Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c^2 x \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {4}{c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^2 x^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {4 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{c^2}+\frac {8 \text {Int}\left (\frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}},x\right )}{3 a^2}+\frac {16}{3} \text {Int}\left (\frac {1}{x \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}},x\right ) \]
-2/3/a/c^2/x/(a^2*x^2+1)/arctan(a*x)^(3/2)+4*FresnelS(2*arctan(a*x)^(1/2)/ Pi^(1/2))*Pi^(1/2)/c^2+4/c^2/(a^2*x^2+1)/arctan(a*x)^(1/2)+4/3/a^2/c^2/x^2 /(a^2*x^2+1)/arctan(a*x)^(1/2)+8/3*Unintegrable(1/x^3/(a^2*c*x^2+c)^2/arct an(a*x)^(1/2),x)/a^2+16/3*Unintegrable(1/x/(a^2*c*x^2+c)^2/arctan(a*x)^(1/ 2),x)
Not integrable
Time = 3.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx \]
Not integrable
Time = 1.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {5503, 27, 5437, 5503, 5505, 4906, 27, 3042, 3786, 3832, 5560}
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.
\(\displaystyle \int \frac {1}{x \arctan (a x)^{5/2} \left (a^2 c x^2+c\right )^2} \, dx\) |
\(\Big \downarrow \) 5503 |
\(\displaystyle -2 a \int \frac {1}{c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}dx-\frac {2 \int \frac {1}{c^2 x^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}dx}{3 a}-\frac {2}{3 a c^2 x \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 a \int \frac {1}{\left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}dx}{c^2}-\frac {2 \int \frac {1}{x^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}dx}{3 a c^2}-\frac {2}{3 a c^2 x \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}\) |
\(\Big \downarrow \) 5437 |
\(\displaystyle -\frac {2 \int \frac {1}{x^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^{3/2}}dx}{3 a c^2}-\frac {2 a \left (-4 a \int \frac {x}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx-\frac {2}{a \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{c^2}-\frac {2}{3 a c^2 x \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}\) |
\(\Big \downarrow \) 5503 |
\(\displaystyle -\frac {2 a \left (-4 a \int \frac {x}{\left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx-\frac {2}{a \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{c^2}-\frac {2 \left (-8 a \int \frac {1}{x \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a x^2 \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{3 a c^2}-\frac {2}{3 a c^2 x \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}\) |
\(\Big \downarrow \) 5505 |
\(\displaystyle -\frac {2 a \left (-\frac {4 \int \frac {a x}{\left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}d\arctan (a x)}{a}-\frac {2}{a \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{c^2}-\frac {2 \left (-8 a \int \frac {1}{x \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a x^2 \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{3 a c^2}-\frac {2}{3 a c^2 x \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {2 a \left (-\frac {4 \int \frac {\sin (2 \arctan (a x))}{2 \sqrt {\arctan (a x)}}d\arctan (a x)}{a}-\frac {2}{a \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{c^2}-\frac {2 \left (-8 a \int \frac {1}{x \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a x^2 \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{3 a c^2}-\frac {2}{3 a c^2 x \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 a \left (-\frac {2 \int \frac {\sin (2 \arctan (a x))}{\sqrt {\arctan (a x)}}d\arctan (a x)}{a}-\frac {2}{a \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{c^2}-\frac {2 \left (-8 a \int \frac {1}{x \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a x^2 \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{3 a c^2}-\frac {2}{3 a c^2 x \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 a \left (-\frac {2 \int \frac {\sin (2 \arctan (a x))}{\sqrt {\arctan (a x)}}d\arctan (a x)}{a}-\frac {2}{a \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{c^2}-\frac {2 \left (-8 a \int \frac {1}{x \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a x^2 \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{3 a c^2}-\frac {2}{3 a c^2 x \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle -\frac {2 a \left (-\frac {4 \int \sin (2 \arctan (a x))d\sqrt {\arctan (a x)}}{a}-\frac {2}{a \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{c^2}-\frac {2 \left (-8 a \int \frac {1}{x \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a x^2 \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{3 a c^2}-\frac {2}{3 a c^2 x \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle -\frac {2 \left (-8 a \int \frac {1}{x \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a x^2 \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{3 a c^2}-\frac {2 a \left (-\frac {2}{a \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}-\frac {2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a}\right )}{c^2}-\frac {2}{3 a c^2 x \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}\) |
\(\Big \downarrow \) 5560 |
\(\displaystyle -\frac {2 \left (-8 a \int \frac {1}{x \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 x^2+1\right )^2 \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a x^2 \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}\right )}{3 a c^2}-\frac {2 a \left (-\frac {2}{a \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}-\frac {2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a}\right )}{c^2}-\frac {2}{3 a c^2 x \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}\) |
3.11.63.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[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_)*((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_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[x^m*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1))), x] + (-Simp[c*((m + 2*q + 2)/(b*(p + 1))) Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] - Simp[m/(b*c*(p + 1)) Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x]) /; F reeQ[{a, b, c, d, e, m}, x] && EqQ[e, c^2*d] && IntegerQ[m] && LtQ[q, -1] & & LtQ[p, -1] && NeQ[m + 2*q + 2, 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])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Unintegrab le[u*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || M atchQ[u, ((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_. )*x)^(m_.)*((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[ u, ((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_.)*x) ^(m_.)*((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, f, m, q}, x]])
Not integrable
Time = 1.76 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Not integrable
Time = 11.87 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\frac {\int \frac {1}{a^{4} x^{5} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 2 a^{2} x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + x \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{2}} \]
Integral(1/(a**4*x**5*atan(a*x)**(5/2) + 2*a**2*x**3*atan(a*x)**(5/2) + x* atan(a*x)**(5/2)), x)/c**2
Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
Not integrable
Time = 221.82 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x \arctan \left (a x\right )^{\frac {5}{2}}} \,d x } \]
Not integrable
Time = 0.64 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]