\(\int \frac {x^3 \cosh (c+d x)}{(a+b x^2)^3} \, dx\) [72]

Optimal result

Integrand size = 19, antiderivative size = 476 \[ \int \frac {x^3 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}+\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 b^3}+\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 b^3}-\frac {3 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}+\frac {3 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac {3 d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 b^3}-\frac {3 d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}}+\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 b^3} \]

[Out]

-1/4*x^2*cosh(d*x+c)/b/(b*x^2+a)^2-1/4*cosh(d*x+c)/b^2/(b*x^2+a)+1/16*d^2*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c
-d*(-a)^(1/2)/b^(1/2))/b^3+1/16*d^2*Chi(-d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c+d*(-a)^(1/2)/b^(1/2))/b^3-1/8*d*x*si
nh(d*x+c)/b^2/(b*x^2+a)+1/16*d^2*Shi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/b^3+1/16*d^2*Shi(d
*x-d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2))/b^3+3/16*d*cosh(c+d*(-a)^(1/2)/b^(1/2))*Shi(d*x-d*(-a)^(
1/2)/b^(1/2))/b^(5/2)/(-a)^(1/2)-3/16*d*cosh(c-d*(-a)^(1/2)/b^(1/2))*Shi(d*x+d*(-a)^(1/2)/b^(1/2))/b^(5/2)/(-a
)^(1/2)-3/16*d*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d*(-a)^(1/2)/b^(1/2))/b^(5/2)/(-a)^(1/2)+3/16*d*Chi(-d*x+d
*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2))/b^(5/2)/(-a)^(1/2)

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {5399, 5397, 5388, 3384, 3379, 3382, 5398, 5401} \[ \int \frac {x^3 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {3 d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}+\frac {3 d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {3 d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {3 d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}+\frac {d^2 \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 b^3}+\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 b^3}-\frac {d^2 \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 b^3}+\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 b^3}-\frac {d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2} \]

[In]

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

[Out]

-1/4*(x^2*Cosh[c + d*x])/(b*(a + b*x^2)^2) - Cosh[c + d*x]/(4*b^2*(a + b*x^2)) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sq
rt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*b^3) + (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(
Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*b^3) - (3*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqr
t[b]])/(16*Sqrt[-a]*b^(5/2)) + (3*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(
16*Sqrt[-a]*b^(5/2)) - (d*x*Sinh[c + d*x])/(8*b^2*(a + b*x^2)) - (3*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhInteg
ral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*Sqrt[-a]*b^(5/2)) - (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqr
t[-a]*d)/Sqrt[b] - d*x])/(16*b^3) - (3*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*
x])/(16*Sqrt[-a]*b^(5/2)) + (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*
b^3)

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])

Rule 5398

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[x^(m - n + 1)*(a + b*
x^n)^(p + 1)*(Sinh[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)*Sinh[c + d*x], x], x] - Dist[d/(b*n*(p + 1)), Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cosh[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 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 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^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac {\int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{2 b}+\frac {d \int \frac {x^2 \sinh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 b} \\ & = -\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac {d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac {d \int \frac {\sinh (c+d x)}{a+b x^2} \, dx}{8 b^2}+\frac {d \int \frac {\sinh (c+d x)}{a+b x^2} \, dx}{4 b^2}+\frac {d^2 \int \frac {x \cosh (c+d x)}{a+b x^2} \, dx}{8 b^2} \\ & = -\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac {d x \sinh (c+d x)}{8 b^2 \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}{8 b^2}+\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}{4 b^2}+\frac {d^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}{8 b^2} \\ & = -\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac {d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 \sqrt {-a} b^2}-\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 \sqrt {-a} b^2}-\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{8 \sqrt {-a} b^2}-\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{8 \sqrt {-a} b^2}-\frac {d^2 \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 b^{5/2}}+\frac {d^2 \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 b^{5/2}} \\ & = -\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac {d x \sinh (c+d x)}{8 b^2 \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}{16 \sqrt {-a} b^2}-\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}{8 \sqrt {-a} b^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} x} \, dx}{16 b^{5/2}}+\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}{16 \sqrt {-a} b^2}+\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}{8 \sqrt {-a} b^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} x} \, dx}{16 b^{5/2}}-\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}{16 \sqrt {-a} b^2}-\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}{8 \sqrt {-a} b^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} x} \, dx}{16 b^{5/2}}-\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}{16 \sqrt {-a} b^2}-\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}{8 \sqrt {-a} b^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} x} \, dx}{16 b^{5/2}} \\ & = -\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}+\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 b^3}+\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 b^3}-\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 \sqrt {-a} b^{5/2}}+\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 \sqrt {-a} b^{5/2}}-\frac {d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\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 \sqrt {-a} b^{5/2}}-\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 b^3}-\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 \sqrt {-a} b^{5/2}}+\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 b^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.06 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.70 \[ \int \frac {x^3 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {\frac {d e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (-3 i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\left (3 i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )}{\sqrt {a}}+\frac {d e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (\left (-3 i \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\left (3 i \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )}{\sqrt {a}}-\frac {4 b \cosh (d x) \left (2 \left (a+2 b x^2\right ) \cosh (c)+d x \left (a+b x^2\right ) \sinh (c)\right )}{\left (a+b x^2\right )^2}-\frac {4 b \left (d x \left (a+b x^2\right ) \cosh (c)+2 \left (a+2 b x^2\right ) \sinh (c)\right ) \sinh (d x)}{\left (a+b x^2\right )^2}}{32 b^3} \]

