3.94 \(\int x^3 (d+e x^2)^2 (a+b \text{csch}^{-1}(c x)) \, dx\)

Optimal. Leaf size=250 \[ \frac{1}{4} d^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b x \left (-c^2 x^2-1\right )^{3/2} \left (6 c^4 d^2-16 c^2 d e+9 e^2\right )}{72 c^7 \sqrt{-c^2 x^2}}-\frac{b x \sqrt{-c^2 x^2-1} \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{24 c^7 \sqrt{-c^2 x^2}}+\frac{b e x \left (-c^2 x^2-1\right )^{5/2} \left (8 c^2 d-9 e\right )}{120 c^7 \sqrt{-c^2 x^2}}-\frac{b e^2 x \left (-c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt{-c^2 x^2}} \]

[Out]

-(b*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*x*Sqrt[-1 - c^2*x^2])/(24*c^7*Sqrt[-(c^2*x^2)]) - (b*(6*c^4*d^2 - 16*c^2*d
*e + 9*e^2)*x*(-1 - c^2*x^2)^(3/2))/(72*c^7*Sqrt[-(c^2*x^2)]) + (b*(8*c^2*d - 9*e)*e*x*(-1 - c^2*x^2)^(5/2))/(
120*c^7*Sqrt[-(c^2*x^2)]) - (b*e^2*x*(-1 - c^2*x^2)^(7/2))/(56*c^7*Sqrt[-(c^2*x^2)]) + (d^2*x^4*(a + b*ArcCsch
[c*x]))/4 + (d*e*x^6*(a + b*ArcCsch[c*x]))/3 + (e^2*x^8*(a + b*ArcCsch[c*x]))/8

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Rubi [A]  time = 0.234657, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {266, 43, 6302, 12, 1251, 771} \[ \frac{1}{4} d^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b x \left (-c^2 x^2-1\right )^{3/2} \left (6 c^4 d^2-16 c^2 d e+9 e^2\right )}{72 c^7 \sqrt{-c^2 x^2}}-\frac{b x \sqrt{-c^2 x^2-1} \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{24 c^7 \sqrt{-c^2 x^2}}+\frac{b e x \left (-c^2 x^2-1\right )^{5/2} \left (8 c^2 d-9 e\right )}{120 c^7 \sqrt{-c^2 x^2}}-\frac{b e^2 x \left (-c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt{-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*x*Sqrt[-1 - c^2*x^2])/(24*c^7*Sqrt[-(c^2*x^2)]) - (b*(6*c^4*d^2 - 16*c^2*d
*e + 9*e^2)*x*(-1 - c^2*x^2)^(3/2))/(72*c^7*Sqrt[-(c^2*x^2)]) + (b*(8*c^2*d - 9*e)*e*x*(-1 - c^2*x^2)^(5/2))/(
120*c^7*Sqrt[-(c^2*x^2)]) - (b*e^2*x*(-1 - c^2*x^2)^(7/2))/(56*c^7*Sqrt[-(c^2*x^2)]) + (d^2*x^4*(a + b*ArcCsch
[c*x]))/4 + (d*e*x^6*(a + b*ArcCsch[c*x]))/3 + (e^2*x^8*(a + b*ArcCsch[c*x]))/8

