3.76 \(\int x^4 (d+e x^2) (a+b \text{csch}^{-1}(c x)) \, dx\)

Optimal. Leaf size=214 \[ \frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^4 \sqrt{-c^2 x^2-1} \left (42 c^2 d-25 e\right )}{840 c^3 \sqrt{-c^2 x^2}}-\frac{b x^2 \sqrt{-c^2 x^2-1} \left (42 c^2 d-25 e\right )}{560 c^5 \sqrt{-c^2 x^2}}-\frac{b x \left (42 c^2 d-25 e\right ) \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{560 c^6 \sqrt{-c^2 x^2}}+\frac{b e x^6 \sqrt{-c^2 x^2-1}}{42 c \sqrt{-c^2 x^2}} \]

[Out]

-(b*(42*c^2*d - 25*e)*x^2*Sqrt[-1 - c^2*x^2])/(560*c^5*Sqrt[-(c^2*x^2)]) + (b*(42*c^2*d - 25*e)*x^4*Sqrt[-1 -
c^2*x^2])/(840*c^3*Sqrt[-(c^2*x^2)]) + (b*e*x^6*Sqrt[-1 - c^2*x^2])/(42*c*Sqrt[-(c^2*x^2)]) + (d*x^5*(a + b*Ar
cCsch[c*x]))/5 + (e*x^7*(a + b*ArcCsch[c*x]))/7 - (b*(42*c^2*d - 25*e)*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(56
0*c^6*Sqrt[-(c^2*x^2)])

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Rubi [A]  time = 0.132, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {14, 6302, 12, 459, 321, 217, 203} \[ \frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^4 \sqrt{-c^2 x^2-1} \left (42 c^2 d-25 e\right )}{840 c^3 \sqrt{-c^2 x^2}}-\frac{b x^2 \sqrt{-c^2 x^2-1} \left (42 c^2 d-25 e\right )}{560 c^5 \sqrt{-c^2 x^2}}-\frac{b x \left (42 c^2 d-25 e\right ) \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{560 c^6 \sqrt{-c^2 x^2}}+\frac{b e x^6 \sqrt{-c^2 x^2-1}}{42 c \sqrt{-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(42*c^2*d - 25*e)*x^2*Sqrt[-1 - c^2*x^2])/(560*c^5*Sqrt[-(c^2*x^2)]) + (b*(42*c^2*d - 25*e)*x^4*Sqrt[-1 -
c^2*x^2])/(840*c^3*Sqrt[-(c^2*x^2)]) + (b*e*x^6*Sqrt[-1 - c^2*x^2])/(42*c*Sqrt[-(c^2*x^2)]) + (d*x^5*(a + b*Ar
cCsch[c*x]))/5 + (e*x^7*(a + b*ArcCsch[c*x]))/7 - (b*(42*c^2*d - 25*e)*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(56
0*c^6*Sqrt[-(c^2*x^2)])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 321

