\(\int \frac {x^2 \cosh (c+d x)}{(a+b x^2)^2} \, dx\) [67]

Optimal result

Integrand size = 19, antiderivative size = 416 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {x \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 \sqrt {-a} b^{3/2}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 \sqrt {-a} b^{3/2}}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^2}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^2}-\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^2}-\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 \sqrt {-a} b^{3/2}}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^2}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 \sqrt {-a} b^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5399, 5389, 3384, 3379, 3382, 5400} \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 \sqrt {-a} b^{3/2}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 \sqrt {-a} b^{3/2}}-\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 \sqrt {-a} b^{3/2}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 \sqrt {-a} b^{3/2}}+\frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^2}+\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^2}-\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^2}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^2}-\frac {x \cosh (c+d x)}{2 b \left (a+b x^2\right )} \]

[In]

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

[Out]

-1/2*(x*Cosh[c + d*x])/(b*(a + b*x^2)) + (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d
*x])/(4*Sqrt[-a]*b^(3/2)) - (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*Sqrt[
-a]*b^(3/2)) + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*b^2) + (d*CoshIn
tegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*b^2) - (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]
*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^2) - (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)
/Sqrt[b] - d*x])/(4*Sqrt[-a]*b^(3/2)) + (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] +
d*x])/(4*b^2) - (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*Sqrt[-a]*b^(3/2))

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 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])

Rule 5400

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])

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.07 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.69 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {\frac {4 b x \cosh (c) \cosh (d x)}{a+b x^2}-\frac {e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (-i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )}{\sqrt {a}}+\frac {e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (-i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{\sqrt {a}}+\frac {4 b x \sinh (c) \sinh (d x)}{a+b x^2}}{8 b^2} \]

[In]

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

[Out]

-1/8*((4*b*x*Cosh[c]*Cosh[d*x])/(a + b*x^2) - (E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(((-I)*Sqrt[b] + Sqrt[a]*d)*E^(((
2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + (I*Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi
[d*((I*Sqrt[a])/Sqrt[b] + x)]))/Sqrt[a] + (E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*(((-I)*Sqrt[b] + Sqrt[a]*d)*E^(((2*I
)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] + (I*Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi[(I
*Sqrt[a]*d)/Sqrt[b] - d*x]))/Sqrt[a] + (4*b*x*Sinh[c]*Sinh[d*x])/(a + b*x^2))/b^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(900\) vs. \(2(318)=636\).

Time = 0.25 (sec) , antiderivative size = 901, normalized size of antiderivative = 2.17

method	result	size
risch	\(-\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) \sqrt {-a b}\, b d \,x^{2}+{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) \sqrt {-a b}\, b d \,x^{2}-{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) \sqrt {-a b}\, b d \,x^{2}-\sqrt {-a b}\, {\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) b d \,x^{2}+{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) b^{2} x^{2}-{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) b^{2} x^{2}+{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) b^{2} x^{2}-{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) b^{2} x^{2}+{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) \sqrt {-a b}\, a d +{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) \sqrt {-a b}\, a d -{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) \sqrt {-a b}\, a d -\sqrt {-a b}\, {\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a d +{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a b -{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a b +2 \,{\mathrm e}^{-d x -c} x b \sqrt {-a b}+{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a b +2 \,{\mathrm e}^{d x +c} x b \sqrt {-a b}-{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a b}{8 b^{2} \left (b \,x^{2}+a \right ) \sqrt {-a b}}\)	\(901\)

[In]

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

[Out]

-1/8*(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*d*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)*b*d*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)*b*d*x^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)*b*d*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)*b^2*x^2-e
xp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*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)*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
)*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*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*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*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*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-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*exp(-d*x-c)*x*b*(-a*b)^(1/2)+exp(-(d*(-a*b)^(1/2)+c*
b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a*b+2*exp(d*x+c)*x*b*(-a*b)^(1/2)-exp(-(-d*(-a*b)^(1/2)+c*b)/b)*
Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a*b)/b^2/(b*x^2+a)/(-a*b)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1162 vs. \(2 (318) = 636\).

Time = 0.27 (sec) , antiderivative size = 1162, normalized size of antiderivative = 2.79 \[ \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(x^2*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 - ((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 + ((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)) - (((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 + ((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 - ((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)) - (((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 - ((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 + ((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)*c
osh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x + c)^2 + ((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 - ((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^3*d*x^2 + a^2*b^2*d)*cosh(d*x + c)^2 - (a
*b^3*d*x^2 + a^2*b^2*d)*sinh(d*x + c)^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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