\(\int \frac {\cosh (c+d x)}{x^2 (a+b x^2)^3} \, dx\) [77]

Optimal result

Integrand size = 19, antiderivative size = 874 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=-\frac {\cosh (c+d x)}{a^3 x}-\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )^2}-\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {15 \sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}+\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 (-a)^{5/2} \sqrt {b}}-\frac {15 \sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{7/2}}-\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 (-a)^{5/2} \sqrt {b}}+\frac {d \text {Chi}(d x) \sinh (c)}{a^3}+\frac {7 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^3}+\frac {7 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^3}+\frac {d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \cosh (c) \text {Shi}(d x)}{a^3}-\frac {7 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^3}-\frac {15 \sqrt {b} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}-\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 (-a)^{5/2} \sqrt {b}}+\frac {7 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^3}-\frac {15 \sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{7/2}}-\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 (-a)^{5/2} \sqrt {b}} \]

[Out]

-cosh(d*x+c)/a^3/x+d*cosh(c)*Shi(d*x)/a^3+7/16*d*cosh(c+d*(-a)^(1/2)/b^(1/2))*Shi(d*x-d*(-a)^(1/2)/b^(1/2))/a^
3+7/16*d*cosh(c-d*(-a)^(1/2)/b^(1/2))*Shi(d*x+d*(-a)^(1/2)/b^(1/2))/a^3+d*Chi(d*x)*sinh(c)/a^3+7/16*d*Chi(d*x+
d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/a^3+7/16*d*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/
2)/b^(1/2))/a^3-1/16*d^2*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c-d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(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))/(-a)^(5/2)/b^(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))/(-a)^(5/2)/b^(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))/(-a)^(5/2)/b^(1/2)-15/16*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c-d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-
a)^(7/2)+15/16*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(7/2)-15/16*Shi(d*x+d*
(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(7/2)+15/16*Shi(d*x-d*(-a)^(1/2)/b^(1/2))*sinh(c
+d*(-a)^(1/2)/b^(1/2))*b^(1/2)/(-a)^(7/2)-1/16*cosh(d*x+c)*b^(1/2)/(-a)^(5/2)/((-a)^(1/2)-x*b^(1/2))^2+1/16*d*
sinh(d*x+c)/(-a)^(5/2)/((-a)^(1/2)-x*b^(1/2))+7/16*cosh(d*x+c)*b^(1/2)/a^3/((-a)^(1/2)-x*b^(1/2))+1/16*cosh(d*
x+c)*b^(1/2)/(-a)^(5/2)/((-a)^(1/2)+x*b^(1/2))^2+1/16*d*sinh(d*x+c)/(-a)^(5/2)/((-a)^(1/2)+x*b^(1/2))-7/16*cos
h(d*x+c)*b^(1/2)/a^3/((-a)^(1/2)+x*b^(1/2))

Rubi [A] (verified)

Time = 2.04 (sec) , antiderivative size = 874, normalized size of antiderivative = 1.00, number of steps used = 60, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5401, 3378, 3384, 3379, 3382, 5389} \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}-\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{5/2} \sqrt {b}}+\frac {\text {Chi}(d x) \sinh (c) d}{a^3}+\frac {7 \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^3}+\frac {7 \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^3}+\frac {\sinh (c+d x) d}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sinh (c+d x) d}{16 (-a)^{5/2} \left (\sqrt {b} x+\sqrt {-a}\right )}+\frac {\cosh (c) \text {Shi}(d x) d}{a^3}-\frac {7 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^3}+\frac {7 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^3}-\frac {\cosh (c+d x)}{a^3 x}+\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {b} x+\sqrt {-a}\right )^2}+\frac {15 \sqrt {b} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}-\frac {15 \sqrt {b} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}}-\frac {15 \sqrt {b} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}-\frac {15 \sqrt {b} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{7/2}} \]

[In]

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

[Out]

-(Cosh[c + d*x]/(a^3*x)) - (Sqrt[b]*Cosh[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] - Sqrt[b]*x)^2) + (7*Sqrt[b]*Cosh[
c + d*x])/(16*a^3*(Sqrt[-a] - Sqrt[b]*x)) + (Sqrt[b]*Cosh[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] + Sqrt[b]*x)^2) -
(7*Sqrt[b]*Cosh[c + d*x])/(16*a^3*(Sqrt[-a] + Sqrt[b]*x)) + (15*Sqrt[b]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIn
tegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(7/2)) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-
a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (15*Sqrt[b]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[
-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2)) - (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b]
+ d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (d*CoshIntegral[d*x]*Sinh[c])/a^3 + (7*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b]
+ d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a^3) + (7*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt
[-a]*d)/Sqrt[b]])/(16*a^3) + (d*Sinh[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] - Sqrt[b]*x)) + (d*Sinh[c + d*x])/(16*
(-a)^(5/2)*(Sqrt[-a] + Sqrt[b]*x)) + (d*Cosh[c]*SinhIntegral[d*x])/a^3 - (7*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*S
inhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^3) - (15*Sqrt[b]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(S
qrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(7/2)) - (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqr
t[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (7*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] +
d*x])/(16*a^3) - (15*Sqrt[b]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a
)^(7/2)) - (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b
])

