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

Optimal. Leaf size=791 \[ \text{result too large to display} \]

[Out]

-Cosh[c + d*x]/(2*a^3*x^2) - (b*Cosh[c + d*x])/(4*a^2*(a + b*x^2)^2) - (b*Cosh[c + d*x])/(a^3*(a + b*x^2)) - (
3*b*Cosh[c]*CoshIntegral[d*x])/a^4 + (d^2*Cosh[c]*CoshIntegral[d*x])/(2*a^3) + (3*b*Cosh[c + (Sqrt[-a]*d)/Sqrt
[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^4) - (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqr
t[-a]*d)/Sqrt[b] - d*x])/(16*a^3) + (3*b*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*
x])/(2*a^4) - (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3) + (9*Sqrt
[b]*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) - (9*Sqrt[b]*d*
CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) - (d*Sinh[c + d*x])/(
2*a^3*x) - (Sqrt[b]*d*Sinh[c + d*x])/(16*a^3*(Sqrt[-a] - Sqrt[b]*x)) + (Sqrt[b]*d*Sinh[c + d*x])/(16*a^3*(Sqrt
[-a] + Sqrt[b]*x)) - (3*b*Sinh[c]*SinhIntegral[d*x])/a^4 + (d^2*Sinh[c]*SinhIntegral[d*x])/(2*a^3) + (9*Sqrt[b
]*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(7/2)) - (3*b*Sinh[c + (
Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^4) + (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*S
inhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^3) + (9*Sqrt[b]*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(
Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2)) + (3*b*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sq
rt[b] + d*x])/(2*a^4) - (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3)

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Rubi [A]  time = 1.86841, antiderivative size = 791, normalized size of antiderivative = 1., number of steps used = 46, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {5293, 3297, 3303, 3298, 3301, 5289, 5280} \[ -\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^3}-\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^3}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}+\frac{3 b \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}+\frac{3 b \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}+\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^3}-\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^3}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}-\frac{3 b \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}+\frac{3 b \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 a^4}-\frac{b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{\sqrt{b} d \sinh (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \sinh (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a^3}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a^3}-\frac{\cosh (c+d x)}{2 a^3 x^2}-\frac{d \sinh (c+d x)}{2 a^3 x}+\frac{9 \sqrt{b} 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)^{7/2}}-\frac{9 \sqrt{b} 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)^{7/2}}+\frac{9 \sqrt{b} 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)^{7/2}}+\frac{9 \sqrt{b} 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)^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cosh[c + d*x]/(2*a^3*x^2) - (b*Cosh[c + d*x])/(4*a^2*(a + b*x^2)^2) - (b*Cosh[c + d*x])/(a^3*(a + b*x^2)) - (
3*b*Cosh[c]*CoshIntegral[d*x])/a^4 + (d^2*Cosh[c]*CoshIntegral[d*x])/(2*a^3) + (3*b*Cosh[c + (Sqrt[-a]*d)/Sqrt
[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^4) - (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqr
t[-a]*d)/Sqrt[b] - d*x])/(16*a^3) + (3*b*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*
x])/(2*a^4) - (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3) + (9*Sqrt
[b]*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) - (9*Sqrt[b]*d*
CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) - (d*Sinh[c + d*x])/(
2*a^3*x) - (Sqrt[b]*d*Sinh[c + d*x])/(16*a^3*(Sqrt[-a] - Sqrt[b]*x)) + (Sqrt[b]*d*Sinh[c + d*x])/(16*a^3*(Sqrt
[-a] + Sqrt[b]*x)) - (3*b*Sinh[c]*SinhIntegral[d*x])/a^4 + (d^2*Sinh[c]*SinhIntegral[d*x])/(2*a^3) + (9*Sqrt[b
]*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(7/2)) - (3*b*Sinh[c + (
Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^4) + (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*S
inhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^3) + (9*Sqrt[b]*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(
Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2)) + (3*b*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sq
rt[b] + d*x])/(2*a^4) - (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3)

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 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 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])

