\(\int \frac {\cosh (c+d x)}{(a+b x^2)^3} \, dx\) [75]

Optimal result

Integrand size = 16, antiderivative size = 856 \[ \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {\cosh (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}-\frac {3 \cosh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )^2}+\frac {3 \cosh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {3 \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^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)^{3/2} b^{3/2}}-\frac {3 \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^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)^{3/2} b^{3/2}}-\frac {3 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^2 b}-\frac {3 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^2 b}+\frac {d \sinh (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \sinh (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {3 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^2 b}-\frac {3 \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 {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)^{3/2} b^{3/2}}-\frac {3 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^2 b}-\frac {3 \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 {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)^{3/2} b^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 856, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5389, 3378, 3384, 3379, 3382} \[ \int \frac {\cosh (c+d x)}{\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)^{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 ) d^2}{16 (-a)^{3/2} b^{3/2}}-\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)^{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 ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac {3 \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^2 b}-\frac {3 \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^2 b}+\frac {\sinh (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sinh (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {b} x+\sqrt {-a}\right )}+\frac {3 \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^2 b}-\frac {3 \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^2 b}-\frac {3 \cosh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {3 \cosh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\cosh (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {\cosh (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )^2}+\frac {3 \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 {3 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {3 \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 {3 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}} \]

[In]

Int[Cosh[c + d*x]/(a + b*x^2)^3,x]

[Out]

-1/16*Cosh[c + d*x]/((-a)^(3/2)*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)^2) - (3*Cosh[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a
] - Sqrt[b]*x)) + Cosh[c + d*x]/(16*(-a)^(3/2)*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)^2) + (3*Cosh[c + d*x])/(16*a^2*S
qrt[b]*(Sqrt[-a] + Sqrt[b]*x)) + (3*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(
16*(-a)^(5/2)*Sqrt[b]) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a
)^(3/2)*b^(3/2)) - (3*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*
Sqrt[b]) - (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2
)) - (3*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a^2*b) - (3*d*CoshInteg
ral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a^2*b) + (d*Sinh[c + d*x])/(16*(-a)^(3/2)*
b*(Sqrt[-a] - Sqrt[b]*x)) + (d*Sinh[c + d*x])/(16*(-a)^(3/2)*b*(Sqrt[-a] + Sqrt[b]*x)) + (3*d*Cosh[c + (Sqrt[-
a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) - (3*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIn
tegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral
[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) - (3*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt
[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b) - (3*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x
])/(16*(-a)^(5/2)*Sqrt[b]) - (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16
*(-a)^(3/2)*b^(3/2))

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5389

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.87 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.46 \[ \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {-e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (3 i b+3 \sqrt {a} \sqrt {b} d-i a d^2\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (-3 i b+3 \sqrt {a} \sqrt {b} d+i a d^2\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (3 i b+3 \sqrt {a} \sqrt {b} d-i a d^2\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (-3 i b+3 \sqrt {a} \sqrt {b} d+i a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )+\frac {4 \sqrt {a} \sqrt {b} \cosh (d x) \left (b x \left (5 a+3 b x^2\right ) \cosh (c)+a d \left (a+b x^2\right ) \sinh (c)\right )}{\left (a+b x^2\right )^2}+\frac {4 \sqrt {a} \sqrt {b} \left (a d \left (a+b x^2\right ) \cosh (c)+b x \left (5 a+3 b x^2\right ) \sinh (c)\right ) \sinh (d x)}{\left (a+b x^2\right )^2}}{32 a^{5/2} b^{3/2}} \]

[In]

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

[Out]

