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 {\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} \]

[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]  time = 1.10, antiderivative size = 746, normalized size of antiderivative = 1.00, 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 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 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 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 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 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])

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])

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*}

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Mathematica [C]  time = 2.67, 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 {Ci}\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 {Ci}\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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fricas [B]  time = 0.48, size = 2047, normalized size = 2.74 \[ \text {result too large to display} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \]

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)

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maple [A]  time = 0.33, size = 1064, normalized size = 1.43 \[ \frac {d^{5} {\mathrm e}^{-d x -c} x^{2}}{16 b \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {d^{4} {\mathrm e}^{-d x -c} x^{3}}{16 a \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {d^{5} {\mathrm e}^{-d x -c} a}{16 b^{2} \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{4} {\mathrm e}^{-d x -c} x}{16 b \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b a \sqrt {-a b}}-\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b a \sqrt {-a b}}+\frac {d^{2} {\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b^{2} \sqrt {-a b}}-\frac {d^{2} {\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b^{2} \sqrt {-a b}}-\frac {d \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b^{2} a}-\frac {d \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b^{2} a}-\frac {d^{2} {\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b^{2} \sqrt {-a b}}+\frac {d^{2} {\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b^{2} \sqrt {-a b}}+\frac {d \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b^{2} a}+\frac {d \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b^{2} a}-\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b a \sqrt {-a b}}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b a \sqrt {-a b}}-\frac {d^{5} {\mathrm e}^{d x +c} x^{2}}{16 b \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {d^{4} {\mathrm e}^{d x +c} x^{3}}{16 a \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{5} {\mathrm e}^{d x +c} a}{16 b^{2} \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{4} {\mathrm e}^{d x +c} x}{16 b \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )} \]

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]

1/16*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)-1/32*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)-1/32/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)-1/16
*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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (d x^{2} e^{\left (2 \, c\right )} + 4 \, x e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} - {\left (d x^{2} - 4 \, x\right )} e^{\left (-d x\right )}}{2 \, {\left (b^{3} d^{2} x^{6} e^{c} + 3 \, a b^{2} d^{2} x^{4} e^{c} + 3 \, a^{2} b d^{2} x^{2} e^{c} + a^{3} d^{2} e^{c}\right )}} + \frac {1}{2} \, \int -\frac {2 \, {\left (3 \, a d x e^{c} - 10 \, b x^{2} e^{c} + 2 \, a e^{c}\right )} e^{\left (d x\right )}}{b^{4} d^{2} x^{8} + 4 \, a b^{3} d^{2} x^{6} + 6 \, a^{2} b^{2} d^{2} x^{4} + 4 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}}\,{d x} + \frac {1}{2} \, \int \frac {2 \, {\left (3 \, a d x + 10 \, b x^{2} - 2 \, a\right )} e^{\left (-d x\right )}}{b^{4} d^{2} x^{8} e^{c} + 4 \, a b^{3} d^{2} x^{6} e^{c} + 6 \, a^{2} b^{2} d^{2} x^{4} e^{c} + 4 \, a^{3} b d^{2} x^{2} e^{c} + a^{4} d^{2} e^{c}}\,{d x} \]

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]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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