Integrand size = 24, antiderivative size = 83 \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\arctan (a x)^{3/2}}{12 a^3 c^3}+\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{64 a^3 c^3}-\frac {\sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{32 a^3 c^3} \] Output:
1/12*arctan(a*x)^(3/2)/a^3/c^3+1/128*2^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/P i^(1/2)*arctan(a*x)^(1/2))/a^3/c^3-1/32*arctan(a*x)^(1/2)*sin(4*arctan(a*x ))/a^3/c^3
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.70 \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {32 \arctan (a x) \left (3 a x \left (-1+a^2 x^2\right )+2 \left (1+a^2 x^2\right )^2 \arctan (a x)\right )-3 \left (1+a^2 x^2\right )^2 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )-3 \left (1+a^2 x^2\right )^2 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )}{768 a^3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}} \] Input:
Integrate[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3,x]
Output:
(32*ArcTan[a*x]*(3*a*x*(-1 + a^2*x^2) + 2*(1 + a^2*x^2)^2*ArcTan[a*x]) - 3 *(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] - 3 *(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(768*a ^3*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])
Time = 0.37 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.87, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5505, 4906, 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.
\(\displaystyle \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (a^2 c x^2+c\right )^3} \, dx\) |
\(\Big \downarrow \) 5505 |
\(\displaystyle \frac {\int \frac {a^2 x^2 \sqrt {\arctan (a x)}}{\left (a^2 x^2+1\right )^2}d\arctan (a x)}{a^3 c^3}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {\int \left (\frac {1}{8} \sqrt {\arctan (a x)}-\frac {1}{8} \sqrt {\arctan (a x)} \cos (4 \arctan (a x))\right )d\arctan (a x)}{a^3 c^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{64} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{12} \arctan (a x)^{3/2}-\frac {1}{32} \sqrt {\arctan (a x)} \sin (4 \arctan (a x))}{a^3 c^3}\) |
Input:
Int[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3,x]
Output:
(ArcTan[a*x]^(3/2)/12 + (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x] ]])/64 - (Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/32)/(a^3*c^3)
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG tQ[p, 0]
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 = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {3 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+32 \arctan \left (a x \right )^{2}-12 \arctan \left (a x \right ) \sin \left (4 \arctan \left (a x \right )\right )}{384 a^{3} c^{3} \sqrt {\arctan \left (a x \right )}}\) | \(66\) |
Input:
int(x^2*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)
Output:
1/384/a^3/c^3*(3*2^(1/2)*arctan(a*x)^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^ (1/2)*arctan(a*x)^(1/2))+32*arctan(a*x)^2-12*arctan(a*x)*sin(4*arctan(a*x) ))/arctan(a*x)^(1/2)
Exception generated. \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2*arctan(a*x)^(1/2)/(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)
\[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \] Input:
integrate(x**2*atan(a*x)**(1/2)/(a**2*c*x**2+c)**3,x)
Output:
Integral(x**2*sqrt(atan(a*x))/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x)/c**3
Exception generated. \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^2*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.24 \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\arctan \left (a x\right )^{\frac {3}{2}}}{12 \, a^{3} c^{3}} - \frac {-\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\arctan \left (a x\right )}\right ) - 8 i \, \sqrt {\arctan \left (a x\right )} e^{\left (4 i \, \arctan \left (a x\right )\right )}}{512 \, a^{3} c^{3}} - \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\arctan \left (a x\right )}\right ) + 8 i \, \sqrt {\arctan \left (a x\right )} e^{\left (-4 i \, \arctan \left (a x\right )\right )}}{512 \, a^{3} c^{3}} \] Input:
integrate(x^2*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x, algorithm="giac")
Output:
1/12*arctan(a*x)^(3/2)/(a^3*c^3) - 1/512*(-(I - 1)*sqrt(2)*sqrt(pi)*erf((I - 1)*sqrt(2)*sqrt(arctan(a*x))) - 8*I*sqrt(arctan(a*x))*e^(4*I*arctan(a*x )))/(a^3*c^3) - 1/512*((I + 1)*sqrt(2)*sqrt(pi)*erf(-(I + 1)*sqrt(2)*sqrt( arctan(a*x))) + 8*I*sqrt(arctan(a*x))*e^(-4*I*arctan(a*x)))/(a^3*c^3)
Timed out. \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {x^2\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \] Input:
int((x^2*atan(a*x)^(1/2))/(c + a^2*c*x^2)^3,x)
Output:
int((x^2*atan(a*x)^(1/2))/(c + a^2*c*x^2)^3, x)
\[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-2 \sqrt {\mathit {atan} \left (a x \right )}\, x +2 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a^{4} x^{4}+4 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right ) a^{2} x^{2}+2 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}}{a^{6} x^{6}+3 a^{4} x^{4}+3 a^{2} x^{2}+1}d x \right )+\left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}\, x}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{5} x^{4}+2 \left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}\, x}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{3} x^{2}+\left (\int \frac {\sqrt {\mathit {atan} \left (a x \right )}\, x}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a}{6 a^{2} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )} \] Input:
int(x^2*atan(a*x)^(1/2)/(a^2*c*x^2+c)^3,x)
Output:
( - 2*sqrt(atan(a*x))*x + 2*int(sqrt(atan(a*x))/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1),x)*a**4*x**4 + 4*int(sqrt(atan(a*x))/(a**6*x**6 + 3*a**4 *x**4 + 3*a**2*x**2 + 1),x)*a**2*x**2 + 2*int(sqrt(atan(a*x))/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1),x) + int((sqrt(atan(a*x))*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 + 2*int((sqrt(atan(a*x))*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 + int((sqrt(atan( a*x))*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)/(6*a**2*c**3*(a**4*x**4 + 2*a**2*x**2 + 1))