\(\int \frac {x^4 \cosh (c+d x)}{(a+b x^2)^2} \, dx\) [65]

Optimal result

Integrand size = 19, antiderivative size = 449 \[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {x \cosh (c+d x)}{2 b^2}-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \sqrt {-a} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}}-\frac {a d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}-\frac {a d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}+\frac {\sinh (c+d x)}{b^2 d}+\frac {a d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}-\frac {3 \sqrt {-a} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {a d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^3}-\frac {3 \sqrt {-a} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}} \]

[Out]

1/2*x*cosh(d*x+c)/b^2-1/2*x^3*cosh(d*x+c)/b/(b*x^2+a)-1/4*a*d*cosh(c+d*(-a)^(1/2)/b^(1/2))*Shi(d*x-d*(-a)^(1/2
)/b^(1/2))/b^3-1/4*a*d*cosh(c-d*(-a)^(1/2)/b^(1/2))*Shi(d*x+d*(-a)^(1/2)/b^(1/2))/b^3+sinh(d*x+c)/b^2/d-1/4*a*
d*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/b^3-1/4*a*d*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c
+d*(-a)^(1/2)/b^(1/2))/b^3-3/4*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c-d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)+3
/4*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c+d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)-3/4*Shi(d*x+d*(-a)^(1/2)/b^(
1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))*(-a)^(1/2)/b^(5/2)+3/4*Shi(d*x-d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^
(1/2))*(-a)^(1/2)/b^(5/2)

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {5399, 5401, 2717, 5389, 3384, 3379, 3382, 5400, 3377} \[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {3 \sqrt {-a} \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}-\frac {a d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}-\frac {a d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}+\frac {a d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}-\frac {a d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\sinh (c+d x)}{b^2 d}+\frac {x \cosh (c+d x)}{2 b^2} \]

[In]

Int[(x^4*Cosh[c + d*x])/(a + b*x^2)^2,x]

[Out]

(x*Cosh[c + d*x])/(2*b^2) - (x^3*Cosh[c + d*x])/(2*b*(a + b*x^2)) + (3*Sqrt[-a]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]
*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) - (3*Sqrt[-a]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegr
al[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2)) - (a*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]
*d)/Sqrt[b]])/(4*b^3) - (a*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*b^3)
+ Sinh[c + d*x]/(b^2*d) + (a*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^3
) - (3*Sqrt[-a]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) - (a*d*Co
sh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^3) - (3*Sqrt[-a]*Sinh[c - (Sqrt[-a
]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2))

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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 5389

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])

Rule 5399

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

Rule 5400

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sinh[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 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}& = -\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx}{2 b}+\frac {d \int \frac {x^3 \sinh (c+d x)}{a+b x^2} \, dx}{2 b} \\ & = -\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \int \left (\frac {\cosh (c+d x)}{b}-\frac {a \cosh (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx}{2 b}+\frac {d \int \left (\frac {x \sinh (c+d x)}{b}-\frac {a x \sinh (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx}{2 b} \\ & = -\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \int \cosh (c+d x) \, dx}{2 b^2}-\frac {(3 a) \int \frac {\cosh (c+d x)}{a+b x^2} \, dx}{2 b^2}+\frac {d \int x \sinh (c+d x) \, dx}{2 b^2}-\frac {(a d) \int \frac {x \sinh (c+d x)}{a+b x^2} \, dx}{2 b^2} \\ & = \frac {x \cosh (c+d x)}{2 b^2}-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \sinh (c+d x)}{2 b^2 d}-\frac {\int \cosh (c+d x) \, dx}{2 b^2}-\frac {(3 a) \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}{2 b^2}-\frac {(a d) \int \left (-\frac {\sinh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sinh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{2 b^2} \\ & = \frac {x \cosh (c+d x)}{2 b^2}-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\sinh (c+d x)}{b^2 d}-\frac {\left (3 \sqrt {-a}\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 b^2}-\frac {\left (3 \sqrt {-a}\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 b^2}+\frac {(a d) \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 b^{5/2}}-\frac {(a d) \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 b^{5/2}} \\ & = \frac {x \cosh (c+d x)}{2 b^2}-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\sinh (c+d x)}{b^2 d}-\frac {\left (3 \sqrt {-a} \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 b^2}-\frac {\left (a 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 b^{5/2}}-\frac {\left (3 \sqrt {-a} \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 b^2}-\frac {\left (a 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 b^{5/2}}-\frac {\left (3 \sqrt {-a} \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 b^2}-\frac {\left (a 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 b^{5/2}}+\frac {\left (3 \sqrt {-a} \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 b^2}+\frac {\left (a 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 b^{5/2}} \\ & = \frac {x \cosh (c+d x)}{2 b^2}-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \sqrt {-a} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}}-\frac {a d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}-\frac {a d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}+\frac {\sinh (c+d x)}{b^2 d}+\frac {a d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}-\frac {3 \sqrt {-a} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {a d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^3}-\frac {3 \sqrt {-a} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.27 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.68 \[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {-\sqrt {a} e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (-3 i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (3 i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+\sqrt {a} e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (-3 i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (3 i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )+4 b \cosh (d x) \left (\frac {a x \cosh (c)}{a+b x^2}+\frac {2 \sinh (c)}{d}\right )+4 b \left (\frac {2 \cosh (c)}{d}+\frac {a x \sinh (c)}{a+b x^2}\right ) \sinh (d x)}{8 b^3} \]

[In]

Integrate[(x^4*Cosh[c + d*x])/(a + b*x^2)^2,x]

[Out]

(-(Sqrt[a]*E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(((-3*I)*Sqrt[b] + Sqrt[a]*d)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegr
alEi[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + ((3*I)*Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)])
) + Sqrt[a]*E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*(((-3*I)*Sqrt[b] + Sqrt[a]*d)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpInte
gralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] + ((3*I)*Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x
]) + 4*b*Cosh[d*x]*((a*x*Cosh[c])/(a + b*x^2) + (2*Sinh[c])/d) + 4*b*((2*Cosh[c])/d + (a*x*Sinh[c])/(a + b*x^2
))*Sinh[d*x])/(8*b^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1037\) vs. \(2(349)=698\).

