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

Optimal. Leaf size=874 \[ \text{result too large to display} \]

[Out]

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

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Rubi [A]  time = 2.68917, antiderivative size = 874, normalized size of antiderivative = 1., number of steps used = 60, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {5293, 3297, 3303, 3298, 3301, 5281} \[ \frac{\cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) d^2}{16 (-a)^{5/2} \sqrt{b}}-\frac{\cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) d^2}{16 (-a)^{5/2} \sqrt{b}}-\frac{\sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) d^2}{16 (-a)^{5/2} \sqrt{b}}-\frac{\sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) d^2}{16 (-a)^{5/2} \sqrt{b}}+\frac{\text{Chi}(d x) \sinh (c) d}{a^3}+\frac{7 \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) d}{16 a^3}+\frac{7 \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) d}{16 a^3}+\frac{\sinh (c+d x) d}{16 (-a)^{5/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sinh (c+d x) d}{16 (-a)^{5/2} \left (\sqrt{b} x+\sqrt{-a}\right )}+\frac{\cosh (c) \text{Shi}(d x) d}{a^3}-\frac{7 \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) d}{16 a^3}+\frac{7 \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) d}{16 a^3}-\frac{\cosh (c+d x)}{a^3 x}+\frac{7 \sqrt{b} \cosh (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{7 \sqrt{b} \cosh (c+d x)}{16 a^3 \left (\sqrt{b} x+\sqrt{-a}\right )}-\frac{\sqrt{b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}-\sqrt{b} x\right )^2}+\frac{\sqrt{b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{b} x+\sqrt{-a}\right )^2}+\frac{15 \sqrt{b} \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{7/2}}-\frac{15 \sqrt{b} \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{7/2}}-\frac{15 \sqrt{b} \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{7/2}}-\frac{15 \sqrt{b} \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 (-a)^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 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 5281

