3.219 \(\int (c+a^2 c x^2)^{5/2} \tan ^{-1}(a x) \, dx\)

Optimal. Leaf size=348 \[ \frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a \sqrt{a^2 c x^2+c}}-\frac{5 c^2 \sqrt{a^2 c x^2+c}}{16 a}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{5 c \left (a^2 c x^2+c\right )^{3/2}}{72 a}-\frac{\left (a^2 c x^2+c\right )^{5/2}}{30 a}+\frac{5}{24} c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x) \]

[Out]

(-5*c^2*Sqrt[c + a^2*c*x^2])/(16*a) - (5*c*(c + a^2*c*x^2)^(3/2))/(72*a) - (c + a^2*c*x^2)^(5/2)/(30*a) + (5*c
^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/24 + (x*(c + a^2*c*x^2)^(
5/2)*ArcTan[a*x])/6 - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a
*Sqrt[c + a^2*c*x^2]) + (((5*I)/16)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/
(a*Sqrt[c + a^2*c*x^2]) - (((5*I)/16)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(
a*Sqrt[c + a^2*c*x^2])

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Rubi [A]  time = 0.194221, antiderivative size = 348, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4878, 4890, 4886} \[ \frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a \sqrt{a^2 c x^2+c}}-\frac{5 c^2 \sqrt{a^2 c x^2+c}}{16 a}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{5 c \left (a^2 c x^2+c\right )^{3/2}}{72 a}-\frac{\left (a^2 c x^2+c\right )^{5/2}}{30 a}+\frac{5}{24} c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x],x]

[Out]

(-5*c^2*Sqrt[c + a^2*c*x^2])/(16*a) - (5*c*(c + a^2*c*x^2)^(3/2))/(72*a) - (c + a^2*c*x^2)^(5/2)/(30*a) + (5*c
^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/24 + (x*(c + a^2*c*x^2)^(
5/2)*ArcTan[a*x])/6 - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a
*Sqrt[c + a^2*c*x^2]) + (((5*I)/16)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/
(a*Sqrt[c + a^2*c*x^2]) - (((5*I)/16)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(
a*Sqrt[c + a^2*c*x^2])

Rule 4878

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

Rule 4890

Int[((a_.) + ArcTan[(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*ArcTan[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 4886

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

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x) \, dx &=-\frac{\left (c+a^2 c x^2\right )^{5/2}}{30 a}+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)+\frac{1}{6} (5 c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx\\ &=-\frac{5 c \left (c+a^2 c x^2\right )^{3/2}}{72 a}-\frac{\left (c+a^2 c x^2\right )^{5/2}}{30 a}+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)+\frac{1}{8} \left (5 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=-\frac{5 c^2 \sqrt{c+a^2 c x^2}}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2}}{72 a}-\frac{\left (c+a^2 c x^2\right )^{5/2}}{30 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)+\frac{1}{16} \left (5 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{5 c^2 \sqrt{c+a^2 c x^2}}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2}}{72 a}-\frac{\left (c+a^2 c x^2\right )^{5/2}}{30 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)+\frac{\left (5 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{16 \sqrt{c+a^2 c x^2}}\\ &=-\frac{5 c^2 \sqrt{c+a^2 c x^2}}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2}}{72 a}-\frac{\left (c+a^2 c x^2\right )^{5/2}}{30 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)-\frac{5 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a \sqrt{c+a^2 c x^2}}+\frac{5 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a \sqrt{c+a^2 c x^2}}-\frac{5 i c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 6.14915, size = 643, normalized size = 1.85 \[ \frac{c^2 \sqrt{a^2 c x^2+c} \left (450 i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-450 i \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+\frac{3}{4} \left (a^2 x^2+1\right )^{5/2}+720 \sqrt{a^2 x^2+1} \left (a x \tan ^{-1}(a x)-1\right )+\frac{55}{8} \left (a^2 x^2+1\right )^3 \cos \left (3 \tan ^{-1}(a x)\right )-\frac{45}{8} \left (a^2 x^2+1\right )^3 \cos \left (5 \tan ^{-1}(a x)\right )+\frac{15}{16} \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x) \left (\frac{156 a x}{\sqrt{a^2 x^2+1}}+30 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-30 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-94 \sin \left (3 \tan ^{-1}(a x)\right )+6 \sin \left (5 \tan ^{-1}(a x)\right )+3 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cos \left (6 \tan ^{-1}(a x)\right )+45 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (2 \tan ^{-1}(a x)\right )+18 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (4 \tan ^{-1}(a x)\right )-3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cos \left (6 \tan ^{-1}(a x)\right )\right )-15 \left (a^2 x^2+1\right )^2 \left (-\frac{2}{\sqrt{a^2 x^2+1}}+3 \tan ^{-1}(a x) \left (-\frac{14 a x}{\sqrt{a^2 x^2+1}}+3 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+2 \sin \left (3 \tan ^{-1}(a x)\right )+4 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (2 \tan ^{-1}(a x)\right )+\left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (4 \tan ^{-1}(a x)\right )\right )-6 \cos \left (3 \tan ^{-1}(a x)\right )\right )+720 \tan ^{-1}(a x) \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right )\right )}{1440 a \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x],x]

