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

Optimal. Leaf size=856 \[ \frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac {3 \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}-\frac {3 \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}+\frac {\sinh (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sinh (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {b} x+\sqrt {-a}\right )}+\frac {3 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^2 b}-\frac {3 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}-\frac {3 \cosh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {3 \cosh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\cosh (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {\cosh (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )^2}+\frac {3 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {3 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {3 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {3 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.26, antiderivative size = 856, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5281, 3297, 3303, 3298, 3301} \[ \frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac {3 \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}-\frac {3 \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}+\frac {\sinh (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sinh (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {b} x+\sqrt {-a}\right )}+\frac {3 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^2 b}-\frac {3 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}-\frac {3 \cosh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {3 \cosh (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\cosh (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {\cosh (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )^2}+\frac {3 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {3 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {3 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {3 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(a + b*x^2)^3,x]

[Out]

-Cosh[c + d*x]/(16*(-a)^(3/2)*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)^2) - (3*Cosh[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a]
- Sqrt[b]*x)) + Cosh[c + d*x]/(16*(-a)^(3/2)*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)^2) + (3*Cosh[c + d*x])/(16*a^2*Sqr
t[b]*(Sqrt[-a] + Sqrt[b]*x)) + (3*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16
*(-a)^(5/2)*Sqrt[b]) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^
(3/2)*b^(3/2)) - (3*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sq
rt[b]) - (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2))
- (3*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a^2*b) - (3*d*CoshIntegra
l[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a^2*b) + (d*Sinh[c + d*x])/(16*(-a)^(3/2)*b*
(Sqrt[-a] - Sqrt[b]*x)) + (d*Sinh[c + d*x])/(16*(-a)^(3/2)*b*(Sqrt[-a] + Sqrt[b]*x)) + (3*d*Cosh[c + (Sqrt[-a]
*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) - (3*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhInte
gral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(
Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) - (3*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-
a]*d)/Sqrt[b] + d*x])/(16*a^2*b) - (3*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])
/(16*(-a)^(5/2)*Sqrt[b]) - (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(
-a)^(3/2)*b^(3/2))

