b^2) - (d*CoshIntegral[
(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a*b^2) - (d*Sinh[c + d*x])/(8*b^2*(a + b*x^2))
+ (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a*b^2) + (Sinh[c + (Sqrt[-a
]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) - (d^2*Sinh[c + (Sqrt[-a]*d)/S
qrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*Sqrt[-a]*b^(5/2)) - (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*Si
nhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a*b^2) + (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)
/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) - (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[
b] + d*x])/(16*Sqrt[-a]*b^(5/2))

Rule 3378

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 3379

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 3382

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 3384

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 5389

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 5396

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[e^m*(a + b*x^
n)^(p + 1)*(Sinh[c + d*x]/(b*n*(p + 1))), x] - Dist[d*(e^m/(b*n*(p + 1))), Int[(a + b*x^n)^(p + 1)*Cosh[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 5399

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])

Rubi steps \begin{align*} \text {integral}& = -\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac {\int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 b}+\frac {d \int \frac {x \sinh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 b} \\ & = -\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac {\int \left (-\frac {b \cosh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \cosh (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {b \cosh (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{4 b}+\frac {d^2 \int \frac {\cosh (c+d x)}{a+b x^2} \, dx}{8 b^2} \\ & = -\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac {\int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 a}-\frac {\int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 a}-\frac {\int \frac {\cosh (c+d x)}{-a b-b^2 x^2} \, dx}{8 a}+\frac {d^2 \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}{8 b^2} \\ & = -\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac {\int \left (-\frac {\sqrt {-a} \cosh (c+d x)}{2 a b \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {-a} \cosh (c+d x)}{2 a b \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{8 a}+\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a b}-\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a b}-\frac {d^2 \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 \sqrt {-a} b^2}-\frac {d^2 \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 \sqrt {-a} b^2} \\ & = -\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac {\int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}+\frac {\int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{3/2} 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}+b x} \, dx}{16 a b}-\frac {\left (d^2 \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}{16 \sqrt {-a} b^2}-\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}-b x} \, dx}{16 a b}-\frac {\left (d^2 \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}{16 \sqrt {-a} b^2}-\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}+b x} \, dx}{16 a b}-\frac {\left (d^2 \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}{16 \sqrt {-a} b^2}+\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}-b x} \, dx}{16 a b}+\frac {\left (d^2 \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}{16 \sqrt {-a} b^2} \\ & = -\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^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 )}{16 a b^2}-\frac {d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}-\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}}+\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}{16 (-a)^{3/2} 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}{16 (-a)^{3/2} 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}{16 (-a)^{3/2} 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}{16 (-a)^{3/2} b} \\ & = -\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}+\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^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 )}{16 a b^2}-\frac {d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}+\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2}+\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.73 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.51 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {-i e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (b-i \sqrt {a} \sqrt {b} d+a d^2\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\left (b+i \sqrt {a} \sqrt {b} d+a d^2\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (i b+\sqrt {a} \sqrt {b} d+i a d^2\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-i \left (b+i \sqrt {a} \sqrt {b} d+a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )-\frac {4 \sqrt {a} \sqrt {b} \cosh (d x) \left (-b x \left (-a+b x^2\right ) \cosh (c)+a d \left (a+b x^2\right ) \sinh (c)\right )}{\left (a+b x^2\right )^2}-\frac {4 \sqrt {a} \sqrt {b} \left (a d \left (a+b x^2\right ) \cosh (c)+b x \left (a-b x^2\right ) \sinh (c)\right ) \sinh (d x)}{\left (a+b x^2\right )^2}}{32 a^{3/2} b^{5/2}} \]

[In]

Integrate[(x^2*Cosh[c + d*x])/(a + b*x^2)^3,x]

[Out]

