3.795 \(\int \frac {\tan ^{-1}(a x)^{3/2}}{(c+a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=219 \[ \frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (a^2 x^2+1\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (a^2 x^2+1\right )}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {3 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a c^3}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a c^3} \]

[Out]

1/4*x*arctan(a*x)^(3/2)/c^3/(a^2*x^2+1)^2+3/8*x*arctan(a*x)^(3/2)/c^3/(a^2*x^2+1)+3/20*arctan(a*x)^(5/2)/a/c^3
-3/1024*FresnelC(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a/c^3-3/32*FresnelC(2*arctan(a*x)^(1/2
)/Pi^(1/2))*Pi^(1/2)/a/c^3-45/256*arctan(a*x)^(1/2)/a/c^3+3/32*arctan(a*x)^(1/2)/a/c^3/(a^2*x^2+1)^2+9/32*arct
an(a*x)^(1/2)/a/c^3/(a^2*x^2+1)

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Rubi [A]  time = 0.29, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4900, 4892, 4930, 4904, 3312, 3304, 3352} \[ \frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (a^2 x^2+1\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (a^2 x^2+1\right )}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {3 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a c^3}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a c^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^3,x]

[Out]

(-45*Sqrt[ArcTan[a*x]])/(256*a*c^3) + (3*Sqrt[ArcTan[a*x]])/(32*a*c^3*(1 + a^2*x^2)^2) + (9*Sqrt[ArcTan[a*x]])
/(32*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^(3/2))/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x]^(3/2))/(8*c^3*(1
+ a^2*x^2)) + (3*ArcTan[a*x]^(5/2))/(20*a*c^3) - (3*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*
a*c^3) - (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a*c^3)

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4892

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

Rule 4900

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b*p*(d + e*x^2)^(q
+ 1)*(a + b*ArcTan[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q +
1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(
p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e
}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

Rule 4904

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

Rule 4930

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

Rubi steps

\begin {align*} \int \frac {\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {3}{64} \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {3 \int \frac {\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}-\frac {(9 a) \int \frac {x \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \operatorname {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}+\frac {\cos (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}-\frac {9 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{64 c}\\ &=-\frac {9 \sqrt {\tan ^{-1}(a x)}}{256 a c^3}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a c^3}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{128 a c^3}-\frac {9 \operatorname {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}\\ &=-\frac {9 \sqrt {\tan ^{-1}(a x)}}{256 a c^3}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \operatorname {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{256 a c^3}-\frac {3 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{64 a c^3}-\frac {9 \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}\\ &=-\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a c^3}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a c^3}-\frac {9 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{128 a c^3}\\ &=-\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a c^3}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a c^3}-\frac {9 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{64 a c^3}\\ &=-\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a c^3}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a c^3}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 142, normalized size = 0.65 \[ \frac {\frac {4 \sqrt {\tan ^{-1}(a x)} \left (192 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2+160 a x \left (3 a^2 x^2+5\right ) \tan ^{-1}(a x)-15 \left (15 a^4 x^4+6 a^2 x^2-17\right )\right )}{\left (a^2 x^2+1\right )^2}-15 \sqrt {2 \pi } C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )-480 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{5120 a c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^3,x]

[Out]

((4*Sqrt[ArcTan[a*x]]*(-15*(-17 + 6*a^2*x^2 + 15*a^4*x^4) + 160*a*x*(5 + 3*a^2*x^2)*ArcTan[a*x] + 192*(1 + a^2
*x^2)^2*ArcTan[a*x]^2))/(1 + a^2*x^2)^2 - 15*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - 480*Sqrt[Pi
]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(5120*a*c^3)

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.77, size = 132, normalized size = 0.60 \[ \frac {-15 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+768 \arctan \left (a x \right )^{3}+1280 \arctan \left (a x \right )^{2} \sin \left (2 \arctan \left (a x \right )\right )+160 \arctan \left (a x \right )^{2} \sin \left (4 \arctan \left (a x \right )\right )-480 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+960 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+60 \cos \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{5120 a \,c^{3} \sqrt {\arctan \left (a x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x)

[Out]

1/5120/a/c^3/arctan(a*x)^(1/2)*(-15*2^(1/2)*arctan(a*x)^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arctan(a*x)
^(1/2))+768*arctan(a*x)^3+1280*arctan(a*x)^2*sin(2*arctan(a*x))+160*arctan(a*x)^2*sin(4*arctan(a*x))-480*arcta
n(a*x)^(1/2)*Pi^(1/2)*FresnelC(2*arctan(a*x)^(1/2)/Pi^(1/2))+960*cos(2*arctan(a*x))*arctan(a*x)+60*cos(4*arcta
n(a*x))*arctan(a*x))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a*x)^(3/2)/(c + a^2*c*x^2)^3,x)

[Out]

int(atan(a*x)^(3/2)/(c + a^2*c*x^2)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)**(3/2)/(a**2*c*x**2+c)**3,x)

[Out]

Integral(atan(a*x)**(3/2)/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x)/c**3

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