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

Optimal. Leaf size=746 \[ \frac{d^2 \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^2}-\frac{d \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{\cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d^2 \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}+\frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{\sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{d \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^2}-\frac{d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2} \]

[Out]

-Cosh[c + d*x]/(16*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[(S
qrt[-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)/Sqr
t[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*Sqrt[-a]*b^(5/2)) - (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*Sinh
Integral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a*b^2) + (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/S
qrt[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))

________________________________________________________________________________________

Rubi [A]  time = 1.10007, antiderivative size = 746, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {5291, 5281, 3297, 3303, 3298, 3301, 5288} \[ \frac{d^2 \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^2}-\frac{d \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{\cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac{d^2 \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}+\frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{\sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{3/2} b^{3/2}}+\frac{d \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a b^2}-\frac{d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a b^2}-\frac{d \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\cosh (c+d x)}{16 a b^{3/2} \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{x \cosh (c+d x)}{4 b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cosh[c + d*x]/(16*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[(S
qrt[-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)/Sqr
t[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*Sqrt[-a]*b^(5/2)) - (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*Sinh
Integral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a*b^2) + (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/S
qrt[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 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 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 3297

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 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 5288

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])

Rubi steps

\begin{align*} \int \frac{x^2 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx &=-\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]  time = 2.50882, size = 932, normalized size = 1.25 \[ \frac{\frac{2 \sqrt{a} b^2 \cosh (c) \cosh (d x) x^3}{\left (b x^2+a\right )^2}+\frac{2 \sqrt{a} b^2 \sinh (c) \sinh (d x) x^3}{\left (b x^2+a\right )^2}-\frac{2 a^{3/2} b d \cosh (d x) \sinh (c) x^2}{\left (b x^2+a\right )^2}-\frac{2 a^{3/2} b d \cosh (c) \sinh (d x) x^2}{\left (b x^2+a\right )^2}-\frac{2 a^{3/2} b \cosh (c) \cosh (d x) x}{\left (b x^2+a\right )^2}-\frac{2 a^{3/2} b \sinh (c) \sinh (d x) x}{\left (b x^2+a\right )^2}-\frac{2 a^{5/2} d \cosh (d x) \sinh (c)}{\left (b x^2+a\right )^2}+\frac{i \text{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\left (a d^2+b\right ) \cosh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )+i \sqrt{a} \sqrt{b} d \sinh \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )}{\sqrt{b}}-\frac{i \text{CosIntegral}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\left (a d^2+b\right ) \cosh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )-i \sqrt{a} \sqrt{b} d \sinh \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )}{\sqrt{b}}-\frac{2 a^{5/2} d \cosh (c) \sinh (d x)}{\left (b x^2+a\right )^2}-i \sqrt{a} d \cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \cosh (c) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )+\frac{i a d^2 \cosh (c) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )}{\sqrt{b}}+i \sqrt{b} \cosh (c) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )-\frac{a d^2 \cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sinh (c) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )}{\sqrt{b}}-\sqrt{b} \cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sinh (c) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )-\sqrt{a} d \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sinh (c) \text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )+i \sqrt{a} d \cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \cosh (c) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )-\frac{i a d^2 \cosh (c) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )}{\sqrt{b}}-i \sqrt{b} \cosh (c) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )-\frac{a d^2 \cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sinh (c) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )}{\sqrt{b}}-\sqrt{b} \cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sinh (c) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )-\sqrt{a} d \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sinh (c) \text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )}{16 a^{3/2} b^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*a^(3/2)*b*x*Cosh[c]*Cosh[d*x])/(a + b*x^2)^2 + (2*Sqrt[a]*b^2*x^3*Cosh[c]*Cosh[d*x])/(a + b*x^2)^2 - (2*a
^(5/2)*d*Cosh[d*x]*Sinh[c])/(a + b*x^2)^2 - (2*a^(3/2)*b*d*x^2*Cosh[d*x]*Sinh[c])/(a + b*x^2)^2 + (I*CosIntegr
al[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x]*((b + a*d^2)*Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]] + I*Sqrt[a]*Sqrt[b]*d*Sinh[c -
(I*Sqrt[a]*d)/Sqrt[b]]))/Sqrt[b] - (I*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x]*((b + a*d^2)*Cosh[c + (I*Sqrt[
a]*d)/Sqrt[b]] - I*Sqrt[a]*Sqrt[b]*d*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]))/Sqrt[b] - (2*a^(5/2)*d*Cosh[c]*Sinh[d*x
])/(a + b*x^2)^2 - (2*a^(3/2)*b*d*x^2*Cosh[c]*Sinh[d*x])/(a + b*x^2)^2 - (2*a^(3/2)*b*x*Sinh[c]*Sinh[d*x])/(a
+ b*x^2)^2 + (2*Sqrt[a]*b^2*x^3*Sinh[c]*Sinh[d*x])/(a + b*x^2)^2 - I*Sqrt[a]*d*Cos[(Sqrt[a]*d)/Sqrt[b]]*Cosh[c
]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] + I*Sqrt[b]*Cosh[c]*Sin[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d
)/Sqrt[b] - I*d*x] + (I*a*d^2*Cosh[c]*Sin[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x])/Sqrt[
b] - Sqrt[b]*Cos[(Sqrt[a]*d)/Sqrt[b]]*Sinh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] - (a*d^2*Cos[(Sqrt[a]*d
)/Sqrt[b]]*Sinh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x])/Sqrt[b] - Sqrt[a]*d*Sin[(Sqrt[a]*d)/Sqrt[b]]*Sinh
[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] + I*Sqrt[a]*d*Cos[(Sqrt[a]*d)/Sqrt[b]]*Cosh[c]*SinIntegral[(Sqrt[
a]*d)/Sqrt[b] + I*d*x] - I*Sqrt[b]*Cosh[c]*Sin[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] -
(I*a*d^2*Cosh[c]*Sin[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])/Sqrt[b] - Sqrt[b]*Cos[(Sq
rt[a]*d)/Sqrt[b]]*Sinh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] - (a*d^2*Cos[(Sqrt[a]*d)/Sqrt[b]]*Sinh[c]*S
inIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])/Sqrt[b] - Sqrt[a]*d*Sin[(Sqrt[a]*d)/Sqrt[b]]*Sinh[c]*SinIntegral[(Sqr
t[a]*d)/Sqrt[b] + I*d*x])/(16*a^(3/2)*b^2)

