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

Optimal. Leaf size=476 \[ -\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}+\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 b^3}+\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 b^3}-\frac {3 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}+\frac {3 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac {3 d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 b^3}-\frac {3 d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}}+\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 b^3} \]

[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)

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Rubi [A]
time = 0.75, antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {5399, 5397, 5388, 3384, 3379, 3382, 5398, 5401} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {\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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.42, size = 648, normalized size = 1.36 \begin {gather*} \frac {-\frac {2 \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 {2 \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}+\frac {3 i d \sinh (c) \left (\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )-\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )\right )}{\sqrt {a} \sqrt {b}}-\frac {i d^2 \sinh (c) \left (\text {CosIntegral}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right ) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )-\text {CosIntegral}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right ) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (-\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )\right )}{b}+\frac {3 d \cosh (c) \left (\text {CosIntegral}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right ) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+\text {CosIntegral}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right ) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )-\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )\right )}{\sqrt {a} \sqrt {b}}+\frac {d^2 \cosh (c) \left (\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )\right )}{b}}{16 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*Cosh[d*x]*(2*(a + 2*b*x^2)*Cosh[c] + d*x*(a + b*x^2)*Sinh[c]))/(a + b*x^2)^2 - (2*(d*x*(a + b*x^2)*Cosh[c
] + 2*(a + 2*b*x^2)*Sinh[c])*Sinh[d*x])/(a + b*x^2)^2 + ((3*I)*d*Sinh[c]*(Cos[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral
[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] - Cos[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] + Sin[(Sq
rt[a]*d)/Sqrt[b]]*(SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] - SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])))/(Sqr
t[a]*Sqrt[b]) - (I*d^2*Sinh[c]*(CosIntegral[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]] - CosInte
gral[(Sqrt[a]*d)/Sqrt[b] + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]] + Cos[(Sqrt[a]*d)/Sqrt[b]]*(-SinIntegral[(Sqrt[a]*d
)/Sqrt[b] - I*d*x] + SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])))/b + (3*d*Cosh[c]*(CosIntegral[-((Sqrt[a]*d)/S
qrt[b]) + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]] + CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x]*Sin[(Sqrt[a]*d)/Sqrt[b]]
- Cos[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[(Sqrt[a]*d)/Sqrt[b] - I*d*x] + SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x
])))/(Sqrt[a]*Sqrt[b]) + (d^2*Cosh[c]*(Cos[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] +
Cos[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x] + Sin[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[(Sqr
t[a]*d)/Sqrt[b] - I*d*x] + SinIntegral[(Sqrt[a]*d)/Sqrt[b] + I*d*x])))/b)/(16*b^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(819\) vs. \(2(374)=748\).
time = 1.10, size = 820, normalized size = 1.72