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 5401

Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cosh[c
+ d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (Eq
Q[n, 2] || EqQ[p, -1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\cosh (c+d x)}{a^3 x^2}-\frac {b \cosh (c+d x)}{a \left (a+b x^2\right )^3}-\frac {b \cosh (c+d x)}{a^2 \left (a+b x^2\right )^2}-\frac {b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{x^2} \, dx}{a^3}-\frac {b \int \frac {\cosh (c+d x)}{a+b x^2} \, dx}{a^3}-\frac {b \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a^2}-\frac {b \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx}{a} \\ & = -\frac {\cosh (c+d x)}{a^3 x}-\frac {b \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}{a^3}-\frac {b \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}{a^2}-\frac {b \int \left (-\frac {b^{3/2} \cosh (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}-b x\right )^3}-\frac {3 b \cosh (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b^{3/2} \cosh (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}+b x\right )^3}-\frac {3 b \cosh (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {3 b \cosh (c+d x)}{8 a^2 \left (-a b-b^2 x^2\right )}\right ) \, dx}{a}+\frac {d \int \frac {\sinh (c+d x)}{x} \, dx}{a^3} \\ & = -\frac {\cosh (c+d x)}{a^3 x}-\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{7/2}}-\frac {b \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{7/2}}+\frac {\left (3 b^2\right ) \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 a^3}+\frac {\left (3 b^2\right ) \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 a^3}+\frac {b^2 \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{4 a^3}+\frac {b^2 \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{4 a^3}+\frac {\left (3 b^2\right ) \int \frac {\cosh (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^3}+\frac {b^2 \int \frac {\cosh (c+d x)}{-a b-b^2 x^2} \, dx}{2 a^3}-\frac {b^{5/2} \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^3} \, dx}{8 (-a)^{5/2}}-\frac {b^{5/2} \int \frac {\cosh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^3} \, dx}{8 (-a)^{5/2}}+\frac {(d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx}{a^3}+\frac {(d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx}{a^3} \\ & = -\frac {\cosh (c+d x)}{a^3 x}-\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right )^2}-\frac {7 \sqrt {b} \cosh (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d \text {Chi}(d x) \sinh (c)}{a^3}+\frac {d \cosh (c) \text {Shi}(d x)}{a^3}+\frac {\left (3 b^2\right ) \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^3}+\frac {b^2 \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}{2 a^3}-\frac {(3 b d) \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a^3}+\frac {(3 b d) \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a^3}-\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{4 a^3}+\frac {(b d) \int \frac {\sinh (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{4 a^3}+\frac {\left (b^{3/2} d\right ) \int \frac {\sinh (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 (-a)^{5/2}}-\frac {\left (b^{3/2} d\right ) \int \frac {\sinh (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 (-a)^{5/2}}-\frac {\left (b \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}{2 (-a)^{7/2}}-\frac {\left (b \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}{2 (-a)^{7/2}}-\frac {\left (b \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}{2 (-a)^{7/2}}+\frac {\left (b \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}{2 (-a)^{7/2}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.87 (sec) , antiderivative size = 613, normalized size of antiderivative = 0.70 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\frac {8 i \sqrt {b} e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+\frac {e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (7 i b+7 \sqrt {a} \sqrt {b} d-i a d^2\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (-7 i b+7 \sqrt {a} \sqrt {b} d+i a d^2\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )}{\sqrt {b}}-8 i \sqrt {b} e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )-\frac {i e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (7 b-7 i \sqrt {a} \sqrt {b} d-a d^2\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (-7 b-7 i \sqrt {a} \sqrt {b} d+a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{\sqrt {b}}-\frac {4 \sqrt {a} \cosh (d x) \left (\left (8 a^2+25 a b x^2+15 b^2 x^4\right ) \cosh (c)+a d x \left (a+b x^2\right ) \sinh (c)\right )}{x \left (a+b x^2\right )^2}-\frac {4 \sqrt {a} \left (a d x \left (a+b x^2\right ) \cosh (c)+\left (8 a^2+25 a b x^2+15 b^2 x^4\right ) \sinh (c)\right ) \sinh (d x)}{x \left (a+b x^2\right )^2}+32 \sqrt {a} d (\text {Chi}(d x) \sinh (c)+\cosh (c) \text {Shi}(d x))}{32 a^{7/2}} \]