Time = 0.42 (sec) , antiderivative size = 1038, normalized size of antiderivative = 2.31

method	result	size
risch	\(\text {Expression too large to display}\)	\(1038\)

[In]

int(x^4*cosh(d*x+c)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

-1/8/d*(exp(-(d*(-a*b)^(1/2)+c*b)/b)*(-a*b)^(1/2)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a*b*d^2*x^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)*a*b*d^2*x^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)*a*b*d^2*x^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)*a*b*d^2*x^2-3*exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+
c)*b+c*b)/b)*a*b^2*d*x^2-3*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a*b^2*d*x^2+3*ex
p((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a*b^2*d*x^2+3*exp(-(-d*(-a*b)^(1/2)+c*b)/b)
*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a*b^2*d*x^2+exp(-(d*(-a*b)^(1/2)+c*b)/b)*(-a*b)^(1/2)*Ei(1,-(d*(-a*b)^
(1/2)-(d*x+c)*b+c*b)/b)*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
)*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)*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)*a^2*d^2-3*exp(-(d*(-a*b)^(1/2)+c*b)/b)
*Ei(1,-(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a^2*b*d-2*(-a*b)^(1/2)*exp(d*x+c)*a*b*d*x-4*(-a*b)^(1/2)*exp(d*x+c)*b
^2*x^2-2*(-a*b)^(1/2)*exp(-d*x-c)*a*b*d*x+4*(-a*b)^(1/2)*exp(-d*x-c)*b^2*x^2-3*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(
1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a^2*b*d+3*exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b
)/b)*a^2*b*d+3*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a^2*b*d-4*(-a*b)^(1/2)*exp
(d*x+c)*a*b+4*(-a*b)^(1/2)*exp(-d*x-c)*a*b)/b^3/(b*x^2+a)/(-a*b)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1179 vs. \(2 (349) = 698\).

Time = 0.27 (sec) , antiderivative size = 1179, normalized size of antiderivative = 2.63 \[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(x^4*cosh(d*x+c)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/8*(4*a*b*d*x*cosh(d*x + c) - (((a*b*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x +
c)^2 - 3*((b^2*x^2 + a*b)*cosh(d*x + c)^2 - (b^2*x^2 + a*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(-a*
d^2/b)) - ((a*b*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x + c)^2 + 3*((b^2*x^2 + a
*b)*cosh(d*x + c)^2 - (b^2*x^2 + a*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(-d*x + sqrt(-a*d^2/b)))*cosh(c + sqr
t(-a*d^2/b)) - (((a*b*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x + c)^2 + 3*((b^2*x
^2 + a*b)*cosh(d*x + c)^2 - (b^2*x^2 + a*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) - ((a*b*
d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x + c)^2 - 3*((b^2*x^2 + a*b)*cosh(d*x + c
)^2 - (b^2*x^2 + a*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(-d*x - sqrt(-a*d^2/b)))*cosh(-c + sqrt(-a*d^2/b)) +
8*(b^2*x^2 + a*b)*sinh(d*x + c) - (((a*b*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x
+ c)^2 - 3*((b^2*x^2 + a*b)*cosh(d*x + c)^2 - (b^2*x^2 + a*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x - sqrt(
-a*d^2/b)) + ((a*b*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x + c)^2 + 3*((b^2*x^2
+ a*b)*cosh(d*x + c)^2 - (b^2*x^2 + a*b)*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*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x + c)^2 + 3*((b^
2*x^2 + a*b)*cosh(d*x + c)^2 - (b^2*x^2 + a*b)*sinh(d*x + c)^2)*sqrt(-a*d^2/b))*Ei(d*x + sqrt(-a*d^2/b)) + ((a
*b*d^2*x^2 + a^2*d^2)*cosh(d*x + c)^2 - (a*b*d^2*x^2 + a^2*d^2)*sinh(d*x + c)^2 - 3*((b^2*x^2 + a*b)*cosh(d*x
+ c)^2 - (b^2*x^2 + a*b)*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^4*d*x^2 + a*b^3*d)*cosh(d*x + c)^2 - (b^4*d*x^2 + a*b^3*d)*sinh(d*x + c)^2)

Sympy [F]

\[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{4} \cosh {\left (c + d x \right )}}{\left (a + b x^{2}\right )^{2}}\, dx \]

[In]

integrate(x**4*cosh(d*x+c)/(b*x**2+a)**2,x)

[Out]

Integral(x**4*cosh(c + d*x)/(a + b*x**2)**2, x)

Maxima [F]

\[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int { \frac {x^{4} \cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]

[In]

integrate(x^4*cosh(d*x+c)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=\text {Exception raised: AttributeError} \]

[In]

integrate(x^4*cosh(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

Exception raised: AttributeError >> type

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^4\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^2} \,d x \]

[In]

int((x^4*cosh(c + d*x))/(a + b*x^2)^2,x)

[Out]

int((x^4*cosh(c + d*x))/(a + b*x^2)^2, x)