\(\int \frac {x \cosh (c+d x)}{(a+b x^2)^2} \, dx\) [68]

Optimal result

Integrand size = 17, antiderivative size = 239 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {\cosh (c+d x)}{2 b \left (a+b x^2\right )}-\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 \sqrt {-a} b^{3/2}}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 \sqrt {-a} b^{3/2}}-\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 \sqrt {-a} b^{3/2}}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 \sqrt {-a} b^{3/2}} \]

[Out]

-1/2*cosh(d*x+c)/b/(b*x^2+a)+1/4*d*cosh(c+d*(-a)^(1/2)/b^(1/2))*Shi(d*x-d*(-a)^(1/2)/b^(1/2))/b^(3/2)/(-a)^(1/
2)-1/4*d*cosh(c-d*(-a)^(1/2)/b^(1/2))*Shi(d*x+d*(-a)^(1/2)/b^(1/2))/b^(3/2)/(-a)^(1/2)-1/4*d*Chi(d*x+d*(-a)^(1
/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/b^(3/2)/(-a)^(1/2)+1/4*d*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a
)^(1/2)/b^(1/2))/b^(3/2)/(-a)^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {5397, 5388, 3384, 3379, 3382} \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 \sqrt {-a} b^{3/2}}+\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 \sqrt {-a} b^{3/2}}-\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 \sqrt {-a} b^{3/2}}-\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 \sqrt {-a} b^{3/2}}-\frac {\cosh (c+d x)}{2 b \left (a+b x^2\right )} \]

[In]

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

[Out]

-1/2*Cosh[c + d*x]/(b*(a + b*x^2)) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]
])/(4*Sqrt[-a]*b^(3/2)) + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*Sqrt[
-a]*b^(3/2)) - (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*Sqrt[-a]*b^(3/2)
) - (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*Sqrt[-a]*b^(3/2))

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 5388

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

Rule 5397

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])

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.82 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.99 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {-\frac {4 \sqrt {b} \cosh (c) \cosh (d x)}{a+b x^2}-\frac {i d e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )}{\sqrt {a}}-\frac {i d e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{\sqrt {a}}-\frac {4 \sqrt {b} \sinh (c) \sinh (d x)}{a+b x^2}}{8 b^{3/2}} \]

[In]

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

[Out]

((-4*Sqrt[b]*Cosh[c]*Cosh[d*x])/(a + b*x^2) - (I*d*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)]))/Sqrt[a] - (I*d*
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]))/Sqrt[a] - (4*Sqrt[b]*Sinh[c]*Sinh[d*x])/(a + b*x^2))/(8*b^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(448\) vs. \(2(181)=362\).

Time = 0.23 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.88

method	result	size
risch	\(-\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) b d \,x^{2}-{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) b d \,x^{2}-{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) b d \,x^{2}+{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) b d \,x^{2}+{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a d -{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a d -{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a d +{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a d +2 \sqrt {-a b}\, {\mathrm e}^{-d x -c}+2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{8 \left (b \,x^{2}+a \right ) b \sqrt {-a b}}\)	\(449\)

[In]

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

[Out]

-1/8*(exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*b*d*x^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*d*x^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*d*x^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*d*x^2+exp((d*(-a*b)^(1/2
)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a*d-exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c
)*b-c*b)/b)*a*d-exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a*d+exp(-(-d*(-a*b)^(1/2)
+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a*d+2*(-a*b)^(1/2)*exp(-d*x-c)+2*(-a*b)^(1/2)*exp(d*x+c))/(b*x
^2+a)/b/(-a*b)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (181) = 362\).

Time = 0.25 (sec) , antiderivative size = 641, normalized size of antiderivative = 2.68 \[ \int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {4 \, a \cosh \left (d x + c\right ) + {\left ({\left ({\left (b x^{2} + a\right )} \cosh \left (d x + c\right )^{2} - {\left (b x^{2} + a\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (b x^{2} + a\right )} \cosh \left (d x + c\right )^{2} - {\left (b x^{2} + a\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\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 ({\left (b x^{2} + a\right )} \cosh \left (d x + c\right )^{2} - {\left (b x^{2} + a\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (b x^{2} + a\right )} \cosh \left (d x + c\right )^{2} - {\left (b x^{2} + a\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\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 ({\left (b x^{2} + a\right )} \cosh \left (d x + c\right )^{2} - {\left (b x^{2} + a\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left ({\left (b x^{2} + a\right )} \cosh \left (d x + c\right )^{2} - {\left (b x^{2} + a\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\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 ({\left (b x^{2} + a\right )} \cosh \left (d x + c\right )^{2} - {\left (b x^{2} + a\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left ({\left (b x^{2} + a\right )} \cosh \left (d x + c\right )^{2} - {\left (b x^{2} + a\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{8 \, {\left ({\left (a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )}} \]

[In]

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

[Out]

-1/8*(4*a*cosh(d*x + c) + (((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*sinh(d*x + c)^2)*sqrt(-a*d^2/b)*Ei(d*x -
sqrt(-a*d^2/b)) - ((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*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)) - (((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*sinh(d*x + c)^2)*sqrt(-a*d^2
/b)*Ei(d*x + sqrt(-a*d^2/b)) - ((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*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)) + (((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*sinh(d*x + c)^2
)*sqrt(-a*d^2/b)*Ei(d*x - sqrt(-a*d^2/b)) + ((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*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)) + (((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*sin
h(d*x + c)^2)*sqrt(-a*d^2/b)*Ei(d*x + sqrt(-a*d^2/b)) + ((b*x^2 + a)*cosh(d*x + c)^2 - (b*x^2 + a)*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*b^2*x^2 + a^2*b)*cosh(d*x + c)^
2 - (a*b^2*x^2 + a^2*b)*sinh(d*x + c)^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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