[In]

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

[Out]

((8*I)*Sqrt[b]*E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sq
rt[b] + x)] - ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)]) + (E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(((7*I)*b + 7*Sqrt[
a]*Sqrt[b]*d - I*a*d^2)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + ((-7*I)*
b + 7*Sqrt[a]*Sqrt[b]*d + I*a*d^2)*ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)]))/Sqrt[b] - (8*I)*Sqrt[b]*E^(-c
- (I*Sqrt[a]*d)/Sqrt[b])*(E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] - ExpInt
egralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]) - (I*E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*((7*b - (7*I)*Sqrt[a]*Sqrt[b]*d - a*
d^2)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] + (-7*b - (7*I)*Sqrt[a]*Sqrt[
b]*d + a*d^2)*ExpIntegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]))/Sqrt[b] - (4*Sqrt[a]*Cosh[d*x]*((8*a^2 + 25*a*b*x^2
+ 15*b^2*x^4)*Cosh[c] + a*d*x*(a + b*x^2)*Sinh[c]))/(x*(a + b*x^2)^2) - (4*Sqrt[a]*(a*d*x*(a + b*x^2)*Cosh[c]
+ (8*a^2 + 25*a*b*x^2 + 15*b^2*x^4)*Sinh[c])*Sinh[d*x])/(x*(a + b*x^2)^2) + 32*Sqrt[a]*d*(CoshIntegral[d*x]*S
inh[c] + Cosh[c]*SinhIntegral[d*x]))/(32*a^(7/2))

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 1178, normalized size of antiderivative = 1.35

method	result	size
risch	\(\text {Expression too large to display}\)	\(1178\)

[In]

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

[Out]

1/16*d^5*exp(-d*x-c)/a^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b*x^2-15/16*exp(-d*x-c)/a^3*x^3*d^4/(b^2*d^4*x^4+
2*a*b*d^4*x^2+a^2*d^4)*b^2+1/16*d^5*exp(-d*x-c)/a/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)-25/16*exp(-d*x-c)*d^4/a^
2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b*x-1/2*exp(-d*x-c)/a/x*d^4/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)-1/32/a^2
*d^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/a^2*d^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)+7/32*d/a^3*exp(-(d*(-a*b)^(1/2)+c*b)/b
)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)+7/32*d/a^3*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c
)*b-c*b)/b)+15/32/a^3/(-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-15/3
2/a^3/(-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+1/2*d/a^3*exp(-c)*Ei
(1,d*x)-1/16*d^5*exp(d*x+c)/a^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b*x^2-15/16*exp(d*x+c)/a^3*x^3*d^4/(b^2*d^
4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b^2-1/16*d^5*exp(d*x+c)/a/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)-25/16*exp(d*x+c)*d^
4/a^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b*x-1/2*exp(d*x+c)/a/x*d^4/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)-1/32/
a^2*d^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/a^2*d^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)-7/32*d/a^3*exp((d*(-a*b)^(1/2)+c*b)/b
)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)-7/32*d/a^3*exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)
*b-c*b)/b)+15/32/a^3/(-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-15/32/a
^3/(-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-1/2*d/a^3*exp(c)*Ei(1,-
d*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2346 vs. \(2 (673) = 1346\).

