Integrand size = 24, antiderivative size = 230 \[ \int \frac {x^4 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {27 \sqrt {\arctan (a x)}}{256 a^5 c^3}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}-\frac {9 \sqrt {\arctan (a x)}}{32 a^5 c^3 \left (1+a^2 x^2\right )}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 x \arctan (a x)^{3/2}}{8 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^{5/2}}{20 a^5 c^3}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{512 a^5 c^3}+\frac {3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{32 a^5 c^3} \] Output:
27/256*arctan(a*x)^(1/2)/a^5/c^3+3/32*x^4*arctan(a*x)^(1/2)/a/c^3/(a^2*x^2 +1)^2-9/32*arctan(a*x)^(1/2)/a^5/c^3/(a^2*x^2+1)-1/4*x^3*arctan(a*x)^(3/2) /a^2/c^3/(a^2*x^2+1)^2-3/8*x*arctan(a*x)^(3/2)/a^4/c^3/(a^2*x^2+1)+3/20*ar ctan(a*x)^(5/2)/a^5/c^3-3/1024*2^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2 )*arctan(a*x)^(1/2))/a^5/c^3+3/32*Pi^(1/2)*FresnelC(2*arctan(a*x)^(1/2)/Pi ^(1/2))/a^5/c^3
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.54 \[ \int \frac {x^4 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\frac {64 \sqrt {\arctan (a x)} \left (15 \left (-15-6 a^2 x^2+17 a^4 x^4\right )-160 a x \left (3+5 a^2 x^2\right ) \arctan (a x)+192 \left (1+a^2 x^2\right )^2 \arctan (a x)^2\right )}{\left (1+a^2 x^2\right )^2}-510 \left (12 \sqrt {\arctan (a x)}+\sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )\right )+90 \sqrt {\arctan (a x)} \left (8+\frac {\Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )}{\sqrt {-i \arctan (a x)}}+\frac {\Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{\sqrt {i \arctan (a x)}}\right )+\frac {225 \left (24 \arctan (a x)-4 i \sqrt {2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+4 i \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )-i \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )+i \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )\right )}{\sqrt {\arctan (a x)}}}{81920 a^5 c^3} \] Input:
Integrate[(x^4*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]
Output:
((64*Sqrt[ArcTan[a*x]]*(15*(-15 - 6*a^2*x^2 + 17*a^4*x^4) - 160*a*x*(3 + 5 *a^2*x^2)*ArcTan[a*x] + 192*(1 + a^2*x^2)^2*ArcTan[a*x]^2))/(1 + a^2*x^2)^ 2 - 510*(12*Sqrt[ArcTan[a*x]] + Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcT an[a*x]]] - 8*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]]) + 90*Sqrt [ArcTan[a*x]]*(8 + Gamma[1/2, (-4*I)*ArcTan[a*x]]/Sqrt[(-I)*ArcTan[a*x]] + Gamma[1/2, (4*I)*ArcTan[a*x]]/Sqrt[I*ArcTan[a*x]]) + (225*(24*ArcTan[a*x] - (4*I)*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + ( 4*I)*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - I*Sqrt[(- I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + I*Sqrt[I*ArcTan[a*x]]*Gam ma[1/2, (4*I)*ArcTan[a*x]]))/Sqrt[ArcTan[a*x]])/(81920*a^5*c^3)
Time = 1.54 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5475, 27, 5471, 5465, 5439, 3042, 3793, 2009, 5505, 3042, 3793, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^4 \arctan (a x)^{3/2}}{\left (a^2 c x^2+c\right )^3} \, dx\) |
| \(\Big \downarrow \) 5475 |
| \(\displaystyle -\frac {3}{64} \int \frac {x^4}{c^3 \left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx+\frac {3 \int \frac {x^2 \arctan (a x)^{3/2}}{c^2 \left (a^2 x^2+1\right )^2}dx}{4 a^2 c}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int \frac {x^2 \arctan (a x)^{3/2}}{\left (a^2 x^2+1\right )^2}dx}{4 a^2 c^3}-\frac {3 \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx}{64 c^3}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 5471 |
| \(\displaystyle -\frac {3 \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx}{64 c^3}+\frac {3 \left (\frac {3 \int \frac {x \sqrt {\arctan (a x)}}{\left (a^2 x^2+1\right )^2}dx}{4 a}+\frac {\arctan (a x)^{5/2}}{5 a^3}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2 c^3}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle -\frac {3 \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx}{64 c^3}+\frac {3 \left (\frac {3 \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx}{4 a}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}+\frac {\arctan (a x)^{5/2}}{5 a^3}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2 c^3}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 5439 |