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6302

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u
= IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCsch[c*x], u, x] - Dist[(b*c*x)/Sqrt[-(c^2*x^2)], Int[Simp
lifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&
!(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]))
|| (ILtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{1}{4} d^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{24 \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=\frac{1}{4} d^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{\sqrt{-1-c^2 x^2}} \, dx}{24 \sqrt{-c^2 x^2}}\\ &=\frac{1}{4} d^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \operatorname{Subst}\left (\int \frac{x \left (6 d^2+8 d e x+3 e^2 x^2\right )}{\sqrt{-1-c^2 x}} \, dx,x,x^2\right )}{48 \sqrt{-c^2 x^2}}\\ &=\frac{1}{4} d^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \operatorname{Subst}\left (\int \left (\frac{-6 c^4 d^2+8 c^2 d e-3 e^2}{c^6 \sqrt{-1-c^2 x}}+\frac{\left (-6 c^4 d^2+16 c^2 d e-9 e^2\right ) \sqrt{-1-c^2 x}}{c^6}+\frac{\left (8 c^2 d-9 e\right ) e \left (-1-c^2 x\right )^{3/2}}{c^6}-\frac{3 e^2 \left (-1-c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{48 \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x \sqrt{-1-c^2 x^2}}{24 c^7 \sqrt{-c^2 x^2}}-\frac{b \left (6 c^4 d^2-16 c^2 d e+9 e^2\right ) x \left (-1-c^2 x^2\right )^{3/2}}{72 c^7 \sqrt{-c^2 x^2}}+\frac{b \left (8 c^2 d-9 e\right ) e x \left (-1-c^2 x^2\right )^{5/2}}{120 c^7 \sqrt{-c^2 x^2}}-\frac{b e^2 x \left (-1-c^2 x^2\right )^{7/2}}{56 c^7 \sqrt{-c^2 x^2}}+\frac{1}{4} d^2 x^4 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \text{csch}^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.402856, size = 159, normalized size = 0.64 \[ \frac{x \left (105 a x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+\frac{b \sqrt{\frac{1}{c^2 x^2}+1} \left (3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )-2 c^4 \left (210 d^2+112 d e x^2+27 e^2 x^4\right )+8 c^2 e \left (56 d+9 e x^2\right )-144 e^2\right )}{c^7}+105 b x^3 \text{csch}^{-1}(c x) \left (6 d^2+8 d e x^2+3 e^2 x^4\right )\right )}{2520} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(105*a*x^3*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + (b*Sqrt[1 + 1/(c^2*x^2)]*(-144*e^2 + 8*c^2*e*(56*d + 9*e*x^2)
- 2*c^4*(210*d^2 + 112*d*e*x^2 + 27*e^2*x^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)))/c^7 + 105*b*x^3*
(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcCsch[c*x]))/2520

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Maple [A]  time = 0.19, size = 214, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{4}} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{8}{x}^{8}}{8}}+{\frac{{c}^{8}de{x}^{6}}{3}}+{\frac{{x}^{4}{c}^{8}{d}^{2}}{4}} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{{\rm arccsch} \left (cx\right ){e}^{2}{c}^{8}{x}^{8}}{8}}+{\frac{{\rm arccsch} \left (cx\right ){c}^{8}de{x}^{6}}{3}}+{\frac{{\rm arccsch} \left (cx\right ){c}^{8}{x}^{4}{d}^{2}}{4}}+{\frac{ \left ({c}^{2}{x}^{2}+1 \right ) \left ( 45\,{c}^{6}{e}^{2}{x}^{6}+168\,{c}^{6}de{x}^{4}+210\,{c}^{6}{d}^{2}{x}^{2}-54\,{c}^{4}{e}^{2}{x}^{4}-224\,{c}^{4}de{x}^{2}-420\,{d}^{2}{c}^{4}+72\,{c}^{2}{e}^{2}{x}^{2}+448\,{c}^{2}de-144\,{e}^{2} \right ) }{2520\,cx}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x^2+d)^2*(a+b*arccsch(c*x)),x)

[Out]

1/c^4*(a/c^4*(1/8*e^2*c^8*x^8+1/3*c^8*d*e*x^6+1/4*x^4*c^8*d^2)+b/c^4*(1/8*arccsch(c*x)*e^2*c^8*x^8+1/3*arccsch
(c*x)*c^8*d*e*x^6+1/4*arccsch(c*x)*c^8*x^4*d^2+1/2520*(c^2*x^2+1)*(45*c^6*e^2*x^6+168*c^6*d*e*x^4+210*c^6*d^2*
x^2-54*c^4*e^2*x^4-224*c^4*d*e*x^2-420*c^4*d^2+72*c^2*e^2*x^2+448*c^2*d*e-144*e^2)/((c^2*x^2+1)/c^2/x^2)^(1/2)
/c/x))

