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

Optimal. Leaf size=730 \[ -\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^3}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^3}+\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^3}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^3}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}+\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 a^2 b}+\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 a^2 b}-\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 a^2 b}+\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 a^2 b}+\frac {\cosh (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {5 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)^{5/2} \sqrt {b}}-\frac {5 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)^{5/2} \sqrt {b}}+\frac {5 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)^{5/2} \sqrt {b}}+\frac {5 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)^{5/2} \sqrt {b}}+\frac {\cosh (c+d x)}{4 a \left (a+b x^2\right )^2} \]

[Out]

Chi(d*x)*cosh(c)/a^3+1/4*cosh(d*x+c)/a/(b*x^2+a)^2+1/2*cosh(d*x+c)/a^2/(b*x^2+a)-1/2*Chi(d*x+d*(-a)^(1/2)/b^(1
/2))*cosh(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^2/
b-1/2*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*cosh(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^2/b+Shi(d*x)*sinh(c)/a^3-1/2*Shi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/
b^(1/2))/a^3+1/16*d^2*Shi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/a^2/b-1/2*Shi(d*x-d*(-a)^(1/2
)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2))/a^3+1/16*d^2*Shi(d*x-d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2)
)/a^2/b-5/16*d*cosh(c+d*(-a)^(1/2)/b^(1/2))*Shi(d*x-d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)+5/16*d*cosh(c-d*(
-a)^(1/2)/b^(1/2))*Shi(d*x+d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)+5/16*d*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(
c-d*(-a)^(1/2)/b^(1/2))/(-a)^(5/2)/b^(1/2)-5/16*d*Chi(-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*sinh(d*x+c)/a^2/b^(1/2)/((-a)^(1/2)-x*b^(1/2))-1/16*d*sinh(d*x+c)/a^2/b^(1/2)/((-a)^
(1/2)+x*b^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 1.68, antiderivative size = 730, normalized size of antiderivative = 1.00, number of steps used = 41, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5293, 3303, 3298, 3301, 5289, 5280, 3297} \[ \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 a^2 b}+\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 a^2 b}-\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^3}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^3}-\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 a^2 b}+\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 a^2 b}+\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^3}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^3}+\frac {\cosh (c+d x)}{2 a^2 \left (a+b x^2\right )}+\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {d \sinh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {\cosh (c) \text {Chi}(d x)}{a^3}+\frac {\sinh (c) \text {Shi}(d x)}{a^3}+\frac {5 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)^{5/2} \sqrt {b}}-\frac {5 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)^{5/2} \sqrt {b}}+\frac {5 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)^{5/2} \sqrt {b}}+\frac {5 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)^{5/2} \sqrt {b}}+\frac {\cosh (c+d x)}{4 a \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cosh[c + d*x]/(4*a*(a + b*x^2)^2) + Cosh[c + d*x]/(2*a^2*(a + b*x^2)) + (Cosh[c]*CoshIntegral[d*x])/a^3 - (Cos
h[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^3) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sqr
t[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) - (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqr
t[-a]*d)/Sqrt[b] + d*x])/(2*a^3) + (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x
])/(16*a^2*b) + (5*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*S
qrt[b]) - (5*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]
) + (d*Sinh[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) - (d*Sinh[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] +
Sqrt[b]*x)) + (Sinh[c]*SinhIntegral[d*x])/a^3 + (5*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/
Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] -
d*x])/(2*a^3) - (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) + (5*
d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (Sinh[c -
(Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^3) + (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]
*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b)

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 5280

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sinh[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 5289

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(e^m*(a + b*x
^n)^(p + 1)*Cosh[c + d*x])/(b*n*(p + 1)), x] - Dist[(d*e^m)/(b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*Sinh[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 5293

