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

Optimal. Leaf size=197 \[ d^2 x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b x \left (120 c^4 d^2-40 c^2 d e+9 e^2\right ) \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{120 c^4 \sqrt{-c^2 x^2}}+\frac{b e x^2 \sqrt{-c^2 x^2-1} \left (40 c^2 d-9 e\right )}{120 c^3 \sqrt{-c^2 x^2}}+\frac{b e^2 x^4 \sqrt{-c^2 x^2-1}}{20 c \sqrt{-c^2 x^2}} \]

[Out]

(b*(40*c^2*d - 9*e)*e*x^2*Sqrt[-1 - c^2*x^2])/(120*c^3*Sqrt[-(c^2*x^2)]) + (b*e^2*x^4*Sqrt[-1 - c^2*x^2])/(20*
c*Sqrt[-(c^2*x^2)]) + d^2*x*(a + b*ArcCsch[c*x]) + (2*d*e*x^3*(a + b*ArcCsch[c*x]))/3 + (e^2*x^5*(a + b*ArcCsc
h[c*x]))/5 - (b*(120*c^4*d^2 - 40*c^2*d*e + 9*e^2)*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(120*c^4*Sqrt[-(c^2*x^2
)])

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Rubi [A]  time = 0.128402, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {194, 6292, 12, 1159, 388, 217, 203} \[ d^2 x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b x \left (120 c^4 d^2-40 c^2 d e+9 e^2\right ) \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{120 c^4 \sqrt{-c^2 x^2}}+\frac{b e x^2 \sqrt{-c^2 x^2-1} \left (40 c^2 d-9 e\right )}{120 c^3 \sqrt{-c^2 x^2}}+\frac{b e^2 x^4 \sqrt{-c^2 x^2-1}}{20 c \sqrt{-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(40*c^2*d - 9*e)*e*x^2*Sqrt[-1 - c^2*x^2])/(120*c^3*Sqrt[-(c^2*x^2)]) + (b*e^2*x^4*Sqrt[-1 - c^2*x^2])/(20*
c*Sqrt[-(c^2*x^2)]) + d^2*x*(a + b*ArcCsch[c*x]) + (2*d*e*x^3*(a + b*ArcCsch[c*x]))/3 + (e^2*x^5*(a + b*ArcCsc
h[c*x]))/5 - (b*(120*c^4*d^2 - 40*c^2*d*e + 9*e^2)*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(120*c^4*Sqrt[-(c^2*x^2
)])

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 6292

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^
2)^p, x]}, Dist[a + b*ArcCsch[c*x], u, x] - Dist[(b*c*x)/Sqrt[-(c^2*x^2)], Int[SimplifyIntegrand[u/(x*Sqrt[-1
- c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 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 1159

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(c^p*x^(4*p - 1)*
(d + e*x^2)^(q + 1))/(e*(4*p + 2*q + 1)), x] + Dist[1/(e*(4*p + 2*q + 1)), Int[(d + e*x^2)^q*ExpandToSum[e*(4*
p + 2*q + 1)*(a + b*x^2 + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 2*q + 1)*x^(4*p), x], x], x] /
; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] &&  !LtQ[
q, -1]