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

Rubi steps

\begin{align*} \int \frac{\cosh (c+d x)}{x^2 \left (a+b x^2\right )^3} \, dx &=\int \left (\frac{\cosh (c+d x)}{a^3 x^2}-\frac{b \cosh (c+d x)}{a \left (a+b x^2\right )^3}-\frac{b \cosh (c+d x)}{a^2 \left (a+b x^2\right )^2}-\frac{b \cosh (c+d x)}{a^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\cosh (c+d x)}{x^2} \, dx}{a^3}-\frac{b \int \frac{\cosh (c+d x)}{a+b x^2} \, dx}{a^3}-\frac{b \int \frac{\cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a^2}-\frac{b \int \frac{\cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx}{a}\\ &=-\frac{\cosh (c+d x)}{a^3 x}-\frac{b \int \left (\frac{\sqrt{-a} \cosh (c+d x)}{2 a \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{-a} \cosh (c+d x)}{2 a \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{a^3}-\frac{b \int \left (-\frac{b \cosh (c+d x)}{4 a \left (\sqrt{-a} \sqrt{b}-b x\right )^2}-\frac{b \cosh (c+d x)}{4 a \left (\sqrt{-a} \sqrt{b}+b x\right )^2}-\frac{b \cosh (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{a^2}-\frac{b \int \left (-\frac{b^{3/2} \cosh (c+d x)}{8 (-a)^{3/2} \left (\sqrt{-a} \sqrt{b}-b x\right )^3}-\frac{3 b \cosh (c+d x)}{16 a^2 \left (\sqrt{-a} \sqrt{b}-b x\right )^2}-\frac{b^{3/2} \cosh (c+d x)}{8 (-a)^{3/2} \left (\sqrt{-a} \sqrt{b}+b x\right )^3}-\frac{3 b \cosh (c+d x)}{16 a^2 \left (\sqrt{-a} \sqrt{b}+b x\right )^2}-\frac{3 b \cosh (c+d x)}{8 a^2 \left (-a b-b^2 x^2\right )}\right ) \, dx}{a}+\frac{d \int \frac{\sinh (c+d x)}{x} \, dx}{a^3}\\ &=-\frac{\cosh (c+d x)}{a^3 x}-\frac{b \int \frac{\cosh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 (-a)^{7/2}}-\frac{b \int \frac{\cosh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 (-a)^{7/2}}+\frac{\left (3 b^2\right ) \int \frac{\cosh (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^2} \, dx}{16 a^3}+\frac{\left (3 b^2\right ) \int \frac{\cosh (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^2} \, dx}{16 a^3}+\frac{b^2 \int \frac{\cosh (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^2} \, dx}{4 a^3}+\frac{b^2 \int \frac{\cosh (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^2} \, dx}{4 a^3}+\frac{\left (3 b^2\right ) \int \frac{\cosh (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^3}+\frac{b^2 \int \frac{\cosh (c+d x)}{-a b-b^2 x^2} \, dx}{2 a^3}-\frac{b^{5/2} \int \frac{\cosh (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^3} \, dx}{8 (-a)^{5/2}}-\frac{b^{5/2} \int \frac{\cosh (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^3} \, dx}{8 (-a)^{5/2}}+\frac{(d \cosh (c)) \int \frac{\sinh (d x)}{x} \, dx}{a^3}+\frac{(d \sinh (c)) \int \frac{\cosh (d x)}{x} \, dx}{a^3}\\ &=-\frac{\cosh (c+d x)}{a^3 x}-\frac{\sqrt{b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}-\sqrt{b} x\right )^2}+\frac{7 \sqrt{b} \cosh (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}+\sqrt{b} x\right )^2}-\frac{7 \sqrt{b} \cosh (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{d \text{Chi}(d x) \sinh (c)}{a^3}+\frac{d \cosh (c) \text{Shi}(d x)}{a^3}+\frac{\left (3 b^2\right ) \int \left (-\frac{\sqrt{-a} \cosh (c+d x)}{2 a b \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{\sqrt{-a} \cosh (c+d x)}{2 a b \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{8 a^3}+\frac{b^2 \int \left (-\frac{\sqrt{-a} \cosh (c+d x)}{2 a b \left (\sqrt{-a}-\sqrt{b} x\right )}-\frac{\sqrt{-a} \cosh (c+d x)}{2 a b \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{2 a^3}-\frac{(3 b d) \int \frac{\sinh (c+d x)}{\sqrt{-a} \sqrt{b}-b x} \, dx}{16 a^3}+\frac{(3 b d) \int \frac{\sinh (c+d x)}{\sqrt{-a} \sqrt{b}+b x} \, dx}{16 a^3}-\frac{(b d) \int \frac{\sinh (c+d x)}{\sqrt{-a} \sqrt{b}-b x} \, dx}{4 a^3}+\frac{(b d) \int \frac{\sinh (c+d x)}{\sqrt{-a} \sqrt{b}+b x} \, dx}{4 a^3}+\frac{\left (b^{3/2} d\right ) \int \frac{\sinh (c+d x)}{\left (\sqrt{-a} \sqrt{b}-b x\right )^2} \, dx}{16 (-a)^{5/2}}-\frac{\left (b^{3/2} d\right ) \int \frac{\sinh (c+d x)}{\left (\sqrt{-a} \sqrt{b}+b x\right )^2} \, dx}{16 (-a)^{5/2}}-\frac{\left (b \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 (-a)^{7/2}}-\frac{\left (b \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 (-a)^{7/2}}-\frac{\left (b \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 (-a)^{7/2}}+\frac{\left (b \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 (-a)^{7/2}}\\ &=-\frac{\cosh (c+d x)}{a^3 x}-\frac{\sqrt{b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}-\sqrt{b} x\right )^2}+\frac{7 \sqrt{b} \cosh (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}+\sqrt{b} x\right )^2}-\frac{7 \sqrt{b} \cosh (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{\sqrt{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)^{7/2}}-\frac{\sqrt{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)^{7/2}}+\frac{d \text{Chi}(d x) \sinh (c)}{a^3}+\frac{d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{d \cosh (c) \text{Shi}(d