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Maxima [A]  time = 1.05541, size = 329, normalized size = 1.32 \begin{align*} \frac{1}{8} \, a e^{2} x^{8} + \frac{1}{3} \, a d e x^{6} + \frac{1}{4} \, a d^{2} x^{4} + \frac{1}{12} \,{\left (3 \, x^{4} \operatorname{arcsch}\left (c x\right ) + \frac{c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 3 \, x \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d^{2} + \frac{1}{45} \,{\left (15 \, x^{6} \operatorname{arcsch}\left (c x\right ) + \frac{3 \, c^{4} x^{5}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 10 \, c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 15 \, x \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d e + \frac{1}{280} \,{\left (35 \, x^{8} \operatorname{arcsch}\left (c x\right ) + \frac{5 \, c^{6} x^{7}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{7}{2}} - 21 \, c^{4} x^{5}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} + 35 \, c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 35 \, x \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 1/12*(3*x^4*arccsch(c*x) + (c^2*x^3*(1/(c^2*x^2) + 1)^(3/2) -
3*x*sqrt(1/(c^2*x^2) + 1))/c^3)*b*d^2 + 1/45*(15*x^6*arccsch(c*x) + (3*c^4*x^5*(1/(c^2*x^2) + 1)^(5/2) - 10*c^
2*x^3*(1/(c^2*x^2) + 1)^(3/2) + 15*x*sqrt(1/(c^2*x^2) + 1))/c^5)*b*d*e + 1/280*(35*x^8*arccsch(c*x) + (5*c^6*x
^7*(1/(c^2*x^2) + 1)^(7/2) - 21*c^4*x^5*(1/(c^2*x^2) + 1)^(5/2) + 35*c^2*x^3*(1/(c^2*x^2) + 1)^(3/2) - 35*x*sq
rt(1/(c^2*x^2) + 1))/c^7)*b*e^2

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Fricas [A]  time = 2.78739, size = 506, normalized size = 2.02 \begin{align*} \frac{315 \, a c^{7} e^{2} x^{8} + 840 \, a c^{7} d e x^{6} + 630 \, a c^{7} d^{2} x^{4} + 105 \,{\left (3 \, b c^{7} e^{2} x^{8} + 8 \, b c^{7} d e x^{6} + 6 \, b c^{7} d^{2} x^{4}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (45 \, b c^{6} e^{2} x^{7} + 6 \,{\left (28 \, b c^{6} d e - 9 \, b c^{4} e^{2}\right )} x^{5} + 2 \,{\left (105 \, b c^{6} d^{2} - 112 \, b c^{4} d e + 36 \, b c^{2} e^{2}\right )} x^{3} - 4 \,{\left (105 \, b c^{4} d^{2} - 112 \, b c^{2} d e + 36 \, b e^{2}\right )} x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2520 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(315*a*c^7*e^2*x^8 + 840*a*c^7*d*e*x^6 + 630*a*c^7*d^2*x^4 + 105*(3*b*c^7*e^2*x^8 + 8*b*c^7*d*e*x^6 + 6
*b*c^7*d^2*x^4)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + (45*b*c^6*e^2*x^7 + 6*(28*b*c^6*d*e - 9*b
*c^4*e^2)*x^5 + 2*(105*b*c^6*d^2 - 112*b*c^4*d*e + 36*b*c^2*e^2)*x^3 - 4*(105*b*c^4*d^2 - 112*b*c^2*d*e + 36*b
*e^2)*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/c^7

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \left (a + b \operatorname{acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x**2+d)**2*(a+b*acsch(c*x)),x)

[Out]

Integral(x**3*(a + b*acsch(c*x))*(d + e*x**2)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}^{2}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x^2 + d)^2*(b*arccsch(c*x) + a)*x^3, x)