(-(E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(((3*I)*b + 3*Sqrt[a]*Sqrt[b]*d - I*a*d^2)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpI
ntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + ((-3*I)*b + 3*Sqrt[a]*Sqrt[b]*d + I*a*d^2)*ExpIntegralEi[d*((I*Sqr
t[a])/Sqrt[b] + x)])) + E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*(((3*I)*b + 3*Sqrt[a]*Sqrt[b]*d - I*a*d^2)*E^(((2*I)*Sq
rt[a]*d)/Sqrt[b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] + ((-3*I)*b + 3*Sqrt[a]*Sqrt[b]*d + I*a*d^2)*E
xpIntegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]) + (4*Sqrt[a]*Sqrt[b]*Cosh[d*x]*(b*x*(5*a + 3*b*x^2)*Cosh[c] + a*d*(
a + b*x^2)*Sinh[c]))/(a + b*x^2)^2 + (4*Sqrt[a]*Sqrt[b]*(a*d*(a + b*x^2)*Cosh[c] + b*x*(5*a + 3*b*x^2)*Sinh[c]
)*Sinh[d*x])/(a + b*x^2)^2)/(32*a^(5/2)*b^(3/2))

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 1064, normalized size of antiderivative = 1.24

method	result	size
risch	\(\text {Expression too large to display}\)	\(1064\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2116 vs. \(2 (655) = 1310\).

Time = 0.29 (sec) , antiderivative size = 2116, normalized size of antiderivative = 2.47 \[ \int \frac {\cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/32*(4*(3*a*b^2*d*x^3 + 5*a^2*b*d*x)*cosh(d*x + c) - ((3*(a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*x
+ c)^2 - 3*(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 - 3*b^3)*x^4
- 3*a^2*b + 2*(a^2*b*d^2 - 3*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 3*b^3)*x^4 - 3*a^2*b + 2*(a
^2*b*d^2 - 3*a*b^2)*x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) - (3*(a*b^2*d^2*x^4 + 2*a^2
*b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - 3*(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 - 3*b^3)*x^4 - 3*a^2*b + 2*(a^2*b*d^2 - 3*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2
- 3*b^3)*x^4 - 3*a^2*b + 2*(a^2*b*d^2 - 3*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)) - ((3*(a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - 3*(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 - 3*b^3)*x^4 - 3*a^2*b + 2*(a^2*b*d
^2 - 3*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 3*b^3)*x^4 - 3*a^2*b + 2*(a^2*b*d^2 - 3*a*b^2)*x^
2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) - (3*(a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*
cosh(d*x + c)^2 - 3*(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 - 3*b
^3)*x^4 - 3*a^2*b + 2*(a^2*b*d^2 - 3*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 3*b^3)*x^4 - 3*a^2*
b + 2*(a^2*b*d^2 - 3*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) - ((3*(a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*
x + c)^2 - 3*(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 - 3*b^3)*x^4
- 3*a^2*b + 2*(a^2*b*d^2 - 3*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 3*b^3)*x^4 - 3*a^2*b + 2*(
a^2*b*d^2 - 3*a*b^2)*x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) + (3*(a*b^2*d^2*x^4 + 2*a^
2*b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - 3*(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 - 3*b^3)*x^4 - 3*a^2*b + 2*(a^2*b*d^2 - 3*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^
2 - 3*b^3)*x^4 - 3*a^2*b + 2*(a^2*b*d^2 - 3*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)) + ((3*(a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)*cosh(d*x + c)^2 - 3*(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 - 3*b^3)*x^4 - 3*a^2*b + 2*(a^2*b*
d^2 - 3*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 3*b^3)*x^4 - 3*a^2*b + 2*(a^2*b*d^2 - 3*a*b^2)*x
^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) + (3*(a*b^2*d^2*x^4 + 2*a^2*b*d^2*x^2 + a^3*d^2)
*cosh(d*x + c)^2 - 3*(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 - 3*
b^3)*x^4 - 3*a^2*b + 2*(a^2*b*d^2 - 3*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 3*b^3)*x^4 - 3*a^2
*b + 2*(a^2*b*d^2 - 3*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^3*b^3*d*x^4 + 2*a^4*b^2*d*x^2 + a^5*b*d)*cosh(d*x + c)^2 - (a^3*b^3*d*x^4 + 2*a^4*b^2*d*x^2 + a
^5*b*d)*sinh(d*x + c)^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(cosh(c + d*x)/(a + b*x^2)^3,x)

[Out]

int(cosh(c + d*x)/(a + b*x^2)^3, x)