3.119 \(\int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt{(1+a^2) c+2 a b c x+b^2 c x^2}} \, dx\)

Optimal. Leaf size=281 \[ \frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{2 b \sqrt{c (a+b x)^2+c}}-\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{2 b \sqrt{c (a+b x)^2+c}}+\frac{\sqrt{c (a+b x)^2+c}}{2 b c}+\frac{(a+b x) \sqrt{c (a+b x)^2+c} \cot ^{-1}(a+b x)}{2 b c}+\frac{i \sqrt{(a+b x)^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right ) \cot ^{-1}(a+b x)}{b \sqrt{c (a+b x)^2+c}} \]

[Out]

Sqrt[c + c*(a + b*x)^2]/(2*b*c) + ((a + b*x)*Sqrt[c + c*(a + b*x)^2]*ArcCot[a + b*x])/(2*b*c) + (I*Sqrt[1 + (a
+ b*x)^2]*ArcCot[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2]) +
((I/2)*Sqrt[1 + (a + b*x)^2]*PolyLog[2, ((-I)*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a
+ b*x)^2]) - ((I/2)*Sqrt[1 + (a + b*x)^2]*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt
[c + c*(a + b*x)^2])

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Rubi [A]  time = 0.3486, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5058, 4953, 261, 4891, 4887} \[ \frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{2 b \sqrt{c (a+b x)^2+c}}-\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{2 b \sqrt{c (a+b x)^2+c}}+\frac{\sqrt{c (a+b x)^2+c}}{2 b c}+\frac{(a+b x) \sqrt{c (a+b x)^2+c} \cot ^{-1}(a+b x)}{2 b c}+\frac{i \sqrt{(a+b x)^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right ) \cot ^{-1}(a+b x)}{b \sqrt{c (a+b x)^2+c}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^2*ArcCot[a + b*x])/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2],x]

[Out]

Sqrt[c + c*(a + b*x)^2]/(2*b*c) + ((a + b*x)*Sqrt[c + c*(a + b*x)^2]*ArcCot[a + b*x])/(2*b*c) + (I*Sqrt[1 + (a
+ b*x)^2]*ArcCot[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2]) +
((I/2)*Sqrt[1 + (a + b*x)^2]*PolyLog[2, ((-I)*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a
+ b*x)^2]) - ((I/2)*Sqrt[1 + (a + b*x)^2]*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt
[c + c*(a + b*x)^2])

Rule 5058

Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_
)^2)^(q_.), x_Symbol] :> Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(C/d^2 + (C*x^2)/d^2)^q*(a + b*ArcCot
[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0]
&& EqQ[2*c*C - B*d, 0]

Rule 4953

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCot[c*x])^p)/(c^2*d*m), x] + (Dist[(b*f*p)/(c*m), Int[((f*x)^(m - 1
)*(a + b*ArcCot[c*x])^(p - 1))/Sqrt[d + e*x^2], x], x] - Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b
*ArcCot[c*x])^p)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && GtQ
[m, 1]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 4891

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcCot[c*x])^p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*
d] && IGtQ[p, 0] &&  !GtQ[d, 0]

Rule 4887

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-2*I*(a + b*ArcCot[c*x])*
ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x] + (-Simp[(I*b*PolyLog[2, -((I*Sqrt[1 + I*c*x])/Sqrt[1
- I*c*x])])/(c*Sqrt[d]), x] + Simp[(I*b*PolyLog[2, (I*Sqrt[1 + I*c*x])/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x]) /;
FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]

Rubi steps

\begin{align*} \int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \cot ^{-1}(x)}{\sqrt{c+c x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \sqrt{c+c (a+b x)^2} \cot ^{-1}(a+b x)}{2 b c}+\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{c+c x^2}} \, dx,x,a+b x\right )}{2 b}-\frac{\operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)}{\sqrt{c+c x^2}} \, dx,x,a+b x\right )}{2 b}\\ &=\frac{\sqrt{c+c (a+b x)^2}}{2 b c}+\frac{(a+b x) \sqrt{c+c (a+b x)^2} \cot ^{-1}(a+b x)}{2 b c}-\frac{\sqrt{1+(a+b x)^2} \operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)}{\sqrt{1+x^2}} \, dx,x,a+b x\right )}{2 b \sqrt{c+c (a+b x)^2}}\\ &=\frac{\sqrt{c+c (a+b x)^2}}{2 b c}+\frac{(a+b x) \sqrt{c+c (a+b x)^2} \cot ^{-1}(a+b x)}{2 b c}+\frac{i \sqrt{1+(a+b x)^2} \cot ^{-1}(a+b x) \tan ^{-1}\left (\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{b \sqrt{c+c (a+b x)^2}}+\frac{i \sqrt{1+(a+b x)^2} \text{Li}_2\left (-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{2 b \sqrt{c+c (a+b x)^2}}-\frac{i \sqrt{1+(a+b x)^2} \text{Li}_2\left (\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{2 b \sqrt{c+c (a+b x)^2}}\\ \end{align*}