((-I)*E^(c - (I*Sqrt[a]*d)/Sqrt[b])*((b - I*Sqrt[a]*Sqrt[b]*d + a*d^2)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegr
alEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] - (b + I*Sqrt[a]*Sqrt[b]*d + a*d^2)*ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b]
+ x)]) + E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*((I*b + Sqrt[a]*Sqrt[b]*d + I*a*d^2)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*Exp
IntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] - I*(b + I*Sqrt[a]*Sqrt[b]*d + a*d^2)*ExpIntegralEi[(I*Sqrt[a]*d)/S
qrt[b] - d*x]) - (4*Sqrt[a]*Sqrt[b]*Cosh[d*x]*(-(b*x*(-a + b*x^2)*Cosh[c]) + a*d*(a + b*x^2)*Sinh[c]))/(a + b*
x^2)^2 - (4*Sqrt[a]*Sqrt[b]*(a*d*(a + b*x^2)*Cosh[c] + b*x*(a - b*x^2)*Sinh[c])*Sinh[d*x])/(a + b*x^2)^2)/(32*
a^(3/2)*b^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2171\) vs. \(2(574)=1148\).

Time = 0.36 (sec) , antiderivative size = 2172, normalized size of antiderivative = 2.91

method	result	size
risch	\(\text {Expression too large to display}\)	\(2172\)

[In]

int(x^2*cosh(d*x+c)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/32/a*(2*exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a^2*b*d^2*x^2-2*exp(-(d*(-a*b)^
(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*(-a*b)^(1/2)*a*b*d*x^2-2*(-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)*a*b*d*x^2-2*exp(d*x+c)*a*b*d*x^2*(-a*b)^(1/2)+2*exp(-d*x-c)
*a*b*d*x^2*(-a*b)^(1/2)+2*b^2*x^3*(-a*b)^(1/2)*exp(d*x+c)-exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*
x+c)*b+c*b)/b)*a*b^2*d^2*x^4+exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a*b^2*d^2*x^
4+exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*(-a*b)^(1/2)*b^2*d*x^4+exp((-d*(-a*b)^(1/
2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*(-a*b)^(1/2)*b^2*d*x^4-2*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(
d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a^2*b*d^2*x^2+2*b^2*x^3*(-a*b)^(1/2)*exp(-d*x-c)+2*exp((d*(-a*b)^(1/2)+c*b)/b
)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*(-a*b)^(1/2)*a*b*d*x^2+2*exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)
^(1/2)+(d*x+c)*b-c*b)/b)*(-a*b)^(1/2)*a*b*d*x^2-2*exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b
+c*b)/b)*a*b^2*x^2+2*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a*b^2*x^2-exp(-(d*(-
a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*(-a*b)^(1/2)*a^2*d-(-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)*a^2*d-2*exp(d*x+c)*a^2*d*(-a*b)^(1/2)+2*exp(-d*x-c)*a^2*d*(-
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)*b^3*x^4+exp((-d*(-a*b)^(1/2)+c*b
)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*b^3*x^4-exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*
b+c*b)/b)*a^3*d^2+exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a^3*d^2-exp((d*(-a*b)^(
1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a^2*b+exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(
d*x+c)*b-c*b)/b)*a^2*b-exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*b^3*x^4+exp(-(-d*(
-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*b^3*x^4-exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b
)^(1/2)-(d*x+c)*b+c*b)/b)*a^3*d^2+exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a^3*d^2
-exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a^2*b+exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1
,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a^2*b-exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*
a*b^2*d^2*x^4+exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a*b^2*d^2*x^4-exp(-(d*(-a*b
)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*(-a*b)^(1/2)*b^2*d*x^4-(-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)*b^2*d*x^4-2*exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1
/2)-(d*x+c)*b+c*b)/b)*a^2*b*d^2*x^2+2*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a^2
*b*d^2*x^2-2*exp(d*x+c)*a*b*x*(-a*b)^(1/2)-2*exp(-d*x-c)*a*b*x*(-a*b)^(1/2)-2*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1
,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a*b^2*x^2+2*exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*
b)/b)*a*b^2*x^2+exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*(-a*b)^(1/2)*a^2*d+exp((-d*
(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*(-a*b)^(1/2)*a^2*d)/b^2/(b^2*x^4+2*a*b*x^2+a^2)/(
-a*b)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2047 vs. \(2 (575) = 1150\).

