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

Optimal. Leaf size=139 \[ -\frac{a+b \text{csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac{b c x \tan ^{-1}\left (\sqrt{-c^2 x^2-1}\right )}{2 d e \sqrt{-c^2 x^2}}+\frac{b c x \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{\sqrt{c^2 d-e}}\right )}{2 d \sqrt{e} \sqrt{-c^2 x^2} \sqrt{c^2 d-e}} \]

[Out]

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

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Rubi [A]  time = 0.150264, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {6300, 446, 86, 63, 205, 208} \[ -\frac{a+b \text{csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac{b c x \tan ^{-1}\left (\sqrt{-c^2 x^2-1}\right )}{2 d e \sqrt{-c^2 x^2}}+\frac{b c x \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{\sqrt{c^2 d-e}}\right )}{2 d \sqrt{e} \sqrt{-c^2 x^2} \sqrt{c^2 d-e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6300

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^(p +
1)*(a + b*ArcCsch[c*x]))/(2*e*(p + 1)), x] - Dist[(b*c*x)/(2*e*(p + 1)*Sqrt[-(c^2*x^2)]), Int[(d + e*x^2)^(p
+ 1)/(x*Sqrt[-1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]

Rule 446

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

Rule 86

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), In
t[(e + f*x)^p/(a + b*x), x], x] - Dist[d/(b*c - a*d), Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] &&  !IntegerQ[p]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

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

Rule 208

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

Rubi steps

\begin{align*} \int \frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=-\frac{a+b \text{csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac{(b c x) \int \frac{1}{x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )} \, dx}{2 e \sqrt{-c^2 x^2}}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac{(b c x) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1-c^2 x} (d+e x)} \, dx,x,x^2\right )}{4 e \sqrt{-c^2 x^2}}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-c^2 x} (d+e x)} \, dx,x,x^2\right )}{4 d \sqrt{-c^2 x^2}}+\frac{(b c x) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1-c^2 x}} \, dx,x,x^2\right )}{4 d e \sqrt{-c^2 x^2}}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac{(b x) \operatorname{Subst}\left (\int \frac{1}{d-\frac{e}{c^2}-\frac{e x^2}{c^2}} \, dx,x,\sqrt{-1-c^2 x^2}\right )}{2 c d \sqrt{-c^2 x^2}}-\frac{(b x) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{-1-c^2 x^2}\right )}{2 c d e \sqrt{-c^2 x^2}}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac{b c x \tan ^{-1}\left (\sqrt{-1-c^2 x^2}\right )}{2 d e \sqrt{-c^2 x^2}}+\frac{b c x \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{-1-c^2 x^2}}{\sqrt{c^2 d-e}}\right )}{2 d \sqrt{c^2 d-e} \sqrt{e} \sqrt{-c^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.719456, size = 271, normalized size = 1.95 \[ -\frac{\frac{2 a}{d+e x^2}+\frac{b \sqrt{e} \log \left (-\frac{4 \left (c d \sqrt{e} x \left (c \sqrt{d}+i \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{e-c^2 d}\right )+i d e\right )}{b \sqrt{e-c^2 d} \left (\sqrt{d}-i \sqrt{e} x\right )}\right )}{d \sqrt{e-c^2 d}}+\frac{b \sqrt{e} \log \left (\frac{4 i \left (d e+c d \sqrt{e} x \left (\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{e-c^2 d}+i c \sqrt{d}\right )\right )}{b \sqrt{e-c^2 d} \left (\sqrt{d}+i \sqrt{e} x\right )}\right )}{d \sqrt{e-c^2 d}}+\frac{2 b \text{csch}^{-1}(c x)}{d+e x^2}-\frac{2 b \sinh ^{-1}\left (\frac{1}{c x}\right )}{d}}{4 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2*a)/(d + e*x^2) + (2*b*ArcCsch[c*x])/(d + e*x^2) - (2*b*ArcSinh[1/(c*x)])/d + (b*Sqrt[e]*Log[(-4*(I*d*e +
c*d*Sqrt[e]*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(b*Sqrt[-(c^2*d) + e]*(Sqrt[d] - I*Sq
rt[e]*x))])/(d*Sqrt[-(c^2*d) + e]) + (b*Sqrt[e]*Log[((4*I)*(d*e + c*d*Sqrt[e]*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e
]*Sqrt[1 + 1/(c^2*x^2)])*x))/(b*Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))])/(d*Sqrt[-(c^2*d) + e]))/(4*e)

