Optimal. Leaf size=561 \[ -\frac {37 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{120 a^2}-\frac {15 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {a^2 c x^2+c}}+\frac {15 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {a^2 c x^2+c}}+\frac {15 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {a^2 c x^2+c}}-\frac {15 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {a^2 c x^2+c}}+\frac {15 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{56 a^2 \sqrt {a^2 c x^2+c}}-\frac {17 c^2 x \sqrt {a^2 c x^2+c}}{420 a}-\frac {15 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{112 a}+\frac {15 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{56 a^2}-\frac {c x \left (a^2 c x^2+c\right )^{3/2}}{140 a}+\frac {\left (a^2 c x^2+c\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac {x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac {\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac {5 c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}+\frac {5 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{84 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.53, antiderivative size = 561, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4930, 4880, 4890, 4888, 4181, 2531, 2282, 6589, 217, 206, 195} \[ -\frac {15 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {a^2 c x^2+c}}+\frac {15 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {a^2 c x^2+c}}+\frac {15 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {a^2 c x^2+c}}-\frac {15 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {a^2 c x^2+c}}-\frac {17 c^2 x \sqrt {a^2 c x^2+c}}{420 a}-\frac {15 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{112 a}+\frac {15 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{56 a^2}+\frac {15 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{56 a^2 \sqrt {a^2 c x^2+c}}-\frac {37 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{120 a^2}-\frac {c x \left (a^2 c x^2+c\right )^{3/2}}{140 a}+\frac {\left (a^2 c x^2+c\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac {x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac {\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac {5 c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}+\frac {5 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{84 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 206
Rule 217
Rule 2282
Rule 2531
Rule 4181
Rule 4880
Rule 4888
Rule 4890
Rule 4930
Rule 6589
Rubi steps
\begin {align*} \int x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3 \, dx &=\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac {3 \int \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2 \, dx}{7 a}\\ &=\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac {c \int \left (c+a^2 c x^2\right )^{3/2} \, dx}{35 a}-\frac {(5 c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx}{14 a}\\ &=-\frac {c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac {\left (3 c^2\right ) \int \sqrt {c+a^2 c x^2} \, dx}{140 a}-\frac {\left (5 c^2\right ) \int \sqrt {c+a^2 c x^2} \, dx}{84 a}-\frac {\left (15 c^2\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx}{56 a}\\ &=-\frac {17 c^2 x \sqrt {c+a^2 c x^2}}{420 a}-\frac {c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac {15 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac {15 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac {\left (3 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{280 a}-\frac {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{168 a}-\frac {\left (15 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{112 a}-\frac {\left (15 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{56 a}\\ &=-\frac {17 c^2 x \sqrt {c+a^2 c x^2}}{420 a}-\frac {c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac {15 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac {15 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac {\left (3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{280 a}-\frac {\left (5 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{168 a}-\frac {\left (15 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{56 a}-\frac {\left (15 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{112 a \sqrt {c+a^2 c x^2}}\\ &=-\frac {17 c^2 x \sqrt {c+a^2 c x^2}}{420 a}-\frac {c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac {15 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac {15 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac {37 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^2}-\frac {\left (15 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{112 a^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {17 c^2 x \sqrt {c+a^2 c x^2}}{420 a}-\frac {c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac {15 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac {15 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac {15 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{56 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac {37 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^2}+\frac {\left (15 c^3 \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 )}{56 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (15 c^3 \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 )}{56 a^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {17 c^2 x \sqrt {c+a^2 c x^2}}{420 a}-\frac {c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac {15 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac {15 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac {15 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{56 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac {37 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^2}-\frac {15 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {c+a^2 