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

Optimal. Leaf size=98 \[ -\frac{a+b \text{csch}^{-1}(c x)}{e (d+e x)}+\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{e}{x}}{c \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d^2+e^2}}\right )}{d \sqrt{c^2 d^2+e^2}}+\frac{b \text{csch}^{-1}(c x)}{d e} \]

[Out]

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

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Rubi [A]  time = 0.15371, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {6290, 1568, 1475, 844, 215, 725, 206} \[ -\frac{a+b \text{csch}^{-1}(c x)}{e (d+e x)}+\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{e}{x}}{c \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d^2+e^2}}\right )}{d \sqrt{c^2 d^2+e^2}}+\frac{b \text{csch}^{-1}(c x)}{d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6290

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

Rule 1568

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Int[x^(m + mn*q
)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; FreeQ[{a, c, d, e, m, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (P
osQ[n2] ||  !IntegerQ[p])

Rule 1475

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x
] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 215

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

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.214515, size = 134, normalized size = 1.37 \[ -\frac{a}{e (d+e x)}-\frac{b \log \left (c x \left (\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d^2+e^2}-c d\right )+e\right )}{d \sqrt{c^2 d^2+e^2}}+\frac{b \log (d+e x)}{d \sqrt{c^2 d^2+e^2}}+\frac{b \sinh ^{-1}\left (\frac{1}{c x}\right )}{d e}-\frac{b \text{csch}^{-1}(c x)}{e (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.305, size = 208, normalized size = 2.1 \begin{align*} -{\frac{ac}{ \left ( cxe+cd \right ) e}}-{\frac{bc{\rm arccsch} \left (cx\right )}{ \left ( cxe+cd \right ) e}}+{\frac{b}{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}{cxed}\sqrt{{c}^{2}{x}^{2}+1}\ln \left ( 2\,{\frac{1}{cxe+cd} \left ( \sqrt{{\frac{{c}^{2}{d}^{2}+{e}^{2}}{{e}^{2}}}}\sqrt{{c}^{2}{x}^{2}+1}e-{c}^{2}dx+e \right ) } \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}{\frac{1}{\sqrt{{\frac{{c}^{2}{d}^{2}+{e}^{2}}{{e}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.68307, size = 741, normalized size = 7.56 \begin{align*} -\frac{a c^{2} d^{3} + a d e^{2} - \sqrt{c^{2} d^{2} + e^{2}}{\left (b e^{2} x + b d e\right )} \log \left (-\frac{c^{3} d^{2} x - c d e +{\left (c^{3} d^{2} + c e^{2}\right )} x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} +{\left (c^{2} d x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + c^{2} d x - e\right )} \sqrt{c^{2} d^{2} + e^{2}}}{e x + d}\right ) -{\left (b c^{2} d^{3} + b d e^{2} +{\left (b c^{2} d^{2} e + b e^{3}\right )} x\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^{3} + b d e^{2} +{\left (b c^{2} d^{2} e + b e^{3}\right )} x\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^{3} + b d e^{2}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c^{2} d^{4} e + d^{2} e^{3} +{\left (c^{2} d^{3} e^{2} + d e^{4}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))/(e*x+d)**2,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 \frac{b \operatorname{arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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