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

Optimal. Leaf size=215 \[ \frac{b \text{PolyLog}\left (2,\frac{\left (e-\sqrt{c^2 d^2+e^2}\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}+\frac{b \text{PolyLog}\left (2,\frac{\left (\sqrt{c^2 d^2+e^2}+e\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}-\frac{b \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )}{2 e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{\left (e-\sqrt{c^2 d^2+e^2}\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{\left (\sqrt{c^2 d^2+e^2}+e\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}-\frac{\log \left (1-e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.389729, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6289, 2518} \[ \frac{b \text{PolyLog}\left (2,\frac{\left (e-\sqrt{c^2 d^2+e^2}\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}+\frac{b \text{PolyLog}\left (2,\frac{\left (\sqrt{c^2 d^2+e^2}+e\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}-\frac{b \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )}{2 e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{\left (e-\sqrt{c^2 d^2+e^2}\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{\left (\sqrt{c^2 d^2+e^2}+e\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}-\frac{\log \left (1-e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{e} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c*x])/(d + e*x),x]

[Out]

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

Rule 6289

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

Rule 2518

Int[Log[v_]*(u_), x_Symbol] :> With[{w = DerivativeDivides[v, u*(1 - v), x]}, Simp[w*PolyLog[2, 1 - v], x] /;
!FalseQ[w]]

Rubi steps

\begin{align*} \int \frac{a+b \text{csch}^{-1}(c x)}{d+e x} \, dx &=\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{\left (e-\sqrt{c^2 d^2+e^2}\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{\left (e+\sqrt{c^2 d^2+e^2}\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )}{e}+\frac{b \int \frac{\log \left (1-\frac{\left (e-\sqrt{c^2 d^2+e^2}\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{\sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{c e}+\frac{b \int \frac{\log \left (1-\frac{\left (e+\sqrt{c^2 d^2+e^2}\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{\sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{c e}-\frac{b \int \frac{\log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )}{\sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{c e}\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{\left (e-\sqrt{c^2 d^2+e^2}\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}+\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{\left (e+\sqrt{c^2 d^2+e^2}\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}-\frac{\left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )}{e}+\frac{b \text{Li}_2\left (\frac{\left (e-\sqrt{c^2 d^2+e^2}\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}+\frac{b \text{Li}_2\left (\frac{\left (e+\sqrt{c^2 d^2+e^2}\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )}{e}-\frac{b \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )}{2 e}\\ \end{align*}

Mathematica [C]  time = 0.729286, size = 506, normalized size = 2.35 \[ \frac{a \log (d+e x)}{e}+\frac{b \left (8 \text{PolyLog}\left (2,\frac{\left (e-\sqrt{c^2 d^2+e^2}\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )+8 \text{PolyLog}\left (2,\frac{\left (\sqrt{c^2 d^2+e^2}+e\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )+4 \text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(c x)}\right )+8 \text{csch}^{-1}(c x) \log \left (\frac{\left (\sqrt{c^2 d^2+e^2}-e\right ) e^{\text{csch}^{-1}(c x)}}{c d}+1\right )+8 \text{csch}^{-1}(c x) \log \left (1-\frac{\left (\sqrt{c^2 d^2+e^2}+e\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )+4 i \pi \log \left (\frac{\left (\sqrt{c^2 d^2+e^2}-e\right ) e^{\text{csch}^{-1}(c x)}}{c d}+1\right )+4 i \pi \log \left (1-\frac{\left (\sqrt{c^2 d^2+e^2}+e\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )+16 i \sin ^{-1}\left (\frac{\sqrt{1+\frac{i e}{c d}}}{\sqrt{2}}\right ) \log \left (\frac{\left (\sqrt{c^2 d^2+e^2}-e\right ) e^{\text{csch}^{-1}(c x)}}{c d}+1\right )-16 i \sin ^{-1}\left (\frac{\sqrt{1+\frac{i e}{c d}}}{\sqrt{2}}\right ) \log \left (1-\frac{\left (\sqrt{c^2 d^2+e^2}+e\right ) e^{\text{csch}^{-1}(c x)}}{c d}\right )-32 \sin ^{-1}\left (\frac{\sqrt{1+\frac{i e}{c d}}}{\sqrt{2}}\right ) \tan ^{-1}\left (\frac{(e+i c d) \cot \left (\frac{1}{4} \left (\pi +2 i \text{csch}^{-1}(c x)\right )\right )}{\sqrt{c^2 d^2+e^2}}\right )-8 \text{csch}^{-1}(c x)^2-4 i \pi \text{csch}^{-1}(c x)-8 \text{csch}^{-1}(c x) \log \left (1-e^{-2 \text{csch}^{-1}(c x)}\right )-4 i \pi \log \left (\frac{d}{x}+e\right )+\pi ^2\right )}{8 e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*Log[d + e*x])/e + (b*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 - 32*ArcSin[Sqrt[1 + (I*e)/(c*d)]/Sqr
t[2]]*ArcTan[((I*c*d + e)*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[c^2*d^2 + e^2]] - 8*ArcCsch[c*x]*Log[1 - E^(-
2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 + ((-e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)] + 8*ArcCsch[c*x]*Log[1
+ ((-e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)] + (16*I)*ArcSin[Sqrt[1 + (I*e)/(c*d)]/Sqrt[2]]*Log[1 + ((
-e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)] + (4*I)*Pi*Log[1 - ((e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])
/(c*d)] + 8*ArcCsch[c*x]*Log[1 - ((e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)] - (16*I)*ArcSin[Sqrt[1 + (I
*e)/(c*d)]/Sqrt[2]]*Log[1 - ((e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)] - (4*I)*Pi*Log[e + d/x] + 4*Poly
Log[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, ((e - Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)] + 8*PolyLog[2, ((
e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)]))/(8*e)

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Maple [F]  time = 0.566, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsch} \left (cx\right )}{ex+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arccsch(c*x) + a)/(e*x + d), 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 )}}{d + e x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acsch(c*x))/(d + e*x), 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}{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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