Rubi steps

\begin{align*} \int \frac{\cosh (c+d x)}{x^3 \left (a+b x^2\right )^3} \, dx &=\int \left (\frac{\cosh (c+d x)}{a^3 x^3}-\frac{3 b \cosh (c+d x)}{a^4 x}+\frac{b^2 x \cosh (c+d x)}{a^2 \left (a+b x^2\right )^3}+\frac{2 b^2 x \cosh (c+d x)}{a^3 \left (a+b x^2\right )^2}+\frac{3 b^2 x \cosh (c+d x)}{a^4 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\cosh (c+d x)}{x^3} \, dx}{a^3}-\frac{(3 b) \int \frac{\cosh (c+d x)}{x} \, dx}{a^4}+\frac{\left (3 b^2\right ) \int \frac{x \cosh (c+d x)}{a+b x^2} \, dx}{a^4}+\frac{\left (2 b^2\right ) \int \frac{x \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a^3}+\frac{b^2 \int \frac{x \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx}{a^2}\\ &=-\frac{\cosh (c+d x)}{2 a^3 x^2}-\frac{b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}+\frac{\left (3 b^2\right ) \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^4}+\frac{d \int \frac{\sinh (c+d x)}{x^2} \, dx}{2 a^3}+\frac{(b d) \int \frac{\sinh (c+d x)}{a+b x^2} \, dx}{a^3}+\frac{(b d) \int \frac{\sinh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 a^2}-\frac{(3 b \cosh (c)) \int \frac{\cosh (d x)}{x} \, dx}{a^4}-\frac{(3 b \sinh (c)) \int \frac{\sinh (d x)}{x} \, dx}{a^4}\\ &=-\frac{\cosh (c+d x)}{2 a^3 x^2}-\frac{b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}-\frac{d \sinh (c+d x)}{2 a^3 x}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}-\frac{\left (3 b^{3/2}\right ) \int \frac{\cosh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 a^4}+\frac{\left (3 b^{3/2}\right ) \int \frac{\cosh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 a^4}+\frac{(b 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}{a^3}+\frac{(b 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^2}+\frac{d^2 \int \frac{\cosh (c+d x)}{x} \, dx}{2 a^3}\\ &=-\frac{\cosh (c+d x)}{2 a^3 x^2}-\frac{b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}-\frac{d \sinh (c+d x)}{2 a^3 x}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}+\frac{(b d) \int \frac{\sinh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 (-a)^{7/2}}+\frac{(b d) \int \frac{\sinh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 (-a)^{7/2}}-\frac{\left (b^2 d\right ) \int \frac{\sinh (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^2} \, dx}{16 a^3}-\frac{\left (b^2 d\right ) \int \frac{\sinh (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^2} \, dx}{16 a^3}-\frac{\left (b^2 d\right ) \int \frac{\sinh (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^3}+\frac{\left (d^2 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx}{2 a^3}+\frac{\left (3 b^{3/2} \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} x} \, dx}{2 a^4}-\frac{\left (3 b^{3/2} \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} x} \, dx}{2 a^4}+\frac{\left (d^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx}{2 a^3}+\frac{\left (3 b^{3/2} \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} x} \, dx}{2 a^4}+\frac{\left (3 b^{3/2} \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} x} \, dx}{2 a^4}\\ &=-\frac{\cosh (c+d x)}{2 a^3 x^2}-\frac{b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a^3}+\frac{3 b \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}+\frac{3 b \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}-\frac{d \sinh (c+d x)}{2 a^3 x}-\frac{\sqrt{b} d \sinh (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \sinh (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a^3}-\frac{3 b \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}+\frac{3 b \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}-\frac{\left (b^2 d\right ) \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^3}+\frac{\left (b d^2\right ) \int \frac{\cosh (c+d x)}{\sqrt{-a} \sqrt{b}-b x} \, dx}{16 a^3}-\frac{\left (b d^2\right ) \int \frac{\cosh (c+d x)}{\sqrt{-a} \sqrt{b}+b x} \, dx}{16 a^3}+\frac{\left (b 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}{2 (-a)^{7/2}}-\frac{\left (b 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}{2 (-a)^{7/2}}+\frac{\left (b 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}{2 (-a)^{7/2}}+\frac{\left (b 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}{2 (-a)^{7/2}}\\ &=-\frac{\cosh (c+d x)}{2 a^3 x^2}-\frac{b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a^3}+\frac{3 b \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}+\frac{3 b \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}+\frac{\sqrt{b} d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{7/2}}-\frac{\sqrt{b} d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{7/2}}-\frac{d \sinh (c+d x)}{2 a^3 x}-\frac{\sqrt{b} d \sinh (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \sinh (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a^3}+\frac{\sqrt{b} d \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{7/2}}-\frac{3 b \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}+\frac{\sqrt{b} d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{7/2}}+\frac{3 b \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}+\frac{(b d) \int \frac{\sinh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 (-a)^{7/2}}+\frac{(b d) \int \frac{\sinh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 (-a)^{7/2}}-\frac{\left (b 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}+\frac{\left (b 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}-\frac{\left (b 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}-\frac{\left (b 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}\\ &=-\frac{\cosh (c+d x)}{2 a^3 x^2}-\frac{b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a^3}+\frac{3 b \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}-\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}+\frac{3 b \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}-\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}+\frac{\sqrt{b} d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{7/2}}-\frac{\sqrt{b} d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 (-a)^{7/2}}-\frac{d \sinh (c+d x)}{2 a^3 x}-\frac{\sqrt{b} d \sinh (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \sinh (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a^3}+\frac{\sqrt{b} d \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 (-a)^{7/2}}-\frac{3 b \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}+\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}+\frac{\sqrt{b} d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 (-a)^{7/2}}+\frac{3 b \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}-\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}+\frac{\left (b 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)^{7/2}}-\frac{\left (b 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)^{7/2}}+\frac{\left (b 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)^{7/2}}+\frac{\left (b 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)^{7/2}}\\ &=-\frac{\cosh (c+d x)}{2 a^3 x^2}-\frac{b \cosh (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac{b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}+\frac{d^2 \cosh (c) \text{Chi}(d x)}{2 a^3}+\frac{3 b \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}-\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}+\frac{3 b \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}-\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}+\frac{9 \sqrt{b} 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)^{7/2}}-\frac{9 \sqrt{b} 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)^{7/2}}-\frac{d \sinh (c+d x)}{2 a^3 x}-\frac{\sqrt{b} d \sinh (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} d \sinh (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}+\frac{d^2 \sinh (c) \text{Shi}(d x)}{2 a^3}+\frac{9 \sqrt{b} 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)^{7/2}}-\frac{3 b \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 a^4}+\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}+\frac{9 \sqrt{b} 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)^{7/2}}+\frac{3 b \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 a^4}-\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}\\ \end{align*}