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Maple [B]  time = 0.273, size = 358, normalized size = 2.6 \begin{align*} -{\frac{a{c}^{2}}{2\,e \left ({c}^{2}{x}^{2}e+{c}^{2}d \right ) }}-{\frac{b{c}^{2}{\rm arccsch} \left (cx\right )}{2\,e \left ({c}^{2}{x}^{2}e+{c}^{2}d \right ) }}+{\frac{b}{2\,cxed}\sqrt{{c}^{2}{x}^{2}+1}{\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}}-{\frac{b}{4\,cxed}\sqrt{{c}^{2}{x}^{2}+1}\ln \left ( 2\,{\frac{1}{cxe+\sqrt{-{c}^{2}de}} \left ( \sqrt{-{\frac{{c}^{2}d-e}{e}}}\sqrt{{c}^{2}{x}^{2}+1}e-\sqrt{-{c}^{2}de}cx+e \right ) } \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}{\frac{1}{\sqrt{-{\frac{{c}^{2}d-e}{e}}}}}}-{\frac{b}{4\,cxed}\sqrt{{c}^{2}{x}^{2}+1}\ln \left ( -2\,{\frac{1}{-cxe+\sqrt{-{c}^{2}de}} \left ( \sqrt{-{\frac{{c}^{2}d-e}{e}}}\sqrt{{c}^{2}{x}^{2}+1}e+\sqrt{-{c}^{2}de}cx+e \right ) } \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}{\frac{1}{\sqrt{-{\frac{{c}^{2}d-e}{e}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c^2*a/e/(c^2*e*x^2+c^2*d)-1/2*c^2*b/e/(c^2*e*x^2+c^2*d)*arccsch(c*x)+1/2/c*b/e*(c^2*x^2+1)^(1/2)/((c^2*x^
2+1)/c^2/x^2)^(1/2)/x/d*arctanh(1/(c^2*x^2+1)^(1/2))-1/4/c*b/e*(c^2*x^2+1)^(1/2)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x
/d/(-(c^2*d-e)/e)^(1/2)*ln(2*((-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x+e)/(c*x*e+(-c^2*d*
e)^(1/2)))-1/4/c*b/e*(c^2*x^2+1)^(1/2)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x/d/(-(c^2*d-e)/e)^(1/2)*ln(-2*((-(c^2*d-e)
/e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(-c*x*e+(-c^2*d*e)^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \,{\left (4 \, c^{2} \int \frac{x}{2 \,{\left (c^{2} e^{2} x^{4} +{\left (c^{2} d e + e^{2}\right )} x^{2} + d e +{\left (c^{2} e^{2} x^{4} +{\left (c^{2} d e + e^{2}\right )} x^{2} + d e\right )} \sqrt{c^{2} x^{2} + 1}\right )}}\,{d x} - \frac{2 \, c^{2} d^{2} \log \left (c\right ) - 2 \,{\left (c^{2} d e - e^{2}\right )} x^{2} \log \left (x\right ) - 2 \, d e \log \left (c\right ) +{\left (c^{2} d e x^{2} + c^{2} d^{2}\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \,{\left (c^{2} d^{2} - d e\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )}{c^{2} d^{3} e - d^{2} e^{2} +{\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}} + \frac{\log \left (e x^{2} + d\right )}{c^{2} d^{2} - d e}\right )} b - \frac{a}{2 \,{\left (e^{2} x^{2} + d e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*c^2*integrate(1/2*x/(c^2*e^2*x^4 + (c^2*d*e + e^2)*x^2 + d*e + (c^2*e^2*x^4 + (c^2*d*e + e^2)*x^2 + d*
e)*sqrt(c^2*x^2 + 1)), x) - (2*c^2*d^2*log(c) - 2*(c^2*d*e - e^2)*x^2*log(x) - 2*d*e*log(c) + (c^2*d*e*x^2 + c
^2*d^2)*log(c^2*x^2 + 1) - 2*(c^2*d^2 - d*e)*log(sqrt(c^2*x^2 + 1) + 1))/(c^2*d^3*e - d^2*e^2 + (c^2*d^2*e^2 -
d*e^3)*x^2) + log(e*x^2 + d)/(c^2*d^2 - d*e))*b - 1/2*a/(e^2*x^2 + d*e)

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Fricas [B]  time = 2.73245, size = 1288, normalized size = 9.27 \begin{align*} \left [-\frac{2 \, a c^{2} d^{2} - 2 \, a d e + \sqrt{-c^{2} d e + e^{2}}{\left (b e x^{2} + b d\right )} \log \left (\frac{c^{2} e x^{2} - c^{2} d - 2 \, \sqrt{-c^{2} d e + e^{2}} c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, e}{e x^{2} + d}\right ) - 2 \,{\left (b c^{2} d^{2} - b d e +{\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + 2 \,{\left (b c^{2} d^{2} - b d e +{\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \,{\left (b c^{2} d^{2} - b d e\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{4 \,{\left (c^{2} d^{3} e - d^{2} e^{2} +{\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}\right )}}, -\frac{a c^{2} d^{2} - a d e + \sqrt{c^{2} d e - e^{2}}{\left (b e x^{2} + b d\right )} \arctan \left (-\frac{\sqrt{c^{2} d e - e^{2}} c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{c^{2} d - e}\right ) -{\left (b c^{2} d^{2} - b d e +{\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) +{\left (b c^{2} d^{2} - b d e +{\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) +{\left (b c^{2} d^{2} - b d e\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \,{\left (c^{2} d^{3} e - d^{2} e^{2} +{\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*a*c^2*d^2 - 2*a*d*e + sqrt(-c^2*d*e + e^2)*(b*e*x^2 + b*d)*log((c^2*e*x^2 - c^2*d - 2*sqrt(-c^2*d*e +
e^2)*c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 2*e)/(e*x^2 + d)) - 2*(b*c^2*d^2 - b*d*e + (b*c^2*d*e - b*e^2)*x^2)*
log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) + 2*(b*c^2*d^2 - b*d*e + (b*c^2*d*e - b*e^2)*x^2)*log(c*x*sqr
t((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + 2*(b*c^2*d^2 - b*d*e)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x
)))/(c^2*d^3*e - d^2*e^2 + (c^2*d^2*e^2 - d*e^3)*x^2), -1/2*(a*c^2*d^2 - a*d*e + sqrt(c^2*d*e - e^2)*(b*e*x^2
+ b*d)*arctan(-sqrt(c^2*d*e - e^2)*c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*d - e)) - (b*c^2*d^2 - b*d*e + (b*c^
2*d*e - b*e^2)*x^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) + (b*c^2*d^2 - b*d*e + (b*c^2*d*e - b*e^2
)*x^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + (b*c^2*d^2 - b*d*e)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2
*x^2)) + 1)/(c*x)))/(c^2*d^3*e - d^2*e^2 + (c^2*d^2*e^2 - d*e^3)*x^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*(a+b*acsch(c*x))/(e*x**2+d)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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