[In]

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

[Out]

((d*E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(((-3*I)*Sqrt[b] + Sqrt[a]*d)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[d*
(((-I)*Sqrt[a])/Sqrt[b] + x)] + ((3*I)*Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)]))/Sqrt[
a] + (d*E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*(((-3*I)*Sqrt[b] + Sqrt[a]*d)*E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegral
Ei[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] + ((3*I)*Sqrt[b] + Sqrt[a]*d)*ExpIntegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]))/
Sqrt[a] - (4*b*Cosh[d*x]*(2*(a + 2*b*x^2)*Cosh[c] + d*x*(a + b*x^2)*Sinh[c]))/(a + b*x^2)^2 - (4*b*(d*x*(a + b
*x^2)*Cosh[c] + 2*(a + 2*b*x^2)*Sinh[c])*Sinh[d*x])/(a + b*x^2)^2)/(32*b^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1546\) vs. \(2(374)=748\).

Time = 0.55 (sec) , antiderivative size = 1547, normalized size of antiderivative = 3.25

method	result	size
risch	\(\text {Expression too large to display}\)	\(1547\)

[In]

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

[Out]

-1/32*(-6*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^2-2*(-a*b)^(1/2)*exp(
-d*x-c)*a*b*d*x+6*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)*b^2*d^2*x^4+(-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^2*d^2*x^4+8*(-a*b)^(1/2)*exp(-d*x-c)*b^2*x^2+4*(-a*b)
^(1/2)*exp(-d*x-c)*a*b+(-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-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)*b^3*d*x^4+3*
exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*b^3*d*x^4-2*(-a*b)^(1/2)*exp(-d*x-c)*b^2*
d*x^3+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*b*d^2*x^2+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)*b^3*d*x^4-3*exp((-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)+
(d*x+c)*b-c*b)/b)*b^3*d*x^4+2*(-a*b)^(1/2)*exp(d*x+c)*b^2*d*x^3+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+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)*b^2*d^2*x^4+(-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^2*d^2*x^4-
6*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+6*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+2*(-a*b)^(1/2)*exp(d*x+c)*a*b*d*x+8*(-a*b)^(1/2)*exp(d*x
+c)*b^2*x^2+4*(-a*b)^(1/2)*exp(d*x+c)*a*b+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^
2-3*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)/(b^2*x^4+2*a*b*x^2+a^2)/b^3/(-a*b)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1620 vs. \(2 (374) = 748\).

Time = 0.29 (sec) , antiderivative size = 1620, normalized size of antiderivative = 3.40 \[ \int \frac {x^3 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

1/2*((d^2*x^3*e^(2*c) + 3*d*x^2*e^(2*c) + 12*x*e^(2*c))*e^(d*x) - (d^2*x^3 - 3*d*x^2 + 12*x)*e^(-d*x))/(b^3*d^
3*x^6*e^c + 3*a*b^2*d^3*x^4*e^c + 3*a^2*b*d^3*x^2*e^c + a^3*d^3*e^c) - 1/2*integrate(6*(3*a*d*x*e^c + (a*d^2*e
^c - 10*b*e^c)*x^2 + 2*a*e^c)*e^(d*x)/(b^4*d^3*x^8 + 4*a*b^3*d^3*x^6 + 6*a^2*b^2*d^3*x^4 + 4*a^3*b*d^3*x^2 + a
^4*d^3), x) + 1/2*integrate(-6*(3*a*d*x - (a*d^2 - 10*b)*x^2 - 2*a)*e^(-d*x)/(b^4*d^3*x^8*e^c + 4*a*b^3*d^3*x^
6*e^c + 6*a^2*b^2*d^3*x^4*e^c + 4*a^3*b*d^3*x^2*e^c + a^4*d^3*e^c), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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