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*((3*(1 + a^2*x^2)^(5/2))/4 + 720*Sqrt[1 + a^2*x^2]*(-1 + a*x*ArcTan[a*x]) + (55*(1 +
a^2*x^2)^3*Cos[3*ArcTan[a*x]])/8 - (45*(1 + a^2*x^2)^3*Cos[5*ArcTan[a*x]])/8 + 720*ArcTan[a*x]*(Log[1 - I*E^(I
*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + (450*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (450*I)*PolyLog[
2, I*E^(I*ArcTan[a*x])] - 15*(1 + a^2*x^2)^2*(-2/Sqrt[1 + a^2*x^2] - 6*Cos[3*ArcTan[a*x]] + 3*ArcTan[a*x]*((-1
4*a*x)/Sqrt[1 + a^2*x^2] + 3*Log[1 - I*E^(I*ArcTan[a*x])] + 4*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])]
- Log[1 + I*E^(I*ArcTan[a*x])]) + Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*
x])]) - 3*Log[1 + I*E^(I*ArcTan[a*x])] + 2*Sin[3*ArcTan[a*x]])) + (15*(1 + a^2*x^2)^3*ArcTan[a*x]*((156*a*x)/S
qrt[1 + a^2*x^2] + 30*Log[1 - I*E^(I*ArcTan[a*x])] + 3*Cos[6*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + 45*Co
s[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + 18*Cos[4*ArcTan[a*x]]*(Log[1
- I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) - 30*Log[1 + I*E^(I*ArcTan[a*x])] - 3*Cos[6*ArcTan[a*x]
]*Log[1 + I*E^(I*ArcTan[a*x])] - 94*Sin[3*ArcTan[a*x]] + 6*Sin[5*ArcTan[a*x]]))/16))/(1440*a*Sqrt[1 + a^2*x^2]
)

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Maple [A]  time = 0.315, size = 225, normalized size = 0.7 \begin{align*}{\frac{{c}^{2} \left ( 120\,\arctan \left ( ax \right ){x}^{5}{a}^{5}-24\,{a}^{4}{x}^{4}+390\,\arctan \left ( ax \right ){x}^{3}{a}^{3}-98\,{a}^{2}{x}^{2}+495\,\arctan \left ( ax \right ) xa-299 \right ) }{720\,a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{5\,{c}^{2}}{16\,a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( \arctan \left ( ax \right ) \ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -\arctan \left ( ax \right ) \ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i{\it dilog} \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +i{\it dilog} \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*c*x^2+c)^(5/2)*arctan(a*x),x)

[Out]

1/720*c^2/a*(c*(a*x-I)*(a*x+I))^(1/2)*(120*arctan(a*x)*x^5*a^5-24*a^4*x^4+390*arctan(a*x)*x^3*a^3-98*a^2*x^2+4
95*arctan(a*x)*x*a-299)-5/16*c^2*(c*(a*x-I)*(a*x+I))^(1/2)*(arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-ar
ctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+I*dilog(1-I*(1+I*a*x)/(
a^2*x^2+1)^(1/2)))/a/(a^2*x^2+1)^(1/2)

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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((a^2*c*x^2+c)^(5/2)*arctan(a*x),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 ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x),x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*sqrt(a^2*c*x^2 + c)*arctan(a*x), 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((a**2*c*x**2+c)**(5/2)*atan(a*x),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x),x, algorithm="giac")

[Out]

Exception raised: TypeError