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

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Mathematica [C]  time = 2.61, size = 1046, normalized size = 1.43 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((12*a^2*Cosh[c + d*x])/(a + b*x^2)^2 + (8*a*b*x^2*Cosh[c + d*x])/(a + b*x^2)^2 + 16*Cosh[c]*CoshIntegral[d*x]
- 8*Cos[(Sqrt[a]*d)/Sqrt[b]]*Cosh[c]*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] + (a*d^2*Cos[(Sqrt[a]*d)/Sqrt[b
]]*Cosh[c]*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])/b - (5*Sqrt[a]*d*Cosh[c]*CosIntegral[(Sqrt[a]*d)/Sqrt[b]
+ I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]])/Sqrt[b] + ((5*I)*Sqrt[a]*d*Cos[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[a]*d)
/Sqrt[b] + I*d*x]*Sinh[c])/Sqrt[b] - (8*I)*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]]*S
inh[c] + (I*a*d^2*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]]*Sinh[c])/b - (CosIntegral[
-((Sqrt[a]*d)/Sqrt[b]) + I*d*x]*((8*b - a*d^2)*Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]] + (5*I)*Sqrt[a]*Sqrt[b]*d*Sinh[
c - (I*Sqrt[a]*d)/Sqrt[b]]))/b - (2*a^2*d*x*Sinh[c + d*x])/(a + b*x^2)^2 - (2*a*b*d*x^3*Sinh[c + d*x])/(a + b*
x^2)^2 + (16*a^2*Sinh[c]*SinhIntegral[d*x])/(a + b*x^2)^2 + (32*a*b*x^2*Sinh[c]*SinhIntegral[d*x])/(a + b*x^2)
^2 + (16*b^2*x^4*Sinh[c]*SinhIntegral[d*x])/(a + b*x^2)^2 + (5*Sqrt[a]*d*Cos[(Sqrt[a]*d)/Sqrt[b]]*Cosh[c]*SinI
ntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x])/Sqrt[b] - 8*Cosh[c]*Sin[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqr
t[b] - I*d*x] + (a*d^2*Cosh[c]*Sin[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x])/b - (8*I)*Co
s[(Sqrt[a]*d)/Sqrt[b]]*Sinh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] + (I*a*d^2*Cos[(Sqrt[a]*d)/Sqrt[b]]*Si
nh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x])/b - ((5*I)*Sqrt[a]*d*Sin[(Sqrt[a]*d)/Sqrt[b]]*Sinh[c]*SinInteg
ral[(Sqrt[a]*d)/Sqrt[b] - I*d*x])/Sqrt[b] + (5*Sqrt[a]*d*Cos[(Sqrt[a]*d)/Sqrt[b]]*Cosh[c]*SinIntegral[(Sqrt[a]
*d)/Sqrt[b] + I*d*x])/Sqrt[b] - 8*Cosh[c]*Sin[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] +
(a*d^2*Cosh[c]*Sin[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])/b + (8*I)*Cos[(Sqrt[a]*d)/Sq
rt[b]]*Sinh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] - (I*a*d^2*Cos[(Sqrt[a]*d)/Sqrt[b]]*Sinh[c]*SinIntegra
l[(Sqrt[a]*d)/Sqrt[b] + I*d*x])/b + ((5*I)*Sqrt[a]*d*Sin[(Sqrt[a]*d)/Sqrt[b]]*Sinh[c]*SinIntegral[(Sqrt[a]*d)/
Sqrt[b] + I*d*x])/Sqrt[b])/(16*a^3)