Rule 388

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

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 \left (d+e x^2\right )^2 \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{15 d^2+10 d e x^2+3 e^2 x^4}{15 \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=d^2 x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{15 d^2+10 d e x^2+3 e^2 x^4}{\sqrt{-1-c^2 x^2}} \, dx}{15 \sqrt{-c^2 x^2}}\\ &=\frac{b e^2 x^4 \sqrt{-1-c^2 x^2}}{20 c \sqrt{-c^2 x^2}}+d^2 x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{(b x) \int \frac{-60 c^2 d^2-\left (40 c^2 d-9 e\right ) e x^2}{\sqrt{-1-c^2 x^2}} \, dx}{60 c \sqrt{-c^2 x^2}}\\ &=\frac{b \left (40 c^2 d-9 e\right ) e x^2 \sqrt{-1-c^2 x^2}}{120 c^3 \sqrt{-c^2 x^2}}+\frac{b e^2 x^4 \sqrt{-1-c^2 x^2}}{20 c \sqrt{-c^2 x^2}}+d^2 x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{\left (b \left (-120 c^4 d^2+\left (40 c^2 d-9 e\right ) e\right ) x\right ) \int \frac{1}{\sqrt{-1-c^2 x^2}} \, dx}{120 c^3 \sqrt{-c^2 x^2}}\\ &=\frac{b \left (40 c^2 d-9 e\right ) e x^2 \sqrt{-1-c^2 x^2}}{120 c^3 \sqrt{-c^2 x^2}}+\frac{b e^2 x^4 \sqrt{-1-c^2 x^2}}{20 c \sqrt{-c^2 x^2}}+d^2 x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{\left (b \left (-120 c^4 d^2+\left (40 c^2 d-9 e\right ) e\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 )}{120 c^3 \sqrt{-c^2 x^2}}\\ &=\frac{b \left (40 c^2 d-9 e\right ) e x^2 \sqrt{-1-c^2 x^2}}{120 c^3 \sqrt{-c^2 x^2}}+\frac{b e^2 x^4 \sqrt{-1-c^2 x^2}}{20 c \sqrt{-c^2 x^2}}+d^2 x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b \left (120 c^4 d^2-40 c^2 d e+9 e^2\right ) x \tan ^{-1}\left (\frac{c x}{\sqrt{-1-c^2 x^2}}\right )}{120 c^4 \sqrt{-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.22926, size = 149, normalized size = 0.76 \[ \frac{c^2 x \left (8 a c^3 \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b e x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 \left (40 d+6 e x^2\right )-9 e\right )\right )+b \left (120 c^4 d^2-40 c^2 d e+9 e^2\right ) \log \left (x \left (\sqrt{\frac{1}{c^2 x^2}+1}+1\right )\right )+8 b c^5 x \text{csch}^{-1}(c x) \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{120 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x*(8*a*c^3*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + b*e*Sqrt[1 + 1/(c^2*x^2)]*x*(-9*e + c^2*(40*d + 6*e*x^2)))
+ 8*b*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcCsch[c*x] + b*(120*c^4*d^2 - 40*c^2*d*e + 9*e^2)*Log[(1 + Sq
rt[1 + 1/(c^2*x^2)])*x])/(120*c^5)

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Maple [A]  time = 0.184, size = 217, normalized size = 1.1 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{5}{x}^{5}}{5}}+{\frac{2\,{c}^{5}de{x}^{3}}{3}}+x{c}^{5}{d}^{2} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{{\rm arccsch} \left (cx\right ){e}^{2}{c}^{5}{x}^{5}}{5}}+{\frac{2\,{\rm arccsch} \left (cx\right ){c}^{5}{x}^{3}de}{3}}+{\rm arccsch} \left (cx\right ){c}^{5}x{d}^{2}+{\frac{1}{120\,cx}\sqrt{{c}^{2}{x}^{2}+1} \left ( 120\,{d}^{2}{c}^{4}{\it Arcsinh} \left ( cx \right ) +6\,{e}^{2}{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}+40\,{c}^{3}dex\sqrt{{c}^{2}{x}^{2}+1}-40\,{c}^{2}de{\it Arcsinh} \left ( cx \right ) -9\,{e}^{2}cx\sqrt{{c}^{2}{x}^{2}+1}+9\,{e}^{2}{\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((e*x^2+d)^2*(a+b*arccsch(c*x)),x)

[Out]