Mathematica [A]  time = 0.900675, size = 207, normalized size = 0.74 \[ -\frac{\sqrt{c \left (a^2+2 a b x+b^2 x^2+1\right )} \left (-4 i \text{PolyLog}\left (2,-e^{i \cot ^{-1}(a+b x)}\right )+4 i \text{PolyLog}\left (2,e^{i \cot ^{-1}(a+b x)}\right )-2 \cot \left (\frac{1}{2} \cot ^{-1}(a+b x)\right )-4 \cot ^{-1}(a+b x) \log \left (1-e^{i \cot ^{-1}(a+b x)}\right )+4 \cot ^{-1}(a+b x) \log \left (1+e^{i \cot ^{-1}(a+b x)}\right )-2 \tan \left (\frac{1}{2} \cot ^{-1}(a+b x)\right )-\cot ^{-1}(a+b x) \csc ^2\left (\frac{1}{2} \cot ^{-1}(a+b x)\right )+\cot ^{-1}(a+b x) \sec ^2\left (\frac{1}{2} \cot ^{-1}(a+b x)\right )\right )}{8 b c (a+b x) \sqrt{\frac{1}{(a+b x)^2}+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + b*x)^2*ArcCot[a + b*x])/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2],x]

[Out]

-(Sqrt[c*(1 + a^2 + 2*a*b*x + b^2*x^2)]*(-2*Cot[ArcCot[a + b*x]/2] - ArcCot[a + b*x]*Csc[ArcCot[a + b*x]/2]^2
- 4*ArcCot[a + b*x]*Log[1 - E^(I*ArcCot[a + b*x])] + 4*ArcCot[a + b*x]*Log[1 + E^(I*ArcCot[a + b*x])] - (4*I)*
PolyLog[2, -E^(I*ArcCot[a + b*x])] + (4*I)*PolyLog[2, E^(I*ArcCot[a + b*x])] + ArcCot[a + b*x]*Sec[ArcCot[a +
b*x]/2]^2 - 2*Tan[ArcCot[a + b*x]/2]))/(8*b*c*(a + b*x)*Sqrt[1 + (a + b*x)^(-2)])

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Maple [A]  time = 1.1, size = 202, normalized size = 0.7 \begin{align*}{\frac{{\rm arccot} \left (bx+a\right )xb+{\rm arccot} \left (bx+a\right )a+1}{2\,cb}\sqrt{c \left ( -i+a+bx \right ) \left ( i+a+bx \right ) }}-{\frac{{\frac{i}{2}}}{cb} \left ( i{\rm arccot} \left (bx+a\right )\ln \left ( 1-{(i+a+bx){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) -i{\rm arccot} \left (bx+a\right )\ln \left ( 1+{(i+a+bx){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) -{\it polylog} \left ( 2,-{(i+a+bx){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) +{\it polylog} \left ( 2,{(i+a+bx){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) \right ) \sqrt{c \left ( -i+a+bx \right ) \left ( i+a+bx \right ) }{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*arccot(b*x+a)/((a^2+1)*c+2*a*b*c*x+b^2*c*x^2)^(1/2),x)

[Out]

1/2*(arccot(b*x+a)*x*b+arccot(b*x+a)*a+1)*(c*(-I+a+b*x)*(I+a+b*x))^(1/2)/c/b-1/2*I*(I*arccot(b*x+a)*ln(1-(I+a+
b*x)/(1+(b*x+a)^2)^(1/2))-I*arccot(b*x+a)*ln(1+(I+a+b*x)/(1+(b*x+a)^2)^(1/2))-polylog(2,-(I+a+b*x)/(1+(b*x+a)^
2)^(1/2))+polylog(2,(I+a+b*x)/(1+(b*x+a)^2)^(1/2)))*(c*(-I+a+b*x)*(I+a+b*x))^(1/2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/
2)/b/c

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*arccot(b*x+a)/((a^2+1)*c+2*a*b*c*x+c*x^2*b^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \operatorname{arccot}\left (b x + a\right )}{\sqrt{b^{2} c x^{2} + 2 \, a b c x +{\left (a^{2} + 1\right )} c}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*arccot(b*x+a)/((a^2+1)*c+2*a*b*c*x+c*x^2*b^2)^(1/2),x, algorithm="fricas")

[Out]

integral((b^2*x^2 + 2*a*b*x + a^2)*arccot(b*x + a)/sqrt(b^2*c*x^2 + 2*a*b*c*x + (a^2 + 1)*c), x)

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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((b*x+a)**2*acot(b*x+a)/((a**2+1)*c+2*a*b*c*x+c*x**2*b**2)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*arccot(b*x+a)/((a^2+1)*c+2*a*b*c*x+c*x^2*b^2)^(1/2),x, algorithm="giac")

[Out]

integrate((b*x + a)^2*arccot(b*x + a)/sqrt(b^2*c*x^2 + 2*a*b*c*x + (a^2 + 1)*c), x)