x)}{a^3}-\frac{\sqrt{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)^{7/2}}-\frac{\sqrt{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)^{7/2}}-\frac{(3 b) \int \frac{\cosh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 (-a)^{7/2}}-\frac{(3 b) \int \frac{\cosh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 (-a)^{7/2}}-\frac{b \int \frac{\cosh (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{4 (-a)^{7/2}}-\frac{b \int \frac{\cosh (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{4 (-a)^{7/2}}-\frac{\left (\sqrt{b} d^2\right ) \int \frac{\cosh (c+d x)}{\sqrt{-a} \sqrt{b}-b x} \, dx}{16 (-a)^{5/2}}-\frac{\left (\sqrt{b} d^2\right ) \int \frac{\cosh (c+d x)}{\sqrt{-a} \sqrt{b}+b x} \, dx}{16 (-a)^{5/2}}+\frac{\left (3 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}+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}+b x} \, dx}{4 a^3}+\frac{\left (3 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}-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}-b x} \, dx}{4 a^3}+\frac{\left (3 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}+b x} \, dx}{16 a^3}+\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}+b x} \, dx}{4 a^3}-\frac{\left (3 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}-b x} \, dx}{16 a^3}-\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}-b x} \, dx}{4 a^3}\\ &=-\frac{\cosh (c+d x)}{a^3 x}-\frac{\sqrt{b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}-\sqrt{b} x\right )^2}+\frac{7 \sqrt{b} \cosh (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}+\sqrt{b} x\right )^2}-\frac{7 \sqrt{b} \cosh (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{\sqrt{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)^{7/2}}-\frac{\sqrt{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)^{7/2}}+\frac{d \text{Chi}(d x) \sinh (c)}{a^3}+\frac{7 d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a^3}+\frac{7 d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a^3}+\frac{d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{d \cosh (c) \text{Shi}(d x)}{a^3}-\frac{7 d \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^3}-\frac{\sqrt{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)^{7/2}}+\frac{7 d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a^3}-\frac{\sqrt{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)^{7/2}}-\frac{\left (3 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}{16 (-a)^{7/2}}-\frac{\left (b \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{4 (-a)^{7/2}}-\frac{\left (\sqrt{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)^{5/2}}-\frac{\left (3 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}{16 (-a)^{7/2}}-\frac{\left (b \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{4 (-a)^{7/2}}-\frac{\left (\sqrt{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)^{5/2}}-\frac{\left (3 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}{16 (-a)^{7/2}}-\frac{\left (b \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{4 (-a)^{7/2}}-\frac{\left (\sqrt{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)^{5/2}}+\frac{\left (3 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}{16 (-a)^{7/2}}+\frac{\left (b \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )\right ) \int \frac{\sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{4 (-a)^{7/2}}+\frac{\left (\sqrt{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)^{5/2}}\\ &=-\frac{\cosh (c+d x)}{a^3 x}-\frac{\sqrt{b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}-\sqrt{b} x\right )^2}+\frac{7 \sqrt{b} \cosh (c+d x)}{16 a^3 \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{b} \cosh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}+\sqrt{b} x\right )^2}-\frac{7 \sqrt{b} \cosh (c+d x)}{16 a^3 \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{15 \sqrt{b} \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{7/2}}+\frac{d^2 \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{5/2} \sqrt{b}}-\frac{15 \sqrt{b} \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{7/2}}-\frac{d^2 \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{5/2} \sqrt{b}}+\frac{d \text{Chi}(d x) \sinh (c)}{a^3}+\frac{7 d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a^3}+\frac{7 d \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 a^3}+\frac{d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{d \sinh (c+d x)}{16 (-a)^{5/2} \left (\sqrt{-a}+\sqrt{b} x\right )}+\frac{d \cosh (c) \text{Shi}(d x)}{a^3}-\frac{7 d \cosh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 a^3}-\frac{15 \sqrt{b} \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{7/2}}-\frac{d^2 \sinh \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 (-a)^{5/2} \sqrt{b}}+\frac{7 d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 a^3}-\frac{15 \sqrt{b} \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{7/2}}-\frac{d^2 \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 (-a)^{5/2} \sqrt{b}}\\ \end{align*}

