\(\int \frac {\arctan (a x)^{5/2}}{(c+a^2 c x^2)^3} \, dx\) [876]

Optimal result

Integrand size = 21, antiderivative size = 296 \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {45 x \sqrt {\arctan (a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac {75 \arctan (a x)^{3/2}}{256 a c^3}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{7/2}}{28 a c^3}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4096 a c^3}+\frac {15 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{128 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{256 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{2048 a c^3} \]

[Out]

-75/256*arctan(a*x)^(3/2)/a/c^3+5/32*arctan(a*x)^(3/2)/a/c^3/(a^2*x^2+1)^2+15/32*arctan(a*x)^(3/2)/a/c^3/(a^2*
x^2+1)+1/4*x*arctan(a*x)^(5/2)/c^3/(a^2*x^2+1)^2+3/8*x*arctan(a*x)^(5/2)/c^3/(a^2*x^2+1)+3/28*arctan(a*x)^(7/2
)/a/c^3+15/8192*FresnelS(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a/c^3+15/128*FresnelS(2*arctan
(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a/c^3-45/128*x*arctan(a*x)^(1/2)/c^3/(a^2*x^2+1)-15/256*sin(2*arctan(a*x))*arct
an(a*x)^(1/2)/a/c^3-15/2048*sin(4*arctan(a*x))*arctan(a*x)^(1/2)/a/c^3

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {5020, 5012, 5050, 5090, 4491, 12, 3386, 3432, 5024, 3393, 3377} \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3 x \arctan (a x)^{5/2}}{8 c^3 \left (a^2 x^2+1\right )}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {15 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {45 x \sqrt {\arctan (a x)}}{128 c^3 \left (a^2 x^2+1\right )}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4096 a c^3}+\frac {15 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{128 a c^3}+\frac {3 \arctan (a x)^{7/2}}{28 a c^3}-\frac {75 \arctan (a x)^{3/2}}{256 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{256 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{2048 a c^3} \]

[In]

Int[ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^3,x]

[Out]

(-45*x*Sqrt[ArcTan[a*x]])/(128*c^3*(1 + a^2*x^2)) - (75*ArcTan[a*x]^(3/2))/(256*a*c^3) + (5*ArcTan[a*x]^(3/2))
/(32*a*c^3*(1 + a^2*x^2)^2) + (15*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^(5/2))/(4*c^3*(
1 + a^2*x^2)^2) + (3*x*ArcTan[a*x]^(5/2))/(8*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(7/2))/(28*a*c^3) + (15*Sqrt[
Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a*c^3) + (15*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqr
t[Pi]])/(128*a*c^3) - (15*Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(256*a*c^3) - (15*Sqrt[ArcTan[a*x]]*Sin[4*ArcT
an[a*x]])/(2048*a*c^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[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 3393

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 3432

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

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 5012

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 5020

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 5024

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 5050

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]

Rule 5090

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

Rubi steps \begin{align*} \text {integral}& = \frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {15}{64} \int \frac {\sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {3 \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c} \\ & = \frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{7/2}}{28 a c^3}-\frac {15 \text {Subst}\left (\int \sqrt {x} \cos ^4(x) \, dx,x,\arctan (a x)\right )}{64 a c^3}-\frac {(15 a) \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c} \\ & = \frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{7/2}}{28 a c^3}-\frac {15 \text {Subst}\left (\int \left (\frac {3 \sqrt {x}}{8}+\frac {1}{2} \sqrt {x} \cos (2 x)+\frac {1}{8} \sqrt {x} \cos (4 x)\right ) \, dx,x,\arctan (a x)\right )}{64 a c^3}-\frac {45 \int \frac {\sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{64 c} \\ & = -\frac {45 x \sqrt {\arctan (a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac {75 \arctan (a x)^{3/2}}{256 a c^3}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{7/2}}{28 a c^3}-\frac {15 \text {Subst}\left (\int \sqrt {x} \cos (4 x) \, dx,x,\arctan (a x)\right )}{512 a c^3}-\frac {15 \text {Subst}\left (\int \sqrt {x} \cos (2 x) \, dx,x,\arctan (a x)\right )}{128 a c^3}+\frac {(45 a) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{256 c} \\ & = -\frac {45 x \sqrt {\arctan (a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac {75 \arctan (a x)^{3/2}}{256 a c^3}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{7/2}}{28 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{256 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{2048 a c^3}+\frac {15 \text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{4096 a c^3}+\frac {15 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{512 a c^3}+\frac {45 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{256 a c^3} \\ & = -\frac {45 x \sqrt {\arctan (a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac {75 \arctan (a x)^{3/2}}{256 a c^3}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{7/2}}{28 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{256 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{2048 a c^3}+\frac {15 \text {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{2048 a c^3}+\frac {15 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{256 a c^3}+\frac {45 \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arctan (a x)\right )}{256 a c^3} \\ & = -\frac {45 x \sqrt {\arctan (a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac {75 \arctan (a x)^{3/2}}{256 a c^3}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{7/2}}{28 a c^3}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4096 a c^3}+\frac {15 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{512 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{256 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{2048 a c^3}+\frac {45 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{512 a c^3} \\ & = -\frac {45 x \sqrt {\arctan (a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac {75 \arctan (a x)^{3/2}}{256 a c^3}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{7/2}}{28 a c^3}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4096 a c^3}+\frac {15 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{512 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{256 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{2048 a c^3}+\frac {45 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{256 a c^3} \\ & = -\frac {45 x \sqrt {\arctan (a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac {75 \arctan (a x)^{3/2}}{256 a c^3}+\frac {5 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \arctan (a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{7/2}}{28 a c^3}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4096 a c^3}+\frac {15 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{128 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (2 \arctan (a x))}{256 a c^3}-\frac {15 \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{2048 a c^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.61 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.97 \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {57120 a x \arctan (a x)+50400 a^3 x^3 \arctan (a x)-38080 \arctan (a x)^2+13440 a^2 x^2 \arctan (a x)^2+33600 a^4 x^4 \arctan (a x)^2-71680 a x \arctan (a x)^3-43008 a^3 x^3 \arctan (a x)^3-12288 \left (1+a^2 x^2\right )^2 \arctan (a x)^4+3360 \sqrt {2} \left (1+a^2 x^2\right )^2 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+3360 \sqrt {2} \left (1+a^2 x^2\right )^2 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )+105 \left (1+a^2 x^2\right )^2 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )+105 \left (1+a^2 x^2\right )^2 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{114688 a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}} \]