| \(\displaystyle -\frac {3 \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx}{64 c^3}+\frac {3 \left (\frac {3 \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}d\arctan (a x)}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}+\frac {\arctan (a x)^{5/2}}{5 a^3}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2 c^3}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx}{64 c^3}+\frac {3 \left (\frac {3 \left (\frac {\int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^2}{\sqrt {\arctan (a x)}}d\arctan (a x)}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}+\frac {\arctan (a x)^{5/2}}{5 a^3}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2 c^3}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle -\frac {3 \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx}{64 c^3}+\frac {3 \left (\frac {3 \left (\frac {\int \left (\frac {\cos (2 \arctan (a x))}{2 \sqrt {\arctan (a x)}}+\frac {1}{2 \sqrt {\arctan (a x)}}\right )d\arctan (a x)}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}+\frac {\arctan (a x)^{5/2}}{5 a^3}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2 c^3}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \int \frac {x^4}{\left (a^2 x^2+1\right )^3 \sqrt {\arctan (a x)}}dx}{64 c^3}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \left (\frac {\arctan (a x)^{5/2}}{5 a^3}+\frac {3 \left (\frac {\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {\arctan (a x)}}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2 c^3}\) |
| \(\Big \downarrow \) 5505 |
| \(\displaystyle -\frac {3 \int \frac {a^4 x^4}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}d\arctan (a x)}{64 a^5 c^3}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \left (\frac {\arctan (a x)^{5/2}}{5 a^3}+\frac {3 \left (\frac {\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {\arctan (a x)}}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \int \frac {\sin (\arctan (a x))^4}{\sqrt {\arctan (a x)}}d\arctan (a x)}{64 a^5 c^3}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \left (\frac {\arctan (a x)^{5/2}}{5 a^3}+\frac {3 \left (\frac {\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {\arctan (a x)}}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2 c^3}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle -\frac {3 \int \left (-\frac {\cos (2 \arctan (a x))}{2 \sqrt {\arctan (a x)}}+\frac {\cos (4 \arctan (a x))}{8 \sqrt {\arctan (a x)}}+\frac {3}{8 \sqrt {\arctan (a x)}}\right )d\arctan (a x)}{64 a^5 c^3}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \left (\frac {\arctan (a x)^{5/2}}{5 a^3}+\frac {3 \left (\frac {\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {\arctan (a x)}}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2 c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )-\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arctan (a x)}\right )}{64 a^5 c^3}+\frac {3 x^4 \sqrt {\arctan (a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^{3/2}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \left (\frac {\arctan (a x)^{5/2}}{5 a^3}+\frac {3 \left (\frac {\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {\arctan (a x)}}{4 a^2}-\frac {\sqrt {\arctan (a x)}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a}-\frac {x \arctan (a x)^{3/2}}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2 c^3}\) |
Input:
Int[(x^4*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]
Output:
(3*x^4*Sqrt[ArcTan[a*x]])/(32*a*c^3*(1 + a^2*x^2)^2) - (x^3*ArcTan[a*x]^(3 /2))/(4*a^2*c^3*(1 + a^2*x^2)^2) - (3*((3*Sqrt[ArcTan[a*x]])/4 + (Sqrt[Pi/ 2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/8 - (Sqrt[Pi]*FresnelC[(2*Sqr t[ArcTan[a*x]])/Sqrt[Pi]])/2))/(64*a^5*c^3) + (3*(-1/2*(x*ArcTan[a*x]^(3/2 ))/(a^2*(1 + a^2*x^2)) + ArcTan[a*x]^(5/2)/(5*a^3) + (3*(-1/2*Sqrt[ArcTan[ a*x]]/(a^2*(1 + a^2*x^2)) + (Sqrt[ArcTan[a*x]] + (Sqrt[Pi]*FresnelC[(2*Sqr t[ArcTan[a*x]])/Sqrt[Pi]])/2)/(4*a^2)))/(4*a)))/(4*a^2*c^3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[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]))
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[d^q/c Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, Ar cTan[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])
