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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.68183, antiderivative size = 730, normalized size of antiderivative = 1., 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 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])

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 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 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 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 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]

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*}

Mathematica [C]  time = 3.262, size = 1558, normalized size = 2.13 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*a^2*b*Cosh[c + d*x] + 8*a*b^2*x^2*Cosh[c + d*x] + 16*b*(a + b*x^2)^2*Cosh[c]*CoshIntegral[d*x] - 8*a^2*b*C
osh[c - (I*Sqrt[a]*d)/Sqrt[b]]*CoshIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + a^3*d^2*Cosh[c - (I*Sqrt[a]*d)/Sqrt
[b]]*CoshIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - 16*a*b^2*x^2*Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]]*CoshIntegral[d*(
(I*Sqrt[a])/Sqrt[b] + x)] + 2*a^2*b*d^2*x^2*Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]]*CoshIntegral[d*((I*Sqrt[a])/Sqrt[b
] + x)] - 8*b^3*x^4*Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]]*CoshIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + a*b^2*d^2*x^4*
Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]]*CoshIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (5*I)*a^(5/2)*Sqrt[b]*d*CoshIntegr
al[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]] - (10*I)*a^(3/2)*b^(3/2)*d*x^2*CoshIntegral[d*
((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]] - (5*I)*Sqrt[a]*b^(5/2)*d*x^4*CoshIntegral[d*((I*Sq
rt[a])/Sqrt[b] + x)]*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]] + (a + b*x^2)^2*CoshIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] +
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]])
- 2*a^2*b*d*x*Sinh[c + d*x] - 2*a*b^2*d*x^3*Sinh[c + d*x] + 16*a^2*b*Sinh[c]*SinhIntegral[d*x] + 32*a*b^2*x^2
*Sinh[c]*SinhIntegral[d*x] + 16*b^3*x^4*Sinh[c]*SinhIntegral[d*x] - (5*I)*a^(5/2)*Sqrt[b]*d*Cosh[c - (I*Sqrt[a
]*d)/Sqrt[b]]*SinhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (10*I)*a^(3/2)*b^(3/2)*d*x^2*Cosh[c - (I*Sqrt[a]*d)/
Sqrt[b]]*SinhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (5*I)*Sqrt[a]*b^(5/2)*d*x^4*Cosh[c - (I*Sqrt[a]*d)/Sqrt[b
]]*SinhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - 8*a^2*b*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[d*((I*Sqrt
[a])/Sqrt[b] + x)] + a^3*d^2*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - 16*a*
b^2*x^2*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + 2*a^2*b*d^2*x^2*Sinh[c - (
I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - 8*b^3*x^4*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]*Si
nhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + a*b^2*d^2*x^4*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[d*((I*Sqr
t[a])/Sqrt[b] + x)] - (5*I)*a^(5/2)*Sqrt[b]*d*Cosh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a]*d)/Sqrt[
b] - d*x] - (10*I)*a^(3/2)*b^(3/2)*d*x^2*Cosh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a]*d)/Sqrt[b] -
d*x] - (5*I)*Sqrt[a]*b^(5/2)*d*x^4*Cosh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] +
8*a^2*b*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - a^3*d^2*Sinh[c + (I*Sqrt[
a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + 16*a*b^2*x^2*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIn
tegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - 2*a^2*b*d^2*x^2*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a]*
d)/Sqrt[b] - d*x] + 8*b^3*x^4*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - a*b^
2*d^2*x^4*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(16*a^3*b*(a + b*x^2)^2)

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Maple [A]  time = 0.105, size = 1090, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.15226, size = 4301, normalized size = 5.89 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3} x}\,{d x} \end{align*}

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)