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

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

[Out]

-(x^2*Cosh[c + d*x])/(4*b*(a + b*x^2)^2) - Cosh[c + d*x]/(4*b^2*(a + b*x^2)) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt
[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*b^3) + (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sq
rt[-a]*d)/Sqrt[b] + d*x])/(16*b^3) - (3*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[
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]]*SinhIntegra
l[(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) - (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)

________________________________________________________________________________________

Rubi [A]  time = 1.06151, antiderivative size = 476, normalized size of antiderivative = 1., 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 = {5291, 5289, 5280, 3303, 3298, 3301, 5290, 5293} \[ \frac{d^2 \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 b^3}+\frac{d^2 \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 b^3}-\frac{3 d \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Chi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}+\frac{3 d \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Chi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \sinh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 b^3}+\frac{d^2 \sinh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 b^3}-\frac{3 d \cosh \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Shi}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{3 d \cosh \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Shi}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac{x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x^2*Cosh[c + d*x])/(4*b*(a + b*x^2)^2) - Cosh[c + d*x]/(4*b^2*(a + b*x^2)) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt
[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*b^3) + (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sq
rt[-a]*d)/Sqrt[b] + d*x])/(16*b^3) - (3*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[
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]]*SinhIntegra
l[(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) - (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 5291

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 5289

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 5280

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 3303

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 3298

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 3301

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 5290

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 5293

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*}

Mathematica [C]  time = 1.82448, size = 648, normalized size = 1.36 \[ \frac{-\frac{i d^2 \sinh (c) \left (\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 ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\right )+\cos \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \left (\text{Si}\left (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )-\text{Si}\left (\frac{\sqrt{a} d}{\sqrt{b}}-i d x\right )\right )\right )}{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 (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )\right )\right )}{b}+\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 (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )\right )\right )}{\sqrt{a} \sqrt{b}}+\frac{3 d \cosh (c) \left (\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 ) \text{CosIntegral}\left (\frac{\sqrt{a} d}{\sqrt{b}}+i d x\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 (i x d+\frac{\sqrt{a} d}{\sqrt{b}}\right )\right )\right )}{\sqrt{a} \sqrt{b}}-\frac{2 \cosh (d x) \left (d x \sinh (c) \left (a+b x^2\right )+2 \cosh (c) \left (a+2 b x^2\right )\right )}{\left (a+b x^2\right )^2}-\frac{2 \sinh (d x) \left (d x \cosh (c) \left (a+b x^2\right )+2 \sinh (c) \left (a+2 b x^2\right )\right )}{\left (a+b x^2\right )^2}}{16 b^2} \]

Warning: Unable to verify antiderivative.

[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]  time = 0.376, size = 820, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 2.4081, size = 3344, normalized size = 7.03 \begin{align*} \text{result too large to display} \end{align*}

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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \end{align*}

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)