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] - Simp[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]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^2)/((d_) + (e_.)*(x_)^2) ^2, x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)), x] + (-Simp[x*((a + b*ArcTan[c*x])^p/(2*c^2*d*(d + e*x^2))), x] + Simp[b*(p/( 2*c)) Int[x*((a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ [{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_)*((d_) + (e_.) *(x_)^2)^(q_), x_Symbol] :> Simp[b*p*(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*Ar cTan[c*x])^(p - 1)/(c*d*m^2)), x] + (-Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(c^2*d*m)), x] + Simp[f^2*((m - 1)/(c^2*d*m)) Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Simp[ b^2*p*((p - 1)/m^2) Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 2) , x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2* q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[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])
Time = 0.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.57
| method | result | size |
| default | \(\frac {-15 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+768 \arctan \left (a x \right )^{3}-1280 \arctan \left (a x \right )^{2} \sin \left (2 \arctan \left (a x \right )\right )+160 \arctan \left (a x \right )^{2} \sin \left (4 \arctan \left (a x \right )\right )+480 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )-960 \arctan \left (a x \right ) \cos \left (2 \arctan \left (a x \right )\right )+60 \arctan \left (a x \right ) \cos \left (4 \arctan \left (a x \right )\right )}{5120 a^{5} c^{3} \sqrt {\arctan \left (a x \right )}}\) | \(132\) |
Input:
int(x^4*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)
Output:
1/5120/a^5/c^3*(-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*arc tan(a*x))+160*arctan(a*x)^2*sin(4*arctan(a*x))+480*arctan(a*x)^(1/2)*Pi^(1 /2)*FresnelC(2*arctan(a*x)^(1/2)/Pi^(1/2))-960*arctan(a*x)*cos(2*arctan(a* x))+60*arctan(a*x)*cos(4*arctan(a*x)))/arctan(a*x)^(1/2)
Exception generated. \[ \int \frac {x^4 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^4*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^4 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{4} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \] Input:
integrate(x**4*atan(a*x)**(3/2)/(a**2*c*x**2+c)**3,x)
Output:
Integral(x**4*atan(a*x)**(3/2)/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1) , x)/c**3
Exception generated. \[ \int \frac {x^4 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^4*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int \frac {x^4 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{4} \arctan \left (a x\right )^{\frac {3}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \] Input:
integrate(x^4*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x, algorithm="giac")
Output:
integrate(x^4*arctan(a*x)^(3/2)/(a^2*c*x^2 + c)^3, x)
Timed out. \[ \int \frac {x^4 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {x^4\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \] Input:
int((x^4*atan(a*x)^(3/2))/(c + a^2*c*x^2)^3,x)
Output:
int((x^4*atan(a*x)^(3/2))/(c + a^2*c*x^2)^3, x)
\[ \int \frac {x^4 \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int(x^4*atan(a*x)^(3/2)/(a^2*c*x^2+c)^3,x)
Output:
(96*sqrt(atan(a*x))*atan(a*x)**2*a**4*x**4 + 192*sqrt(atan(a*x))*atan(a*x) **2*a**2*x**2 + 96*sqrt(atan(a*x))*atan(a*x)**2 - 400*sqrt(atan(a*x))*atan (a*x)*a**3*x**3 - 240*sqrt(atan(a*x))*atan(a*x)*a*x + 150*sqrt(atan(a*x))* a**4*x**4 - 90*sqrt(atan(a*x)) + 45*int(sqrt(atan(a*x))/(atan(a*x)*a**6*x* *6 + 3*atan(a*x)*a**4*x**4 + 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**5*x* *4 + 90*int(sqrt(atan(a*x))/(atan(a*x)*a**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**3*x**2 + 45*int(sqrt(atan(a*x))/ (atan(a*x)*a**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3*atan(a*x)*a**2*x**2 + ata n(a*x)),x)*a - 75*int((sqrt(atan(a*x))*x**4)/(atan(a*x)*a**6*x**6 + 3*atan (a*x)*a**4*x**4 + 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**9*x**4 - 150*in t((sqrt(atan(a*x))*x**4)/(atan(a*x)*a**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3* atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**7*x**2 - 75*int((sqrt(atan(a*x))*x* *4)/(atan(a*x)*a**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**5)/(640*a**5*c**3*(a**4*x**4 + 2*a**2*x**2 + 1))