Integrand size = 24, antiderivative size = 129 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=-\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {4 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 a^2 c \sqrt {c+a^2 c x^2}} \] Output:
-2/3*x/a/c/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(3/2)-4/3/a^2/c/(a^2*c*x^2+c)^( 1/2)/arctan(a*x)^(1/2)-4/3*2^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*FresnelS(2^( 1/2)/Pi^(1/2)*arctan(a*x)^(1/2))/a^2/c/(a^2*c*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.96 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=-\frac {2 \left (a x+2 \arctan (a x)-i \sqrt {1+a^2 x^2} (-i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )+i \sqrt {1+a^2 x^2} (i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )\right )}{3 a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}} \] Input:
Integrate[x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]
Output:
(-2*(a*x + 2*ArcTan[a*x] - I*Sqrt[1 + a^2*x^2]*((-I)*ArcTan[a*x])^(3/2)*Ga mma[1/2, (-I)*ArcTan[a*x]] + I*Sqrt[1 + a^2*x^2]*(I*ArcTan[a*x])^(3/2)*Gam ma[1/2, I*ArcTan[a*x]]))/(3*a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))
Time = 0.80 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5477, 5437, 5506, 5505, 3042, 3786, 3832}
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.
\(\displaystyle \int \frac {x}{\arctan (a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 5477 |
\(\displaystyle \frac {2 \int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^{3/2}}dx}{3 a}-\frac {2 x}{3 a c \arctan (a x)^{3/2} \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 5437 |
\(\displaystyle \frac {2 \left (-2 a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \sqrt {\arctan (a x)}}dx-\frac {2}{a c \sqrt {\arctan (a x)} \sqrt {a^2 c x^2+c}}\right )}{3 a}-\frac {2 x}{3 a c \arctan (a x)^{3/2} \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 5506 |
\(\displaystyle \frac {2 \left (-\frac {2 a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{3/2} \sqrt {\arctan (a x)}}dx}{c \sqrt {a^2 c x^2+c}}-\frac {2}{a c \sqrt {\arctan (a x)} \sqrt {a^2 c x^2+c}}\right )}{3 a}-\frac {2 x}{3 a c \arctan (a x)^{3/2} \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 5505 |
\(\displaystyle \frac {2 \left (-\frac {2 \sqrt {a^2 x^2+1} \int \frac {a x}{\sqrt {a^2 x^2+1} \sqrt {\arctan (a x)}}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {2}{a c \sqrt {\arctan (a x)} \sqrt {a^2 c x^2+c}}\right )}{3 a}-\frac {2 x}{3 a c \arctan (a x)^{3/2} \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\sin (\arctan (a x))}{\sqrt {\arctan (a x)}}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {2}{a c \sqrt {\arctan (a x)} \sqrt {a^2 c x^2+c}}\right )}{3 a}-\frac {2 x}{3 a c \arctan (a x)^{3/2} \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle \frac {2 \left (-\frac {4 \sqrt {a^2 x^2+1} \int \frac {a x}{\sqrt {a^2 x^2+1}}d\sqrt {\arctan (a x)}}{a c \sqrt {a^2 c x^2+c}}-\frac {2}{a c \sqrt {\arctan (a x)} \sqrt {a^2 c x^2+c}}\right )}{3 a}-\frac {2 x}{3 a c \arctan (a x)^{3/2} \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {2 \left (-\frac {2 \sqrt {2 \pi } \sqrt {a^2 x^2+1} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a c \sqrt {a^2 c x^2+c}}-\frac {2}{a c \sqrt {\arctan (a x)} \sqrt {a^2 c x^2+c}}\right )}{3 a}-\frac {2 x}{3 a c \arctan (a x)^{3/2} \sqrt {a^2 c x^2+c}}\) |
Input:
Int[x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]
Output:
(-2*x)/(3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + (2*(-2/(a*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (2*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS [Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a*c*Sqrt[c + a^2*c*x^2])))/(3*a)
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[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]
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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_S ymbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1))), x] - Simp[2*c*((q + 1)/(b*(p + 1))) Int[x*(d + e*x^2)^q*(a + b*Arc Tan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && LtQ[p, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*ArcT an[c*x])^(p + 1)/(b*c*d*(p + 1))), x] - Simp[f*(m/(b*c*(p + 1))) Int[(f*x )^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[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])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[d^(q + 1/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]) Int[x^m*(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && !(I ntegerQ[q] || GtQ[d, 0])
\[\int \frac {x}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}d x\]
Input:
int(x/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x)
Output:
int(x/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x)
Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(x/(a**2*c*x**2+c)**(3/2)/atan(a*x)**(5/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(5/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \] Input:
int(x/(atan(a*x)^(5/2)*(c + a^2*c*x^2)^(3/2)),x)
Output:
int(x/(atan(a*x)^(5/2)*(c + a^2*c*x^2)^(3/2)), x)
\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\frac {2 \sqrt {c}\, \left (-2 \mathit {atan} \left (a x \right )^{2} \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}\, x}{\mathit {atan} \left (a x \right ) a^{4} x^{4}+2 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{4} x^{2}-2 \mathit {atan} \left (a x \right )^{2} \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}\, x}{\mathit {atan} \left (a x \right ) a^{4} x^{4}+2 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{2}-2 \sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}\, \mathit {atan} \left (a x \right )-\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}\, a x \right )}{3 \mathit {atan} \left (a x \right )^{2} a^{2} c^{2} \left (a^{2} x^{2}+1\right )} \] Input:
int(x/(a^2*c*x^2+c)^(3/2)/atan(a*x)^(5/2),x)
Output:
(2*sqrt(c)*( - 2*atan(a*x)**2*int((sqrt(a**2*x**2 + 1)*sqrt(atan(a*x))*x)/ (atan(a*x)*a**4*x**4 + 2*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**4*x**2 - 2 *atan(a*x)**2*int((sqrt(a**2*x**2 + 1)*sqrt(atan(a*x))*x)/(atan(a*x)*a**4* x**4 + 2*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**2 - 2*sqrt(a**2*x**2 + 1)* sqrt(atan(a*x))*atan(a*x) - sqrt(a**2*x**2 + 1)*sqrt(atan(a*x))*a*x))/(3*a tan(a*x)**2*a**2*c**2*(a**2*x**2 + 1))