Time = 0.29 (sec) , antiderivative size = 2346, normalized size of antiderivative = 2.68 \[ \int \frac {\cosh (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/32*(4*(15*a*b^2*d*x^4 + 25*a^2*b*d*x^2 + 8*a^3*d)*cosh(d*x + c) - ((7*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^
3*d^2*x)*cosh(d*x + c)^2 - 7*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*sinh(d*x + c)^2 - (((a*b^2*d^2 - 15
*b^3)*x^5 + 2*(a^2*b*d^2 - 15*a*b^2)*x^3 + (a^3*d^2 - 15*a^2*b)*x)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 15*b^3)*x^5
+ 2*(a^2*b*d^2 - 15*a*b^2)*x^3 + (a^3*d^2 - 15*a^2*b)*x)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^
2/b)) - (7*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*cosh(d*x + c)^2 - 7*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3
+ a^3*d^2*x)*sinh(d*x + c)^2 + (((a*b^2*d^2 - 15*b^3)*x^5 + 2*(a^2*b*d^2 - 15*a*b^2)*x^3 + (a^3*d^2 - 15*a^2*b
)*x)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 15*b^3)*x^5 + 2*(a^2*b*d^2 - 15*a*b^2)*x^3 + (a^3*d^2 - 15*a^2*b)*x)*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)) - 16*((a*b^2*d^2*x^5 + 2*a^2*
b*d^2*x^3 + a^3*d^2*x)*Ei(d*x) - (a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*Ei(-d*x))*cosh(c) - ((7*(a*b^2*
d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*cosh(d*x + c)^2 - 7*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*sinh(
d*x + c)^2 + (((a*b^2*d^2 - 15*b^3)*x^5 + 2*(a^2*b*d^2 - 15*a*b^2)*x^3 + (a^3*d^2 - 15*a^2*b)*x)*cosh(d*x + c)
^2 - ((a*b^2*d^2 - 15*b^3)*x^5 + 2*(a^2*b*d^2 - 15*a*b^2)*x^3 + (a^3*d^2 - 15*a^2*b)*x)*sinh(d*x + c)^2)*sqrt(
-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) - (7*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*cosh(d*x + c)^2 - 7*(a*
b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*sinh(d*x + c)^2 - (((a*b^2*d^2 - 15*b^3)*x^5 + 2*(a^2*b*d^2 - 15*a*
b^2)*x^3 + (a^3*d^2 - 15*a^2*b)*x)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 15*b^3)*x^5 + 2*(a^2*b*d^2 - 15*a*b^2)*x^3
+ (a^3*d^2 - 15*a^2*b)*x)*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^3 + a^3*d^2*x)*sinh(d*x + c) - ((7*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*cosh(d*x +
c)^2 - 7*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*sinh(d*x + c)^2 - (((a*b^2*d^2 - 15*b^3)*x^5 + 2*(a^2*
b*d^2 - 15*a*b^2)*x^3 + (a^3*d^2 - 15*a^2*b)*x)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 15*b^3)*x^5 + 2*(a^2*b*d^2 - 1
5*a*b^2)*x^3 + (a^3*d^2 - 15*a^2*b)*x)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) + (7*(a*b^2*d
^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*cosh(d*x + c)^2 - 7*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*sinh(d
*x + c)^2 + (((a*b^2*d^2 - 15*b^3)*x^5 + 2*(a^2*b*d^2 - 15*a*b^2)*x^3 + (a^3*d^2 - 15*a^2*b)*x)*cosh(d*x + c)^
2 - ((a*b^2*d^2 - 15*b^3)*x^5 + 2*(a^2*b*d^2 - 15*a*b^2)*x^3 + (a^3*d^2 - 15*a^2*b)*x)*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)) - 16*((a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2
*x)*Ei(d*x) + (a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*Ei(-d*x))*sinh(c) + ((7*(a*b^2*d^2*x^5 + 2*a^2*b*d
^2*x^3 + a^3*d^2*x)*cosh(d*x + c)^2 - 7*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*sinh(d*x + c)^2 + (((a*b
^2*d^2 - 15*b^3)*x^5 + 2*(a^2*b*d^2 - 15*a*b^2)*x^3 + (a^3*d^2 - 15*a^2*b)*x)*cosh(d*x + c)^2 - ((a*b^2*d^2 -
15*b^3)*x^5 + 2*(a^2*b*d^2 - 15*a*b^2)*x^3 + (a^3*d^2 - 15*a^2*b)*x)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x +
sqrt(-a*d^2/b)) + (7*(a*b^2*d^2*x^5 + 2*a^2*b*d^2*x^3 + a^3*d^2*x)*cosh(d*x + c)^2 - 7*(a*b^2*d^2*x^5 + 2*a^2
*b*d^2*x^3 + a^3*d^2*x)*sinh(d*x + c)^2 - (((a*b^2*d^2 - 15*b^3)*x^5 + 2*(a^2*b*d^2 - 15*a*b^2)*x^3 + (a^3*d^2
- 15*a^2*b)*x)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 15*b^3)*x^5 + 2*(a^2*b*d^2 - 15*a*b^2)*x^3 + (a^3*d^2 - 15*a^2
*b)*x)*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^4*b^2*d*x^5
+ 2*a^5*b*d*x^3 + a^6*d*x)*cosh(d*x + c)^2 - (a^4*b^2*d*x^5 + 2*a^5*b*d*x^3 + a^6*d*x)*sinh(d*x + c)^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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