Mathematica [C]  time = 3.37809, size = 1359, normalized size = 1.55 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*a^(5/2)*Sqrt[b]*Cosh[c + d*x] - 50*a^(3/2)*b^(3/2)*x^2*Cosh[c + d*x] - 30*Sqrt[a]*b^(5/2)*x^4*Cosh[c + d*
x] + 16*a^(5/2)*Sqrt[b]*d*x*CoshIntegral[d*x]*Sinh[c] + 32*a^(3/2)*b^(3/2)*d*x^3*CoshIntegral[d*x]*Sinh[c] + 1
6*Sqrt[a]*b^(5/2)*d*x^5*CoshIntegral[d*x]*Sinh[c] + x*(a + b*x^2)^2*CoshIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*
((-I)*(15*b - a*d^2)*Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]] + 7*Sqrt[a]*Sqrt[b]*d*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]) +
x*(a + b*x^2)^2*CoshIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(I*(15*b - a*d^2)*Cosh[c + (I*Sqrt[a]*d)/Sqrt[b]]
+ 7*Sqrt[a]*Sqrt[b]*d*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]) - 2*a^(5/2)*Sqrt[b]*d*x*Sinh[c + d*x] - 2*a^(3/2)*b^(3
/2)*d*x^3*Sinh[c + d*x] + 16*a^(5/2)*Sqrt[b]*d*x*Cosh[c]*SinhIntegral[d*x] + 32*a^(3/2)*b^(3/2)*d*x^3*Cosh[c]*
SinhIntegral[d*x] + 16*Sqrt[a]*b^(5/2)*d*x^5*Cosh[c]*SinhIntegral[d*x] + 7*a^(5/2)*Sqrt[b]*d*x*Cosh[c - (I*Sqr
t[a]*d)/Sqrt[b]]*SinhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + 14*a^(3/2)*b^(3/2)*d*x^3*Cosh[c - (I*Sqrt[a]*d)/S
qrt[b]]*SinhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + 7*Sqrt[a]*b^(5/2)*d*x^5*Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]]*Si
nhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (15*I)*a^2*b*x*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[d*((I*Sq
rt[a])/Sqrt[b] + x)] + I*a^3*d^2*x*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] -
(30*I)*a*b^2*x^3*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + (2*I)*a^2*b*d^2*
x^3*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (15*I)*b^3*x^5*Sinh[c - (I*Sqr
t[a]*d)/Sqrt[b]]*SinhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*a*b^2*d^2*x^5*Sinh[c - (I*Sqrt[a]*d)/Sqrt[b]]*S
inhIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - 7*a^(5/2)*Sqrt[b]*d*x*Cosh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[
(I*Sqrt[a]*d)/Sqrt[b] - d*x] - 14*a^(3/2)*b^(3/2)*d*x^3*Cosh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a
]*d)/Sqrt[b] - d*x] - 7*Sqrt[a]*b^(5/2)*d*x^5*Cosh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a]*d)/Sqrt[
b] - d*x] - (15*I)*a^2*b*x*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + I*a^3*d
^2*x*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - (30*I)*a*b^2*x^3*Sinh[c + (I*
Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + (2*I)*a^2*b*d^2*x^3*Sinh[c + (I*Sqrt[a]*d)/Sqr
t[b]]*SinhIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - (15*I)*b^3*x^5*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[
(I*Sqrt[a]*d)/Sqrt[b] - d*x] + I*a*b^2*d^2*x^5*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinhIntegral[(I*Sqrt[a]*d)/Sqrt
[b] - d*x])/(16*a^(7/2)*Sqrt[b]*x*(a + b*x^2)^2)

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Maple [A]  time = 0.128, size = 1178, normalized size = 1.4 \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^2/(b*x^2+a)^3,x)

[Out]

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

________________________________________________________________________________________

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^2/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.17239, size = 4894, normalized size = 5.6 \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^2/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

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

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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**2/(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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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