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int x^4 \left (d+e x^2\right ) \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{x^4 \left (7 d+5 e x^2\right )}{35 \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{x^4 \left (7 d+5 e x^2\right )}{\sqrt{-1-c^2 x^2}} \, dx}{35 \sqrt{-c^2 x^2}}\\ &=\frac{b e x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b c \left (42 d-\frac{25 e}{c^2}\right ) x\right ) \int \frac{x^4}{\sqrt{-1-c^2 x^2}} \, dx}{210 \sqrt{-c^2 x^2}}\\ &=\frac{b \left (42 c^2 d-25 e\right ) x^4 \sqrt{-1-c^2 x^2}}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{\left (b \left (42 d-\frac{25 e}{c^2}\right ) x\right ) \int \frac{x^2}{\sqrt{-1-c^2 x^2}} \, dx}{280 c \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (42 c^2 d-25 e\right ) x^2 \sqrt{-1-c^2 x^2}}{560 c^5 \sqrt{-c^2 x^2}}+\frac{b \left (42 c^2 d-25 e\right ) x^4 \sqrt{-1-c^2 x^2}}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b \left (42 d-\frac{25 e}{c^2}\right ) x\right ) \int \frac{1}{\sqrt{-1-c^2 x^2}} \, dx}{560 c^3 \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (42 c^2 d-25 e\right ) x^2 \sqrt{-1-c^2 x^2}}{560 c^5 \sqrt{-c^2 x^2}}+\frac{b \left (42 c^2 d-25 e\right ) x^4 \sqrt{-1-c^2 x^2}}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b \left (42 d-\frac{25 e}{c^2}\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1-c^2 x^2}}\right )}{560 c^3 \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (42 c^2 d-25 e\right ) x^2 \sqrt{-1-c^2 x^2}}{560 c^5 \sqrt{-c^2 x^2}}+\frac{b \left (42 c^2 d-25 e\right ) x^4 \sqrt{-1-c^2 x^2}}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{5} d x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b \left (42 c^2 d-25 e\right ) x \tan ^{-1}\left (\frac{c x}{\sqrt{-1-c^2 x^2}}\right )}{560 c^6 \sqrt{-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.232764, size = 138, normalized size = 0.64 \[ \frac{48 a c^7 x^5 \left (7 d+5 e x^2\right )+b c^2 x^2 \sqrt{\frac{1}{c^2 x^2}+1} \left (c^4 \left (84 d x^2+40 e x^4\right )-2 c^2 \left (63 d+25 e x^2\right )+75 e\right )+3 b \left (42 c^2 d-25 e\right ) \log \left (x \left (\sqrt{\frac{1}{c^2 x^2}+1}+1\right )\right )+48 b c^7 x^5 \text{csch}^{-1}(c x) \left (7 d+5 e x^2\right )}{1680 c^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(48*a*c^7*x^5*(7*d + 5*e*x^2) + b*c^2*Sqrt[1 + 1/(c^2*x^2)]*x^2*(75*e - 2*c^2*(63*d + 25*e*x^2) + c^4*(84*d*x^
2 + 40*e*x^4)) + 48*b*c^7*x^5*(7*d + 5*e*x^2)*ArcCsch[c*x] + 3*b*(42*c^2*d - 25*e)*Log[(1 + Sqrt[1 + 1/(c^2*x^
2)])*x])/(1680*c^7)

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Maple [A]  time = 0.196, size = 211, normalized size = 1. \begin{align*}{\frac{1}{{c}^{5}} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{7}{x}^{7}}{7}}+{\frac{{c}^{7}{x}^{5}d}{5}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{{\rm arccsch} \left (cx\right )e{c}^{7}{x}^{7}}{7}}+{\frac{{\rm arccsch} \left (cx\right ){c}^{7}{x}^{5}d}{5}}+{\frac{1}{1680\,cx}\sqrt{{c}^{2}{x}^{2}+1} \left ( 40\,e{c}^{5}{x}^{5}\sqrt{{c}^{2}{x}^{2}+1}+84\,{c}^{5}d{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}-50\,e{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}-126\,{c}^{3}dx\sqrt{{c}^{2}{x}^{2}+1}+126\,{c}^{2}d{\it Arcsinh} \left ( cx \right ) +75\,ecx\sqrt{{c}^{2}{x}^{2}+1}-75\,e{\it Arcsinh} \left ( cx \right ) \right ){\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^4*(e*x^2+d)*(a+b*arccsch(c*x)),x)

[Out]

1/c^5*(a/c^2*(1/7*e*c^7*x^7+1/5*c^7*x^5*d)+b/c^2*(1/7*arccsch(c*x)*e*c^7*x^7+1/5*arccsch(c*x)*c^7*x^5*d+1/1680
*(c^2*x^2+1)^(1/2)*(40*e*c^5*x^5*(c^2*x^2+1)^(1/2)+84*c^5*d*x^3*(c^2*x^2+1)^(1/2)-50*e*c^3*x^3*(c^2*x^2+1)^(1/
2)-126*c^3*d*x*(c^2*x^2+1)^(1/2)+126*c^2*d*arcsinh(c*x)+75*e*c*x*(c^2*x^2+1)^(1/2)-75*e*arcsinh(c*x))/((c^2*x^
2+1)/c^2/x^2)^(1/2)/c/x))