Rule 3297

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

Rule 3298

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

Rule 3301

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

Rule 3303

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

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

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Mathematica [C]  time = 2.60, size = 933, normalized size = 1.09 \[ \frac {\frac {6 b^{5/2} \cosh (c) \cosh (d x) x^3}{\left (b x^2+a\right )^2}+\frac {6 b^{5/2} \sinh (c) \sinh (d x) x^3}{\left (b x^2+a\right )^2}+\frac {2 a b^{3/2} d \cosh (d x) \sinh (c) x^2}{\left (b x^2+a\right )^2}+\frac {2 a b^{3/2} d \cosh (c) \sinh (d x) x^2}{\left (b x^2+a\right )^2}+\frac {10 a b^{3/2} \cosh (c) \cosh (d x) x}{\left (b x^2+a\right )^2}+\frac {10 a b^{3/2} \sinh (c) \sinh (d x) x}{\left (b x^2+a\right )^2}+\frac {2 a^2 \sqrt {b} d \cosh (d x) \sinh (c)}{\left (b x^2+a\right )^2}+\frac {\text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (i \left (3 b-a d^2\right ) \cosh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )-3 \sqrt {a} \sqrt {b} d \sinh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )}{\sqrt {a}}+\frac {i \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\left (a d^2-3 b\right ) \cosh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )+3 i \sqrt {a} \sqrt {b} d \sinh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )}{\sqrt {a}}+\frac {2 a^2 \sqrt {b} d \cosh (c) \sinh (d x)}{\left (b x^2+a\right )^2}-3 i \sqrt {b} d \cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \cosh (c) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-i \sqrt {a} d^2 \cosh (c) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\frac {3 i b \cosh (c) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )}{\sqrt {a}}+\sqrt {a} d^2 \cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sinh (c) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-\frac {3 b \cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sinh (c) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )}{\sqrt {a}}-3 \sqrt {b} d \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sinh (c) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+3 i \sqrt {b} d \cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \cosh (c) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+i \sqrt {a} d^2 \cosh (c) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-\frac {3 i b \cosh (c) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )}{\sqrt {a}}+\sqrt {a} d^2 \cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sinh (c) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-\frac {3 b \cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sinh (c) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )}{\sqrt {a}}-3 \sqrt {b} d \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \sinh (c) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )}{16 a^2 b^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((10*a*b^(3/2)*x*Cosh[c]*Cosh[d*x])/(a + b*x^2)^2 + (6*b^(5/2)*x^3*Cosh[c]*Cosh[d*x])/(a + b*x^2)^2 + (2*a^2*S
qrt[b]*d*Cosh[d*x]*Sinh[c])/(a + b*x^2)^2 + (2*a*b^(3/2)*d*x^2*Cosh[d*x]*Sinh[c])/(a + b*x^2)^2 + (CosIntegral
[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x]*(I*(3*b - a*d^2)*Cosh[c - (I*Sqrt[a]*d)/Sqrt[b]] - 3*Sqrt[a]*Sqrt[b]*d*Sinh[c
- (I*Sqrt[a]*d)/Sqrt[b]]))/Sqrt[a] + (I*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x]*((-3*b + a*d^2)*Cosh[c + (I*
Sqrt[a]*d)/Sqrt[b]] + (3*I)*Sqrt[a]*Sqrt[b]*d*Sinh[c + (I*Sqrt[a]*d)/Sqrt[b]]))/Sqrt[a] + (2*a^2*Sqrt[b]*d*Cos
h[c]*Sinh[d*x])/(a + b*x^2)^2 + (2*a*b^(3/2)*d*x^2*Cosh[c]*Sinh[d*x])/(a + b*x^2)^2 + (10*a*b^(3/2)*x*Sinh[c]*
Sinh[d*x])/(a + b*x^2)^2 + (6*b^(5/2)*x^3*Sinh[c]*Sinh[d*x])/(a + b*x^2)^2 - (3*I)*Sqrt[b]*d*Cos[(Sqrt[a]*d)/S
qrt[b]]*Cosh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] + ((3*I)*b*Cosh[c]*Sin[(Sqrt[a]*d)/Sqrt[b]]*SinIntegr
al[(Sqrt[a]*d)/Sqrt[b] - I*d*x])/Sqrt[a] - I*Sqrt[a]*d^2*Cosh[c]*Sin[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]
*d)/Sqrt[b] - I*d*x] - (3*b*Cos[(Sqrt[a]*d)/Sqrt[b]]*Sinh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x])/Sqrt[a]
+ Sqrt[a]*d^2*Cos[(Sqrt[a]*d)/Sqrt[b]]*Sinh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] - 3*Sqrt[b]*d*Sin[(Sq
rt[a]*d)/Sqrt[b]]*Sinh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] + (3*I)*Sqrt[b]*d*Cos[(Sqrt[a]*d)/Sqrt[b]]*
Cosh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] - ((3*I)*b*Cosh[c]*Sin[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt
[a]*d)/Sqrt[b] + I*d*x])/Sqrt[a] + I*Sqrt[a]*d^2*Cosh[c]*Sin[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[a]*d)/Sqrt
[b] + I*d*x] - (3*b*Cos[(Sqrt[a]*d)/Sqrt[b]]*Sinh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])/Sqrt[a] + Sqrt[
a]*d^2*Cos[(Sqrt[a]*d)/Sqrt[b]]*Sinh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] - 3*Sqrt[b]*d*Sin[(Sqrt[a]*d)
/Sqrt[b]]*Sinh[c]*SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])/(16*a^2*b^(3/2))

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fricas [B]  time = 0.66, size = 2116, normalized size = 2.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.15, size = 1064, normalized size = 1.24 \[ -\frac {d^{5} {\mathrm e}^{-d x -c} x^{2}}{16 a \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {3 d^{4} {\mathrm e}^{-d x -c} b \,x^{3}}{16 a^{2} \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{5} {\mathrm e}^{-d x -c}}{16 b \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {5 d^{4} {\mathrm e}^{-d x -c} x}{16 a \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {d^{2} {\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b a \sqrt {-a b}}-\frac {d^{2} {\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b a \sqrt {-a b}}-\frac {3 d \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b \,a^{2}}-\frac {3 d \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b \,a^{2}}-\frac {3 \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 a^{2} \sqrt {-a b}}+\frac {3 \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 a^{2} \sqrt {-a b}}+\frac {d^{5} {\mathrm e}^{d x +c} x^{2}}{16 a \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {3 d^{4} {\mathrm e}^{d x +c} b \,x^{3}}{16 a^{2} \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {d^{5} {\mathrm e}^{d x +c}}{16 b \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {5 d^{4} {\mathrm e}^{d x +c} x}{16 a \left (b^{2} d^{4} x^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {d^{2} {\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b a \sqrt {-a b}}-\frac {d^{2} {\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b a \sqrt {-a b}}+\frac {3 d \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 b \,a^{2}}+\frac {3 d \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 b \,a^{2}}-\frac {3 \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{32 a^{2} \sqrt {-a b}}+\frac {3 \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{32 a^{2} \sqrt {-a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)/(a + b*x^2)^3,x)

[Out]

int(cosh(c + d*x)/(a + b*x^2)^3, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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