\(\int \frac {\cosh (c+d x)}{x^3 (a+b x^2)} \, dx\) [64]

Optimal result

Integrand size = 19, antiderivative size = 270 \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=-\frac {\cosh (c+d x)}{2 a x^2}-\frac {b \cosh (c) \text {Chi}(d x)}{a^2}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a}+\frac {b \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2}-\frac {d \sinh (c+d x)}{2 a x}-\frac {b \sinh (c) \text {Shi}(d x)}{a^2}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a}-\frac {b \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2} \]

[Out]

-b*Chi(d*x)*cosh(c)/a^2+1/2*d^2*Chi(d*x)*cosh(c)/a-1/2*cosh(d*x+c)/a/x^2+1/2*b*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*c
osh(c-d*(-a)^(1/2)/b^(1/2))/a^2+1/2*b*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c+d*(-a)^(1/2)/b^(1/2))/a^2-b*Shi(d*
x)*sinh(c)/a^2+1/2*d^2*Shi(d*x)*sinh(c)/a-1/2*d*sinh(d*x+c)/a/x+1/2*b*Shi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(
-a)^(1/2)/b^(1/2))/a^2+1/2*b*Shi(d*x-d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2))/a^2

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5401, 3378, 3384, 3379, 3382} \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=-\frac {b \cosh (c) \text {Chi}(d x)}{a^2}+\frac {b \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}-\frac {b \sinh (c) \text {Shi}(d x)}{a^2}-\frac {b \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a}-\frac {\cosh (c+d x)}{2 a x^2}-\frac {d \sinh (c+d x)}{2 a x} \]

[In]

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

[Out]

-1/2*Cosh[c + d*x]/(a*x^2) - (b*Cosh[c]*CoshIntegral[d*x])/a^2 + (d^2*Cosh[c]*CoshIntegral[d*x])/(2*a) + (b*Co
sh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^2) + (b*Cosh[c - (Sqrt[-a]*d)/Sqrt
[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2) - (d*Sinh[c + d*x])/(2*a*x) - (b*Sinh[c]*SinhIntegral[d
*x])/a^2 + (d^2*Sinh[c]*SinhIntegral[d*x])/(2*a) - (b*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)
/Sqrt[b] - d*x])/(2*a^2) + (b*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2)

Rule 3378

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 3379

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 3382

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 3384

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 5401

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.80 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.86 \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\frac {b e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+b e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )-\frac {2 a \cosh (d x) (\cosh (c)+d x \sinh (c))}{x^2}-\frac {2 a (d x \cosh (c)+\sinh (c)) \sinh (d x)}{x^2}+2 \left (-2 b+a d^2\right ) (\cosh (c) \text {Chi}(d x)+\sinh (c) \text {Shi}(d x))}{4 a^2} \]

[In]

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

[Out]

(b*E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]
+ ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)]) + b*E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*(E^(((2*I)*Sqrt[a]*d)/Sqrt[b]
)*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] + ExpIntegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]) - (2*a*Cosh[d*x]
*(Cosh[c] + d*x*Sinh[c]))/x^2 - (2*a*(d*x*Cosh[c] + Sinh[c])*Sinh[d*x])/x^2 + 2*(-2*b + a*d^2)*(Cosh[c]*CoshIn
tegral[d*x] + Sinh[c]*SinhIntegral[d*x]))/(4*a^2)

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.22

method	result	size
risch	\(\frac {d \,{\mathrm e}^{-d x -c}}{4 a x}-\frac {{\mathrm e}^{-d x -c}}{4 a \,x^{2}}-\frac {b \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 a^{2}}-\frac {b \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 a^{2}}-\frac {d^{2} {\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{4 a}+\frac {{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b}{2 a^{2}}-\frac {d \,{\mathrm e}^{d x +c}}{4 a x}-\frac {{\mathrm e}^{d x +c}}{4 a \,x^{2}}-\frac {b \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{4 a^{2}}-\frac {b \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{4 a^{2}}-\frac {d^{2} {\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{4 a}+\frac {{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right ) b}{2 a^{2}}\)	\(330\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (222) = 444\).

Time = 0.28 (sec) , antiderivative size = 583, normalized size of antiderivative = 2.16 \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=-\frac {2 \, a d x \sinh \left (d x + c\right ) + 2 \, a \cosh \left (d x + c\right ) - {\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (a d^{2} - 2 \, b\right )} x^{2} {\rm Ei}\left (d x\right ) + {\left (a d^{2} - 2 \, b\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - {\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (a d^{2} - 2 \, b\right )} x^{2} {\rm Ei}\left (d x\right ) - {\left (a d^{2} - 2 \, b\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) + {\left ({\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (b x^{2} \cosh \left (d x + c\right )^{2} - b x^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{4 \, {\left (a^{2} x^{2} \cosh \left (d x + c\right )^{2} - a^{2} x^{2} \sinh \left (d x + c\right )^{2}\right )}} \]

[In]

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

[Out]

-1/4*(2*a*d*x*sinh(d*x + c) + 2*a*cosh(d*x + c) - ((b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei(d*x - sq
rt(-a*d^2/b)) + (b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei(-d*x + sqrt(-a*d^2/b)))*cosh(c + sqrt(-a*d^
2/b)) - ((a*d^2 - 2*b)*x^2*Ei(d*x) + (a*d^2 - 2*b)*x^2*Ei(-d*x))*cosh(c) - ((b*x^2*cosh(d*x + c)^2 - b*x^2*sin
h(d*x + c)^2)*Ei(d*x + sqrt(-a*d^2/b)) + (b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei(-d*x - sqrt(-a*d^2
/b)))*cosh(-c + sqrt(-a*d^2/b)) - ((b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei(d*x - sqrt(-a*d^2/b)) -
(b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei(-d*x + sqrt(-a*d^2/b)))*sinh(c + sqrt(-a*d^2/b)) - ((a*d^2
- 2*b)*x^2*Ei(d*x) - (a*d^2 - 2*b)*x^2*Ei(-d*x))*sinh(c) + ((b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei
(d*x + sqrt(-a*d^2/b)) - (b*x^2*cosh(d*x + c)^2 - b*x^2*sinh(d*x + c)^2)*Ei(-d*x - sqrt(-a*d^2/b)))*sinh(-c +
sqrt(-a*d^2/b)))/(a^2*x^2*cosh(d*x + c)^2 - a^2*x^2*sinh(d*x + c)^2)

Sympy [F]

\[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{x^{3} \left (a + b x^{2}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh (c+d x)}{x^3 \left (a+b x^2\right )} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x^3\,\left (b\,x^2+a\right )} \,d x \]

[In]

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

[Out]

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