[In]

Integrate[ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^3,x]

[Out]

-1/114688*(57120*a*x*ArcTan[a*x] + 50400*a^3*x^3*ArcTan[a*x] - 38080*ArcTan[a*x]^2 + 13440*a^2*x^2*ArcTan[a*x]
^2 + 33600*a^4*x^4*ArcTan[a*x]^2 - 71680*a*x*ArcTan[a*x]^3 - 43008*a^3*x^3*ArcTan[a*x]^3 - 12288*(1 + a^2*x^2)
^2*ArcTan[a*x]^4 + 3360*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + 3360*S
qrt[2]*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + 105*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTa
n[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + 105*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]]
)/(a*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])

Maple [A] (verified)

Time = 2.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.57

\[\frac {6144 \arctan \left (a x \right )^{\frac {7}{2}} \sqrt {\pi }+14336 \arctan \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, \sin \left (2 \arctan \left (a x \right )\right )+1792 \arctan \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, \sin \left (4 \arctan \left (a x \right )\right )+105 \pi \sqrt {2}\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+17920 \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \cos \left (2 \arctan \left (a x \right )\right )+1120 \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \cos \left (4 \arctan \left (a x \right )\right )+6720 \pi \,\operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )-13440 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \sin \left (2 \arctan \left (a x \right )\right )-420 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \sin \left (4 \arctan \left (a x \right )\right )}{57344 c^{3} a \sqrt {\pi }}\]

[In]

int(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^3,x)

[Out]

1/57344/c^3/a/Pi^(1/2)*(6144*arctan(a*x)^(7/2)*Pi^(1/2)+14336*arctan(a*x)^(5/2)*Pi^(1/2)*sin(2*arctan(a*x))+17
92*arctan(a*x)^(5/2)*Pi^(1/2)*sin(4*arctan(a*x))+105*Pi*2^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))
+17920*arctan(a*x)^(3/2)*Pi^(1/2)*cos(2*arctan(a*x))+1120*arctan(a*x)^(3/2)*Pi^(1/2)*cos(4*arctan(a*x))+6720*P
i*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2))-13440*arctan(a*x)^(1/2)*Pi^(1/2)*sin(2*arctan(a*x))-420*arctan(a*x)^(
1/2)*Pi^(1/2)*sin(4*arctan(a*x)))

Fricas [F(-2)]

Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(arctan(a*x)^(5/2)/(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)

Sympy [F]

\[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]

[In]

integrate(atan(a*x)**(5/2)/(a**2*c*x**2+c)**3,x)

[Out]

Integral(atan(a*x)**(5/2)/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x)/c**3

Maxima [F(-2)]

Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F]

\[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]

[In]

integrate(arctan(a*x)^(5/2)/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

[In]

int(atan(a*x)^(5/2)/(c + a^2*c*x^2)^3,x)

[Out]

int(atan(a*x)^(5/2)/(c + a^2*c*x^2)^3, x)