Optimal. Leaf size=444 \[ \frac {2 x^3}{27 a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {x^3 \tan ^{-1}(a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {22 \tan ^{-1}(a x)}{9 a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {22 x}{9 a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {x \tan ^{-1}(a x)^2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 x^2 \tan ^{-1}(a x)}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.77, antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4964, 4890, 4888, 4181, 2531, 2282, 6589, 4898, 191, 4944, 4938, 4930} \[ \frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {22 x}{9 a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {x \tan ^{-1}(a x)^2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {22 \tan ^{-1}(a x)}{9 a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 x^3}{27 a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {x^3 \tan ^{-1}(a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 x^2 \tan ^{-1}(a x)}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 191
Rule 2282
Rule 2531
Rule 4181
Rule 4888
Rule 4890
Rule 4898
Rule 4930
Rule 4938
Rule 4944
Rule 4964
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^4 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\frac {\int \frac {x^2 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a^2}+\frac {\int \frac {x^2 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2 c}\\ &=-\frac {x^3 \tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {x^3 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{3 a}+\frac {\int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a^4 c^2}-\frac {\int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c}\\ &=\frac {2 x^3}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 x^2 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \tan ^{-1}(a x)}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^2}{a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c}+\frac {4 \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 a^3 c}+\frac {\sqrt {1+a^2 x^2} \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{a^4 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x^3}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \tan ^{-1}(a x)}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^2}{a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {4 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 a^4 c}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x^3}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x}{9 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \tan ^{-1}(a x)}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^2}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x^3}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x}{9 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \tan ^{-1}(a x)}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^2}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (2 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x^3}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x}{9 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \tan ^{-1}(a x)}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^2}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x^3}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x}{9 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \tan ^{-1}(a x)}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^2}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.59, size = 239, normalized size = 0.54 \[ \frac {\sqrt {c \left (a^2 x^2+1\right )} \left (-\frac {270 \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}-\frac {135 a x \left (\tan ^{-1}(a x)^2-2\right )}{\sqrt {a^2 x^2+1}}+216 i \tan ^{-1}(a x) \left (\text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )-\text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )\right )-216 \left (\text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )-\text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )\right )+108 \tan ^{-1}(a x)^2 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right )+\left (9 \tan ^{-1}(a x)^2-2\right ) \sin \left (3 \tan ^{-1}(a x)\right )+6 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )\right )}{108 a^5 c^3 \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x^{4} \arctan \left (a x\right )^{2}}{a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 3.12, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \arctan \left (a x \right )^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________