Time = 0.28 (sec) , antiderivative size = 2047, normalized size of antiderivative = 2.74 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(x^2*cosh(d*x+c)/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/32*(4*(a*b^2*d*x^3 - a^2*b*d*x)*cosh(d*x + c) - (((a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^
2 - (a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x + c)^2 + ((a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b +
2*(a^2*b*d^2 + a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*
x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) - ((a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*
cosh(d*x + c)^2 - (a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x + c)^2 - ((a^3*d^2 + (a*b^2*d^2 + b^3)*
x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b
*d^2 + a*b^2)*x^2)*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)) - (((a
*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - (a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d
*x + c)^2 - ((a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2
+ (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a
*d^2/b)) - ((a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - (a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a
^3*d^2)*sinh(d*x + c)^2 + ((a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*cosh(d*x + c)
^2 - (a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*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)) - 4*(a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x + c) - (((a*b^2*d^2*
x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - (a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x + c)^2
+ ((a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*
d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b))
+ ((a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - (a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*s
inh(d*x + c)^2 - ((a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3
*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(-d*x + s
qrt(-a*d^2/b)))*sinh(c + sqrt(-a*d^2/b)) + (((a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - (a*
b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x + c)^2 - ((a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*
b*d^2 + a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*si
nh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) + ((a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*
x + c)^2 - (a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*sinh(d*x + c)^2 + ((a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a
^2*b + 2*(a^2*b*d^2 + a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 + b^3)*x^4 + a^2*b + 2*(a^2*b*d^2 +
a*b^2)*x^2)*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*b^4*d
*x^4 + 2*a^3*b^3*d*x^2 + a^4*b^2*d)*cosh(d*x + c)^2 - (a^2*b^4*d*x^4 + 2*a^3*b^3*d*x^2 + a^4*b^2*d)*sinh(d*x +
c)^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(x**2*cosh(d*x+c)/(b*x**2+a)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {x^{2} \cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]

[In]

integrate(x^2*cosh(d*x+c)/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/2*((d*x^2*e^(2*c) + 4*x*e^(2*c))*e^(d*x) - (d*x^2 - 4*x)*e^(-d*x))/(b^3*d^2*x^6*e^c + 3*a*b^2*d^2*x^4*e^c +
3*a^2*b*d^2*x^2*e^c + a^3*d^2*e^c) + 1/2*integrate(-2*(3*a*d*x*e^c - 10*b*x^2*e^c + 2*a*e^c)*e^(d*x)/(b^4*d^2*
x^8 + 4*a*b^3*d^2*x^6 + 6*a^2*b^2*d^2*x^4 + 4*a^3*b*d^2*x^2 + a^4*d^2), x) + 1/2*integrate(2*(3*a*d*x + 10*b*x
^2 - 2*a)*e^(-d*x)/(b^4*d^2*x^8*e^c + 4*a*b^3*d^2*x^6*e^c + 6*a^2*b^2*d^2*x^4*e^c + 4*a^3*b*d^2*x^2*e^c + a^4*
d^2*e^c), x)

Giac [F]

\[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {x^{2} \cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]

[In]

integrate(x^2*cosh(d*x+c)/(b*x^2+a)^3,x, algorithm="giac")

[Out]

integrate(x^2*cosh(d*x + c)/(b*x^2 + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int \frac {x^2\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^3} \,d x \]

[In]

int((x^2*cosh(c + d*x))/(a + b*x^2)^3,x)

[Out]

int((x^2*cosh(c + d*x))/(a + b*x^2)^3, x)