method	result	size
risch	\(-\frac {d^{4} {\mathrm e}^{-d x -c} x^{2}}{4 b \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{2} {\mathrm e}^{-\frac {-d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, \frac {d \sqrt {-a b}+b \left (d x +c \right )-b c}{b}\right )}{32 b^{3}}-\frac {d^{2} {\mathrm e}^{-\frac {d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, -\frac {d \sqrt {-a b}-b \left (d x +c \right )+b c}{b}\right )}{32 b^{3}}-\frac {3 d \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, \frac {d \sqrt {-a b}+b \left (d x +c \right )-b c}{b}\right )}{32 b^{2} \sqrt {-a b}}+\frac {3 d \,{\mathrm e}^{-\frac {d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, -\frac {d \sqrt {-a b}-b \left (d x +c \right )+b c}{b}\right )}{32 b^{2} \sqrt {-a b}}+\frac {d^{5} {\mathrm e}^{-d x -c} x^{3}}{16 b \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {d^{5} {\mathrm e}^{-d x -c} a x}{16 b^{2} \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{4} {\mathrm e}^{-d x -c} a}{8 b^{2} \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{4} {\mathrm e}^{d x +c} x^{2}}{4 b \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{2} {\mathrm e}^{\frac {d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, \frac {d \sqrt {-a b}-b \left (d x +c \right )+b c}{b}\right )}{32 b^{3}}-\frac {d^{2} {\mathrm e}^{\frac {-d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, -\frac {d \sqrt {-a b}+b \left (d x +c \right )-b c}{b}\right )}{32 b^{3}}-\frac {3 d \,{\mathrm e}^{\frac {d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, \frac {d \sqrt {-a b}-b \left (d x +c \right )+b c}{b}\right )}{32 b^{2} \sqrt {-a b}}+\frac {3 d \,{\mathrm e}^{\frac {-d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, -\frac {d \sqrt {-a b}+b \left (d x +c \right )-b c}{b}\right )}{32 b^{2} \sqrt {-a b}}-\frac {d^{5} {\mathrm e}^{d x +c} x^{3}}{16 b \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{5} {\mathrm e}^{d x +c} a x}{16 b^{2} \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{4} {\mathrm e}^{d x +c} a}{8 b^{2} \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}\)	\(820\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*d^4*exp(-d*x-c)/b/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*x^2-1/32*d^2/b^3*exp(-(-d*(-a*b)^(1/2)+b*c)/b)*Ei(1
,(d*(-a*b)^(1/2)+b*(d*x+c)-b*c)/b)-1/32*d^2/b^3*exp(-(d*(-a*b)^(1/2)+b*c)/b)*Ei(1,-(d*(-a*b)^(1/2)-b*(d*x+c)+b
*c)/b)-3/32*d/b^2/(-a*b)^(1/2)*exp(-(-d*(-a*b)^(1/2)+b*c)/b)*Ei(1,(d*(-a*b)^(1/2)+b*(d*x+c)-b*c)/b)+3/32*d/b^2
/(-a*b)^(1/2)*exp(-(d*(-a*b)^(1/2)+b*c)/b)*Ei(1,-(d*(-a*b)^(1/2)-b*(d*x+c)+b*c)/b)+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)*x^3+1/16*d^5*exp(-d*x-c)*a/b^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*x-1/8*d^4*e
xp(-d*x-c)*a/b^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)-1/4*d^4*exp(d*x+c)/b/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*
x^2-1/32*d^2/b^3*exp((d*(-a*b)^(1/2)+b*c)/b)*Ei(1,(d*(-a*b)^(1/2)-b*(d*x+c)+b*c)/b)-1/32*d^2/b^3*exp((-d*(-a*b
)^(1/2)+b*c)/b)*Ei(1,-(d*(-a*b)^(1/2)+b*(d*x+c)-b*c)/b)-3/32*d/b^2/(-a*b)^(1/2)*exp((d*(-a*b)^(1/2)+b*c)/b)*Ei
(1,(d*(-a*b)^(1/2)-b*(d*x+c)+b*c)/b)+3/32*d/b^2/(-a*b)^(1/2)*exp((-d*(-a*b)^(1/2)+b*c)/b)*Ei(1,-(d*(-a*b)^(1/2
)+b*(d*x+c)-b*c)/b)-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)*x^3-1/16*d^5*exp(d*x+c)*a/b^2/(b
^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)*x-1/8*d^4*exp(d*x+c)*a/b^2/(b^2*d^4*x^4+2*a*b*d^4*x^2+a^2*d^4)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1620 vs. \(2 (374) = 748\).
time = 0.44, size = 1620, normalized size = 3.40 \begin {gather*} -\frac {8 \, {\left (2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right ) - {\left ({\left ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} - 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\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 ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} - 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) + 4 \, {\left (a b^{2} d x^{3} + a^{2} b d x\right )} \sinh \left (d x + c\right ) - {\left ({\left ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} - 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\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 ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} - 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{32 \, {\left ({\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \sinh \left (d x + c\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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