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

Optimal. Leaf size=115 \[ d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b x \left (6 c^2 d-e\right ) \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{6 c^2 \sqrt{-c^2 x^2}}+\frac{b e x^2 \sqrt{-c^2 x^2-1}}{6 c \sqrt{-c^2 x^2}} \]

[Out]

(b*e*x^2*Sqrt[-1 - c^2*x^2])/(6*c*Sqrt[-(c^2*x^2)]) + d*x*(a + b*ArcCsch[c*x]) + (e*x^3*(a + b*ArcCsch[c*x]))/
3 - (b*(6*c^2*d - e)*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(6*c^2*Sqrt[-(c^2*x^2)])

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Rubi [A]  time = 0.0556023, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {6292, 12, 388, 217, 203} \[ d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b x \left (6 c^2 d-e\right ) \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{6 c^2 \sqrt{-c^2 x^2}}+\frac{b e x^2 \sqrt{-c^2 x^2-1}}{6 c \sqrt{-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*x^2*Sqrt[-1 - c^2*x^2])/(6*c*Sqrt[-(c^2*x^2)]) + d*x*(a + b*ArcCsch[c*x]) + (e*x^3*(a + b*ArcCsch[c*x]))/
3 - (b*(6*c^2*d - e)*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(6*c^2*Sqrt[-(c^2*x^2)])

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 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 ) \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{3 d+e x^2}{3 \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{3 d+e x^2}{\sqrt{-1-c^2 x^2}} \, dx}{3 \sqrt{-c^2 x^2}}\\ &=\frac{b e x^2 \sqrt{-1-c^2 x^2}}{6 c \sqrt{-c^2 x^2}}+d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b c \left (6 d-\frac{e}{c^2}\right ) x\right ) \int \frac{1}{\sqrt{-1-c^2 x^2}} \, dx}{6 \sqrt{-c^2 x^2}}\\ &=\frac{b e x^2 \sqrt{-1-c^2 x^2}}{6 c \sqrt{-c^2 x^2}}+d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b c \left (6 d-\frac{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 )}{6 \sqrt{-c^2 x^2}}\\ &=\frac{b e x^2 \sqrt{-1-c^2 x^2}}{6 c \sqrt{-c^2 x^2}}+d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b \left (6 d-\frac{e}{c^2}\right ) x \tan ^{-1}\left (\frac{c x}{\sqrt{-1-c^2 x^2}}\right )}{6 \sqrt{-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.262972, size = 135, normalized size = 1.17 \[ a d x+\frac{1}{3} a e x^3+\frac{b d x \sqrt{\frac{1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{b e x^2 \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}}{6 c}-\frac{b e \log \left (x \left (\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right )\right )}{6 c^3}+b d x \text{csch}^{-1}(c x)+\frac{1}{3} b e x^3 \text{csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*d*x + (a*e*x^3)/3 + (b*e*x^2*Sqrt[(1 + c^2*x^2)/(c^2*x^2)])/(6*c) + b*d*x*ArcCsch[c*x] + (b*e*x^3*ArcCsch[c*
x])/3 + (b*d*Sqrt[1 + 1/(c^2*x^2)]*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (b*e*Log[x*(1 + Sqrt[(1 + c^2*x^2)/(c^2
*x^2)])])/(6*c^3)

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Maple [A]  time = 0.209, size = 126, normalized size = 1.1 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{3}{x}^{3}}{3}}+x{c}^{3}d \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{{\rm arccsch} \left (cx\right )e{c}^{3}{x}^{3}}{3}}+{\rm arccsch} \left (cx\right ){c}^{3}dx+{\frac{1}{6\,cx}\sqrt{{c}^{2}{x}^{2}+1} \left ( 6\,{c}^{2}d{\it Arcsinh} \left ( cx \right ) +ecx\sqrt{{c}^{2}{x}^{2}+1}-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((e*x^2+d)*(a+b*arccsch(c*x)),x)

[Out]

1/c*(a/c^2*(1/3*e*c^3*x^3+x*c^3*d)+b/c^2*(1/3*arccsch(c*x)*e*c^3*x^3+arccsch(c*x)*c^3*d*x+1/6*(c^2*x^2+1)^(1/2
)*(6*c^2*d*arcsinh(c*x)+e*c*x*(c^2*x^2+1)^(1/2)-e*arcsinh(c*x))/((c^2*x^2+1)/c^2/x^2)^(1/2)/c/x))

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Maxima [A]  time = 1.02023, size = 200, normalized size = 1.74 \begin{align*} \frac{1}{3} \, a e x^{3} + \frac{1}{12} \,{\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 e + a d 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 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*e*x^3 + 1/12*(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*e + a*d*x + 1/2*(2*c*x*arccsch(c*x) + log(sq
rt(1/(c^2*x^2) + 1) + 1) - log(sqrt(1/(c^2*x^2) + 1) - 1))*b*d/c

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Fricas [B]  time = 2.95878, size = 544, normalized size = 4.73 \begin{align*} \frac{2 \, a c^{3} e x^{3} + b c^{2} e x^{2} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 6 \, a c^{3} d x + 2 \,{\left (3 \, b c^{3} d + b c^{3} e\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) -{\left (6 \, b c^{2} d - b e\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 2 \,{\left (3 \, b c^{3} d + b c^{3} e\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \,{\left (b c^{3} e x^{3} + 3 \, b c^{3} d x - 3 \, b c^{3} d - b c^{3} e\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{6 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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