Mathematica [C]  time = 3.71688, size = 998, normalized size = 1.26 \[ -\frac{-i a \sinh (c) \left (\text{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right )-\text{CosIntegral}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right )+\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )-\text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )\right )\right ) d^2+a \cosh (c) \left (\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right )+\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )+\sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )+\text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )\right )\right ) d^2-9 i \sqrt{a} \sqrt{b} \sinh (c) \left (\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right )-\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )+\sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )-\text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )\right )\right ) d-9 \sqrt{a} \sqrt{b} \cosh (c) \left (\text{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right )+\text{CosIntegral}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right )-\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )+\text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )\right )\right ) d+\frac{2 a \cosh (d x) \left (2 \left (6 b^2 x^4+9 a b x^2+2 a^2\right ) \cosh (c)+d x \left (3 b^2 x^4+7 a b x^2+4 a^2\right ) \sinh (c)\right )}{x^2 \left (b x^2+a\right )^2}+\frac{2 a \left (d x \left (3 b^2 x^4+7 a b x^2+4 a^2\right ) \cosh (c)+2 \left (6 b^2 x^4+9 a b x^2+2 a^2\right ) \sinh (c)\right ) \sinh (d x)}{x^2 \left (b x^2+a\right )^2}+8 \left (6 b-a d^2\right ) (\cosh (c) \text{Chi}(d x)+\sinh (c) \text{Shi}(d x))+24 i b \sinh (c) \left (\text{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right )-\text{CosIntegral}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right ) \sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right )+\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )-\text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )\right )\right )-24 b \cosh (c) \left (\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (i d x-\frac{\sqrt{a} d}{\sqrt{b}}\right )+\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )+\sin \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )+\text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )\right )\right )}{16 a^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((2*a*Cosh[d*x]*(2*(2*a^2 + 9*a*b*x^2 + 6*b^2*x^4)*Cosh[c] + d*x*(4*a^2 + 7*a*b*x^2 + 3*b^2*x^4)*Sinh[c]))/(x
^2*(a + b*x^2)^2) + (2*a*(d*x*(4*a^2 + 7*a*b*x^2 + 3*b^2*x^4)*Cosh[c] + 2*(2*a^2 + 9*a*b*x^2 + 6*b^2*x^4)*Sinh
[c])*Sinh[d*x])/(x^2*(a + b*x^2)^2) + 8*(6*b - a*d^2)*(Cosh[c]*CoshIntegral[d*x] + Sinh[c]*SinhIntegral[d*x])
- (9*I)*Sqrt[a]*Sqrt[b]*d*Sinh[c]*(Cos[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] - Cos[
(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] + Sin[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[(Sqrt[a]
*d)/Sqrt[b] - I*d*x] - SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])) + (24*I)*b*Sinh[c]*(CosIntegral[-((Sqrt[a]*d
)/Sqrt[b]) + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]] - CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b
]] + Cos[(Sqrt[a]*d)/Sqrt[b]]*(-SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] + SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I
*d*x])) - I*a*d^2*Sinh[c]*(CosIntegral[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]] - CosIntegral[
(Sqrt[a]*d)/Sqrt[b] + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]] + Cos[(Sqrt[a]*d)/Sqrt[b]]*(-SinIntegral[(Sqrt[a]*d)/Sqr
t[b] - I*d*x] + SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])) - 9*Sqrt[a]*Sqrt[b]*d*Cosh[c]*(CosIntegral[-((Sqrt[
a]*d)/Sqrt[b]) + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]] + CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x]*Sin[(Sqrt[a]*d)/Sq
rt[b]] - Cos[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] + SinIntegral[(Sqrt[a]*d)/Sqrt[b]
+ I*d*x])) - 24*b*Cosh[c]*(Cos[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] + Cos[(Sqrt[a]
*d)/Sqrt[b]]*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] + Sin[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[(Sqrt[a]*d)/Sqrt
[b] - I*d*x] + SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])) + a*d^2*Cosh[c]*(Cos[(Sqrt[a]*d)/Sqrt[b]]*CosIntegra
l[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] + Cos[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] + Sin[(S
qrt[a]*d)/Sqrt[b]]*(SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] + SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])))/(16
*a^4)