c x^2}}+\frac {15 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (15 i c^3 \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 )}{56 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (15 i c^3 \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 )}{56 a^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {17 c^2 x \sqrt {c+a^2 c x^2}}{420 a}-\frac {c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac {15 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac {15 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac {15 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{56 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac {37 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^2}-\frac {15 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {c+a^2 c x^2}}+\frac {15 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (15 c^3 \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 )}{56 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (15 c^3 \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 )}{56 a^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {17 c^2 x \sqrt {c+a^2 c x^2}}{420 a}-\frac {c x \left (c+a^2 c x^2\right )^{3/2}}{140 a}+\frac {15 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{56 a^2}+\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{84 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac {15 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{56 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac {15 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{56 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac {37 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^2}-\frac {15 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {c+a^2 c x^2}}+\frac {15 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {c+a^2 c x^2}}+\frac {15 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {c+a^2 c x^2}}-\frac {15 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{56 a^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 6.09, size = 718, normalized size = 1.28 \[ \frac {c^2 \sqrt {a^2 c x^2+c} \left (64 \left (-259 \tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )-309 i \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+309 i \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )+309 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )-309 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )+309 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )+53760 \left (-\tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )-i \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+i \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )+\text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )-\text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )+i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )-112 \left (\left (a^2 x^2+1\right )^{5/2} \left (\frac {48 a x}{\left (a^2 x^2+1\right )^2}+\tan ^{-1}(a x)^2 \left (6 \sin \left (2 \tan ^{-1}(a x)\right )-33 \sin \left (4 \tan ^{-1}(a x)\right )\right )+32 \tan ^{-1}(a x)^3 \left (5 \cos \left (2 \tan ^{-1}(a x)\right )-1\right )+6 \tan ^{-1}(a x) \left (36 \cos \left (2 \tan ^{-1}(a x)\right )+11 \cos \left (4 \tan ^{-1}(a x)\right )+25\right )\right )+48 \left (-10 \tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )-11 i \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+11 i \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )+11 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )-11 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )+11 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )\right )+\left (a^2 x^2+1\right )^{7/2} \left (\frac {8 \tan ^{-1}(a x) \left (764 \cos \left (2 \tan ^{-1}(a x)\right )+309 \cos \left (4 \tan ^{-1}(a x)\right )+647\right )}{a^2 x^2+1}-3 \tan ^{-1}(a x)^2 \left (211 \sin \left (2 \tan ^{-1}(a x)\right )-60 \sin \left (4 \tan ^{-1}(a x)\right )+103 \sin \left (6 \tan ^{-1}(a x)\right )\right )+4 \left (101 \sin \left (2 \tan ^{-1}(a x)\right )+88 \sin \left (4 \tan ^{-1}(a x)\right )+25 \sin \left (6 \tan ^{-1}(a x)\right )\right )+64 \tan ^{-1}(a x)^3 \left (-28 \cos \left (2 \tan ^{-1}(a x)\right )+35 \cos \left (4 \tan ^{-1}(a x)\right )+57\right )\right )+4480 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x) \left (4 \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )+6 \cos \left (2 \tan ^{-1}(a x)\right )+6\right )\right )}{53760 a^2 \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{3} + c^{2} x\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.37, size = 477, normalized size = 0.85 \[ \frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (240 \arctan \left (a x \right )^{3} x^{6} a^{6}-120 \arctan \left (a x \right )^{2} x^{5} a^{5}+720 \arctan \left (a x \right )^{3} x^{4} a^{4}+48 \arctan \left (a x \right ) x^{4} a^{4}-390 \arctan \left (a x \right )^{2} x^{3} a^{3}+720 \arctan \left (a x \right )^{3} x^{2} a^{2}-12 a^{3} x^{3}+196 \arctan \left (a x \right ) a^{2} x^{2}-495 \arctan \left (a x \right )^{2} x a +240 \arctan \left (a x \right )^{3}-80 a x +598 \arctan \left (a x \right )\right )}{1680 a^{2}}+\frac {5 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}+6 i \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{112 a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {5 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (-i \arctan \left (a x \right )^{3}+3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{112 a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {37 i c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{60 a^{2} \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x \arctan \left (a x\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________