1/c*(a/c^4*(1/5*e^2*c^5*x^5+2/3*c^5*d*e*x^3+x*c^5*d^2)+b/c^4*(1/5*arccsch(c*x)*e^2*c^5*x^5+2/3*arccsch(c*x)*c^
5*x^3*d*e+arccsch(c*x)*c^5*x*d^2+1/120*(c^2*x^2+1)^(1/2)*(120*d^2*c^4*arcsinh(c*x)+6*e^2*c^3*x^3*(c^2*x^2+1)^(
1/2)+40*c^3*d*e*x*(c^2*x^2+1)^(1/2)-40*c^2*d*e*arcsinh(c*x)-9*e^2*c*x*(c^2*x^2+1)^(1/2)+9*e^2*arcsinh(c*x))/((
c^2*x^2+1)/c^2/x^2)^(1/2)/c/x))

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Maxima [A]  time = 1.0372, size = 387, normalized size = 1.96 \begin{align*} \frac{1}{5} \, a e^{2} x^{5} + \frac{2}{3} \, a d e x^{3} + \frac{1}{6} \,{\left (4 \, x^{3} \operatorname{arcsch}\left (c x\right ) + \frac{\frac{2 \, \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c^{2}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac{\log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac{\log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d e + \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 e^{2} + a d^{2} x + \frac{{\left (2 \, c x \operatorname{arcsch}\left (c x\right ) + \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d^{2}}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*e^2*x^5 + 2/3*a*d*e*x^3 + 1/6*(4*x^3*arccsch(c*x) + (2*sqrt(1/(c^2*x^2) + 1)/(c^2*(1/(c^2*x^2) + 1) - c^
2) - log(sqrt(1/(c^2*x^2) + 1) + 1)/c^2 + log(sqrt(1/(c^2*x^2) + 1) - 1)/c^2)/c)*b*d*e + 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(sqrt(1/(c^2*x^2) + 1) - 1)/c^4)/c)*b*e^2 + a*d^2*x
+ 1/2*(2*c*x*arccsch(c*x) + log(sqrt(1/(c^2*x^2) + 1) + 1) - log(sqrt(1/(c^2*x^2) + 1) - 1))*b*d^2/c

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Fricas [B]  time = 3.81338, size = 792, normalized size = 4.02 \begin{align*} \frac{24 \, a c^{5} e^{2} x^{5} + 80 \, a c^{5} d e x^{3} + 120 \, a c^{5} d^{2} x + 8 \,{\left (15 \, b c^{5} d^{2} + 10 \, b c^{5} d e + 3 \, b c^{5} e^{2}\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) -{\left (120 \, b c^{4} d^{2} - 40 \, b c^{2} d e + 9 \, b e^{2}\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 8 \,{\left (15 \, b c^{5} d^{2} + 10 \, b c^{5} d e + 3 \, b c^{5} e^{2}\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 8 \,{\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x - 15 \, b c^{5} d^{2} - 10 \, b c^{5} d e - 3 \, b c^{5} e^{2}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (6 \, b c^{4} e^{2} x^{4} +{\left (40 \, b c^{4} d e - 9 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{120 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(24*a*c^5*e^2*x^5 + 80*a*c^5*d*e*x^3 + 120*a*c^5*d^2*x + 8*(15*b*c^5*d^2 + 10*b*c^5*d*e + 3*b*c^5*e^2)*l
og(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) - (120*b*c^4*d^2 - 40*b*c^2*d*e + 9*b*e^2)*log(c*x*sqrt((c^2*x
^2 + 1)/(c^2*x^2)) - c*x) - 8*(15*b*c^5*d^2 + 10*b*c^5*d*e + 3*b*c^5*e^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)
) - c*x - 1) + 8*(3*b*c^5*e^2*x^5 + 10*b*c^5*d*e*x^3 + 15*b*c^5*d^2*x - 15*b*c^5*d^2 - 10*b*c^5*d*e - 3*b*c^5*
e^2)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + (6*b*c^4*e^2*x^4 + (40*b*c^4*d*e - 9*b*c^2*e^2)*x^2)
*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/c^5

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

[Out]

Integral((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 )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)