________________________________________________________________________________________

Maple [B]  time = 0.152, size = 1294, normalized size = 1.6 \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^3/(b*x^2+a)^3,x)

[Out]

3/16*d^5*exp(-d*x-c)/a^3*x^3/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b^2+7/16*d^5*exp(-d*x-c)/a^2*x/(b^2*d^4*x^4+2
*a*b*d^4*x^2+a^2*d^4)*b-3/4*d^4*exp(-d*x-c)/a^3*x^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b^2+9/32*d/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-9/32*d/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+1/32*d^2/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)+1/32*d^2/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
)-3/4/a^4*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*b-3/4/a^4*exp(-(d*(-a*b)^(1/2)-c*
b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*b-1/4*d^2/a^3*exp(-c)*Ei(1,d*x)-3/16*d^5*exp(d*x+c)/a^3*x^3/(b^2
*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b^2-7/16*d^5*exp(d*x+c)/a^2*x/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b-3/4*d^4*ex
p(d*x+c)/a^3*x^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b^2-9/32*d/a^3/(-a*b)^(1/2)*exp((d*(-a*b)^(1/2)+c*b)/b)*E
i(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*b+3/2/a^4*b*exp(c)*Ei(1,-d*x)+9/32*d/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+1/32*d^2/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)+1/32*d^2/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)-3/4/a
^4*exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*b-3/4/a^4*exp((d*(-a*b)^(1/2)-c*b)/b)*
Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*b+1/4*d^5*exp(-d*x-c)/a/x/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)-9/8*d^4*e
xp(-d*x-c)/a^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b-1/4*d^4*exp(-d*x-c)/a/x^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*
d^4)+3/2/a^4*exp(-c)*Ei(1,d*x)*b-1/4*d^2/a^3*exp(c)*Ei(1,-d*x)-1/4*d^5*exp(d*x+c)/a/x/(b^2*d^4*x^4+2*a*b*d^4*x
^2+a^2*d^4)-9/8*d^4*exp(d*x+c)/a^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*b-1/4*d^4*exp(d*x+c)/a/x^2/(b^2*d^4*x^4
+2*a*b*d^4*x^2+a^2*d^4)