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fricas [B]  time = 0.55, size = 2076, normalized size = 2.84 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(8*(2*a*b^2*x^2 + 3*a^2*b)*cosh(d*x + c) + (((a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d^2
- 8*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d^2 - 8*a*b^2)*x^2)*
sinh(d*x + c)^2 + 5*((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*sinh(d*
x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) + ((a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*
d^2 - 8*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d^2 - 8*a*b^2)*x
^2)*sinh(d*x + c)^2 - 5*((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*sin
h(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*((b^3*x^4 + 2*a*b^2*x^2
+ a^2*b)*Ei(d*x) + (b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*Ei(-d*x))*cosh(c) + (((a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 -
8*a^2*b + 2*(a^2*b*d^2 - 8*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^
2*b*d^2 - 8*a*b^2)*x^2)*sinh(d*x + c)^2 - 5*((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 + 2*a*
b^2*x^2 + a^2*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) + ((a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x
^4 - 8*a^2*b + 2*(a^2*b*d^2 - 8*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2
*(a^2*b*d^2 - 8*a*b^2)*x^2)*sinh(d*x + c)^2 + 5*((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 +
2*a*b^2*x^2 + a^2*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)) - 4
*(a*b^2*d*x^3 + a^2*b*d*x)*sinh(d*x + c) + (((a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d^2 - 8*a
*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d^2 - 8*a*b^2)*x^2)*sinh(
d*x + c)^2 + 5*((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*sinh(d*x + c
)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) - ((a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d^2 -
8*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d^2 - 8*a*b^2)*x^2)*s
inh(d*x + c)^2 - 5*((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 + 2*a*b^2*x^2 + a^2*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)) + 16*((b^3*x^4 + 2*a*b^2*x^2 + a^
2*b)*Ei(d*x) - (b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*Ei(-d*x))*sinh(c) - (((a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^
2*b + 2*(a^2*b*d^2 - 8*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2*b*d
^2 - 8*a*b^2)*x^2)*sinh(d*x + c)^2 - 5*((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 + 2*a*b^2*x
^2 + a^2*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) - ((a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 -
8*a^2*b + 2*(a^2*b*d^2 - 8*a*b^2)*x^2)*cosh(d*x + c)^2 - (a^3*d^2 + (a*b^2*d^2 - 8*b^3)*x^4 - 8*a^2*b + 2*(a^2
*b*d^2 - 8*a*b^2)*x^2)*sinh(d*x + c)^2 + 5*((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*cosh(d*x + c)^2 - (b^3*x^4 + 2*a*b
^2*x^2 + a^2*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^3*b
^3*x^4 + 2*a^4*b^2*x^2 + a^5*b)*cosh(d*x + c)^2 - (a^3*b^3*x^4 + 2*a^4*b^2*x^2 + a^5*b)*sinh(d*x + c)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.18, size = 1090, normalized size = 1.49 \[ \frac {{\mathrm e}^{-d x -c} d^{2} \left (\left (d x +c \right )^{3} b -3 \left (d x +c \right )^{2} b c +\left (d x +c \right ) a \,d^{2}+3 \left (d x +c \right ) b \,c^{2}-a c \,d^{2}-b \,c^{3}+4 \left (d x +c \right )^{2} b -8 \left (d x +c \right ) b c +6 a \,d^{2}+4 b \,c^{2}\right )}{16 a^{2} \left (\left (d x +c \right )^{4} b^{2}-4 \left (d x +c \right )^{3} c \,b^{2}+2 \left (d x +c \right )^{2} a b \,d^{2}+6 \left (d x +c \right )^{2} c^{2} b^{2}-4 a b \left (d x +c \right ) c \,d^{2}-4 b^{2} \left (d x +c \right ) c^{3}+a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}\right )}-\frac {{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2 a^{3}}-\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 ) d^{2}}{32 b \,a^{2}}-\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 ) d^{2}}{32 b \,a^{2}}-\frac {5 \,{\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 ) d}{32 a^{2} \sqrt {-a b}}+\frac {5 \,{\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 ) d}{32 a^{2} \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 )}{4 a^{3}}+\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 )}{4 a^{3}}-\frac {{\mathrm e}^{d x +c} d^{2} \left (\left (d x +c \right )^{3} b -3 \left (d x +c \right )^{2} b c +\left (d x +c \right ) a \,d^{2}+3 \left (d x +c \right ) b \,c^{2}-a c \,d^{2}-b \,c^{3}-4 \left (d x +c \right )^{2} b +8 \left (d x +c \right ) b c -6 a \,d^{2}-4 b \,c^{2}\right )}{16 a^{2} \left (\left (d x +c \right )^{4} b^{2}-4 \left (d x +c \right )^{3} c \,b^{2}+2 \left (d x +c \right )^{2} a b \,d^{2}+6 \left (d x +c \right )^{2} c^{2} b^{2}-4 a b \left (d x +c \right ) c \,d^{2}-4 b^{2} \left (d x +c \right ) c^{3}+a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}\right )}-\frac {{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2 a^{3}}-\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 ) d^{2}}{32 b \,a^{2}}-\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 ) d^{2}}{32 b \,a^{2}}-\frac {5 \,{\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 ) d}{32 a^{2} \sqrt {-a b}}+\frac {5 \,{\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 ) d}{32 a^{2} \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 )}{4 a^{3}}+\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 )}{4 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*exp(-d*x-c)*d^2*((d*x+c)^3*b-3*(d*x+c)^2*b*c+(d*x+c)*a*d^2+3*(d*x+c)*b*c^2-a*c*d^2-b*c^3+4*(d*x+c)^2*b-8*
(d*x+c)*b*c+6*a*d^2+4*b*c^2)/a^2/((d*x+c)^4*b^2-4*(d*x+c)^3*c*b^2+2*(d*x+c)^2*a*b*d^2+6*(d*x+c)^2*c^2*b^2-4*a*
b*(d*x+c)*c*d^2-4*b^2*(d*x+c)*c^3+a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)-1/2/a^3*exp(-c)*Ei(1,d*x)-1/32/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)*d^2-1/32/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)*d^2-5/32/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)*d+5/32/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)*d+1/4/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)+1/4/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)-1/16*exp(d*x+c)*d^2*((d*x+c)^3*b-3*(d*x+c)^2*b*c+(d*x+c)*a*d^
2+3*(d*x+c)*b*c^2-a*c*d^2-b*c^3-4*(d*x+c)^2*b+8*(d*x+c)*b*c-6*a*d^2-4*b*c^2)/a^2/((d*x+c)^4*b^2-4*(d*x+c)^3*c*
b^2+2*(d*x+c)^2*a*b*d^2+6*(d*x+c)^2*c^2*b^2-4*a*b*(d*x+c)*c*d^2-4*b^2*(d*x+c)*c^3+a^2*d^4+2*a*b*c^2*d^2+b^2*c^
4)-1/2/a^3*exp(c)*Ei(1,-d*x)-1/32/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)*d
^2-1/32/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)*d^2-5/32/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)*d+5/32/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)*d+1/4/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)+1/4/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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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