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Maple [A]  time = 0.108, size = 1064, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32/b/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)-1/32/b/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)-1/32*d/b^2/a*exp(-(d*(-a*b)^(1/2)+c*b)/b
)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)+1/32*d^2/b^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)-1/32*d^2/b^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)-1/32*d/b^2/a*exp((d*(-a*b)^(1/2)-c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)+1/16*d^5*exp(-d*x-c)
/b/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*x^2+1/16*d^4*exp(-d*x-c)/a/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*x^3+1/16
*d^5*exp(-d*x-c)*a/b^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)-1/16*d^4*exp(-d*x-c)/b/(b^2*d^4*x^4+2*a*b*d^4*x^2+a
^2*d^4)*x-1/32/b/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)+1/32/b/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)-1/32*d^2/b^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)+1/32*d^2/b^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)+1/32*d/b^2/a*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-
(d*x+c)*b+c*b)/b)+1/32*d/b^2/a*exp(-(d*(-a*b)^(1/2)-c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)-1/16*d^5*e
xp(d*x+c)/b/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*x^2+1/16*d^4*exp(d*x+c)/a/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*
x^3-1/16*d^5*exp(d*x+c)*a/b^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)-1/16*d^4*exp(d*x+c)/b/(b^2*d^4*x^4+2*a*b*d^4
*x^2+a^2*d^4)*x

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.39734, size = 4169, normalized size = 5.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)