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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^3/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.20699, size = 4891, normalized size = 6.18 \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^3/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

-1/32*(8*(6*a*b^2*x^4 + 9*a^2*b*x^2 + 2*a^3)*cosh(d*x + c) + ((((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a
*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x
^4 + (a^3*d^2 - 24*a^2*b)*x^2)*sinh(d*x + c)^2 + 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*cosh(d*x + c)^2 - (b^3
*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) + (((a*b^2*d^2 - 24*
b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^
6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*sinh(d*x + c)^2 - 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2
*b*x^2)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^4 + a^2*b*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)) - 8*(((a*b^2*d^2 - 6*b^3)*x^6 + 2*(a^2*b*d^2 - 6*a*b^2)*x^4 + (a^3*d^2 -
6*a^2*b)*x^2)*Ei(d*x) + ((a*b^2*d^2 - 6*b^3)*x^6 + 2*(a^2*b*d^2 - 6*a*b^2)*x^4 + (a^3*d^2 - 6*a^2*b)*x^2)*Ei(-
d*x))*cosh(c) + ((((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*cosh(d*
x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*sinh(d*x + c)^
2 - 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*sinh(d*x + c)
^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) + (((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^
3*d^2 - 24*a^2*b)*x^2)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 -
24*a^2*b)*x^2)*sinh(d*x + c)^2 + 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*
x^4 + a^2*b*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*(3*
a*b^2*d*x^5 + 7*a^2*b*d*x^3 + 4*a^3*d*x)*sinh(d*x + c) + ((((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2
)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 +
(a^3*d^2 - 24*a^2*b)*x^2)*sinh(d*x + c)^2 + 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*cosh(d*x + c)^2 - (b^3*x^6
+ 2*a*b^2*x^4 + a^2*b*x^2)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*d^2/b)) - (((a*b^2*d^2 - 24*b^3)
*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 +
2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*sinh(d*x + c)^2 - 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x
^2)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^4 + a^2*b*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)) - 8*(((a*b^2*d^2 - 6*b^3)*x^6 + 2*(a^2*b*d^2 - 6*a*b^2)*x^4 + (a^3*d^2 - 6*a^
2*b)*x^2)*Ei(d*x) - ((a*b^2*d^2 - 6*b^3)*x^6 + 2*(a^2*b*d^2 - 6*a*b^2)*x^4 + (a^3*d^2 - 6*a^2*b)*x^2)*Ei(-d*x)
)*sinh(c) - ((((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*cosh(d*x +
c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*a^2*b)*x^2)*sinh(d*x + c)^2 -
9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*sinh(d*x + c)^2)*
sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) - (((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^
2 - 24*a^2*b)*x^2)*cosh(d*x + c)^2 - ((a*b^2*d^2 - 24*b^3)*x^6 + 2*(a^2*b*d^2 - 24*a*b^2)*x^4 + (a^3*d^2 - 24*
a^2*b)*x^2)*sinh(d*x + c)^2 + 9*((b^3*x^6 + 2*a*b^2*x^4 + a^2*b*x^2)*cosh(d*x + c)^2 - (b^3*x^6 + 2*a*b^2*x^4
+ a^2*b*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^4*b^2*
x^6 + 2*a^5*b*x^4 + a^6*x^2)*cosh(d*x + c)^2 - (a^4*b^2*x^6 + 2*a^5*b*x^4 + a^6*x^2)*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**3/(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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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