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Maxima [A]  time = 1.0222, size = 390, normalized size = 1.82 \begin{align*} \frac{1}{7} \, a e x^{7} + \frac{1}{5} \, a d x^{5} + \frac{1}{80} \,{\left (16 \, x^{5} \operatorname{arcsch}\left (c x\right ) - \frac{\frac{2 \,{\left (3 \,{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{4}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{2} - 2 \, c^{4}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} + c^{4}} - \frac{3 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac{3 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d + \frac{1}{672} \,{\left (96 \, x^{7} \operatorname{arcsch}\left (c x\right ) + \frac{\frac{2 \,{\left (15 \,{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 40 \,{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{6}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{3} - 3 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{2} + 3 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} - c^{6}} - \frac{15 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} + \frac{15 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*e*x^7 + 1/5*a*d*x^5 + 1/80*(16*x^5*arccsch(c*x) - (2*(3*(1/(c^2*x^2) + 1)^(3/2) - 5*sqrt(1/(c^2*x^2) + 1
))/(c^4*(1/(c^2*x^2) + 1)^2 - 2*c^4*(1/(c^2*x^2) + 1) + c^4) - 3*log(sqrt(1/(c^2*x^2) + 1) + 1)/c^4 + 3*log(sq
rt(1/(c^2*x^2) + 1) - 1)/c^4)/c)*b*d + 1/672*(96*x^7*arccsch(c*x) + (2*(15*(1/(c^2*x^2) + 1)^(5/2) - 40*(1/(c^
2*x^2) + 1)^(3/2) + 33*sqrt(1/(c^2*x^2) + 1))/(c^6*(1/(c^2*x^2) + 1)^3 - 3*c^6*(1/(c^2*x^2) + 1)^2 + 3*c^6*(1/
(c^2*x^2) + 1) - c^6) - 15*log(sqrt(1/(c^2*x^2) + 1) + 1)/c^6 + 15*log(sqrt(1/(c^2*x^2) + 1) - 1)/c^6)/c)*b*e

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Fricas [A]  time = 3.32047, size = 680, normalized size = 3.18 \begin{align*} \frac{240 \, a c^{7} e x^{7} + 336 \, a c^{7} d x^{5} + 48 \,{\left (7 \, b c^{7} d + 5 \, b c^{7} e\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 3 \,{\left (42 \, b c^{2} d - 25 \, b e\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 48 \,{\left (7 \, b c^{7} d + 5 \, b c^{7} e\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 48 \,{\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5} - 7 \, b c^{7} d - 5 \, b c^{7} e\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (40 \, b c^{6} e x^{6} + 2 \,{\left (42 \, b c^{6} d - 25 \, b c^{4} e\right )} x^{4} - 3 \,{\left (42 \, b c^{4} d - 25 \, b c^{2} e\right )} x^{2}\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{1680 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(240*a*c^7*e*x^7 + 336*a*c^7*d*x^5 + 48*(7*b*c^7*d + 5*b*c^7*e)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) -
c*x + 1) - 3*(42*b*c^2*d - 25*b*e)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x) - 48*(7*b*c^7*d + 5*b*c^7*e)*
log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + 48*(5*b*c^7*e*x^7 + 7*b*c^7*d*x^5 - 7*b*c^7*d - 5*b*c^7*e)*
log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + (40*b*c^6*e*x^6 + 2*(42*b*c^6*d - 25*b*c^4*e)*x^4 - 3*(42
*b*c^4*d - 25*b*c^2*e)*x^2)*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^{4} \left (a + b \operatorname{acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x**2+d)*(a+b*acsch(c*x)),x)

[Out]

Integral(x**4*(a + b*acsch(c*x))*(d + e*x**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x^2 + d)*(b*arccsch(c*x) + a)*x^4, x)