Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=-\frac {1}{a^5 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {2}{a^5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{4 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \text {Si}(3 \arctan (a x))}{4 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^2},x\right )}{a^4 c^2} \]
-1/a^5/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)+2/a^5/c^2/arctan(a*x)/(a^2*c*x^2+ c)^(1/2)+5/4*Si(arctan(a*x))*(a^2*x^2+1)^(1/2)/a^5/c^2/(a^2*c*x^2+c)^(1/2) -3/4*Si(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/a^5/c^2/(a^2*c*x^2+c)^(1/2)+Unint egrable(1/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x)/a^4/c^2
Not integrable
Time = 11.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx \]
Not integrable
Time = 2.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {5499, 5499, 5437, 5506, 5505, 3042, 3780, 4906, 2009, 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.
\(\displaystyle \int \frac {x^4}{\arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 5499 |
\(\displaystyle \frac {\int \frac {x^2}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx}{a^2 c}-\frac {\int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx}{a^2}\) |
\(\Big \downarrow \) 5499 |
\(\displaystyle \frac {\frac {\int \frac {1}{\sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{a^2 c}-\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx}{a^2}}{a^2 c}-\frac {\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx}{a^2 c}-\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx}{a^2}}{a^2}\) |
\(\Big \downarrow \) 5437 |
\(\displaystyle \frac {\frac {\int \frac {1}{\sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{a^2 c}-\frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-3 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}}{a^2}\) |
\(\Big \downarrow \) 5506 |
\(\displaystyle \frac {\frac {\int \frac {1}{\sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{a^2 c}-\frac {-\frac {a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}dx}{c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {-\frac {a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}dx}{c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}}{a^2}\) |
\(\Big \downarrow \) 5505 |
\(\displaystyle \frac {\frac {\int \frac {1}{\sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{a^2 c}-\frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {a x}{\sqrt {a^2 x^2+1} \arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {a x}{\sqrt {a^2 x^2+1} \arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {1}{\sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{a^2 c}-\frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {\sin (\arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {\sin (\arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}}{a^2}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \frac {\frac {\int \frac {1}{\sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{a^2 c}-\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}}{a^2}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {\frac {\int \frac {1}{\sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{a^2 c}-\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \left (\frac {a x}{4 \sqrt {a^2 x^2+1} \arctan (a x)}+\frac {\sin (3 \arctan (a x))}{4 \arctan (a x)}\right )d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}}{a^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\int \frac {1}{\sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{a^2 c}-\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \text {Si}(\arctan (a x))+\frac {1}{4} \text {Si}(3 \arctan (a x))\right )}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}}{a^2}\) |
\(\Big \downarrow \) 5560 |
\(\displaystyle \frac {\frac {\int \frac {1}{\sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{a^2 c}-\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \text {Si}(\arctan (a x))+\frac {1}{4} \text {Si}(3 \arctan (a x))\right )}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}}{a^2}\) |
3.6.88.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
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[1/e Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*Ar cTan[c*x])^p, x], x] - Simp[d/e Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcTan [c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ [p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[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[((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 = 17.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {x^{4}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{2}}d x\]
Not integrable
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.58 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}} \,d x } \]
integral(sqrt(a^2*c*x^2 + c)*x^4/((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3 *x^2 + c^3)*arctan(a*x)^2), x)
Not integrable
Time = 6.93 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {x^{4}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]
Not integrable
Time = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}} \,d x } \]
Not integrable
Time = 167.80 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}} \,d x } \]
Not integrable
Time = 0.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {x^4}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]