Optimal. Leaf size=1027 \[ -\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^3-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^3}{4 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {ArcTan}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \tanh ^{-1}\left (e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a c^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,-e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {45 i a c^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {45 i a c^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {6 i a c^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 a c^3 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {45 a c^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {45 a c^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {6 a c^3 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {45 i a c^3 \sqrt {1+a^2 x^2} \text {PolyLog}\left (4,-i e^{i \text {ArcTan}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {45 i a c^3 \sqrt {1+a^2 x^2} \text {PolyLog}\left (4,i e^{i \text {ArcTan}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.43, antiderivative size = 1027, normalized size of antiderivative = 1.00, number of steps
used = 56, number of rules used = 15, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5070,
5064, 5078, 5076, 4268, 2611, 2320, 6724, 5010, 5008, 4266, 6744, 5000, 5006, 4998}
\begin {gather*} -\frac {15 i a \sqrt {a^2 x^2+1} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^3 c^3}{4 \sqrt {a^2 c x^2+c}}-\frac {11 i a \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {ArcTan}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {6 a \sqrt {a^2 x^2+1} \text {ArcTan}(a x)^2 \tanh ^{-1}\left (e^{i \text {ArcTan}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {6 i a \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (-e^{i \text {ArcTan}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {45 i a \sqrt {a^2 x^2+1} \text {ArcTan}(a x)^2 \text {Li}_2\left (-i e^{i \text {ArcTan}(a x)}\right ) c^3}{8 \sqrt {a^2 c x^2+c}}-\frac {45 i a \sqrt {a^2 x^2+1} \text {ArcTan}(a x)^2 \text {Li}_2\left (i e^{i \text {ArcTan}(a x)}\right ) c^3}{8 \sqrt {a^2 c x^2+c}}-\frac {6 i a \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (e^{i \text {ArcTan}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {11 i a \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right ) c^3}{2 \sqrt {a^2 c x^2+c}}-\frac {11 i a \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right ) c^3}{2 \sqrt {a^2 c x^2+c}}-\frac {6 a \sqrt {a^2 x^2+1} \text {Li}_3\left (-e^{i \text {ArcTan}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {45 a \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_3\left (-i e^{i \text {ArcTan}(a x)}\right ) c^3}{4 \sqrt {a^2 c x^2+c}}+\frac {45 a \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_3\left (i e^{i \text {ArcTan}(a x)}\right ) c^3}{4 \sqrt {a^2 c x^2+c}}+\frac {6 a \sqrt {a^2 x^2+1} \text {Li}_3\left (e^{i \text {ArcTan}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {45 i a \sqrt {a^2 x^2+1} \text {Li}_4\left (-i e^{i \text {ArcTan}(a x)}\right ) c^3}{4 \sqrt {a^2 c x^2+c}}+\frac {45 i a \sqrt {a^2 x^2+1} \text {Li}_4\left (i e^{i \text {ArcTan}(a x)}\right ) c^3}{4 \sqrt {a^2 c x^2+c}}+\frac {7}{8} a^2 x \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^3 c^2-\frac {\sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^3 c^2}{x}-\frac {21}{8} a \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^2 c^2+\frac {1}{4} a^2 x \sqrt {a^2 c x^2+c} \text {ArcTan}(a x) c^2-\frac {1}{4} a \sqrt {a^2 c x^2+c} c^2+\frac {1}{4} a^2 x \left (a^2 c x^2+c\right )^{3/2} \text {ArcTan}(a x)^3 c-\frac {1}{4} a \left (a^2 c x^2+c\right )^{3/2} \text {ArcTan}(a x)^2 c \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2320
Rule 2611
Rule 4266
Rule 4268
Rule 4998
Rule 5000
Rule 5006
Rule 5008
Rule 5010
Rule 5064
Rule 5070
Rule 5076
Rule 5078
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3}{x^2} \, dx &=c \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{x^2} \, dx+\left (a^2 c\right ) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx\\ &=-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+c^2 \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x^2} \, dx+\frac {1}{2} \left (a^2 c^2\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx+\frac {1}{4} \left (3 a^2 c^2\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx+\left (a^2 c^2\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=-\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+c^3 \int \frac {\tan ^{-1}(a x)^3}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{4} \left (a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (3 a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{2} \left (a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{4} \left (9 a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\left (3 a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\left (3 a c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{8 \sqrt {c+a^2 c x^2}}+\frac {\left (a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}+\frac {\left (9 a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 \sqrt {c+a^2 c x^2}}+\frac {\left (a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (9 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 \sqrt {c+a^2 c x^2}}+\frac {\left (9 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {45 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {45 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt {c+a^2 c x^2}}+\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (9 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {\left (9 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {45 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {45 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {6 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {45 a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {45 a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}}-\frac {\left (6 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (9 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 \sqrt {c+a^2 c x^2}}-\frac {\left (9 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {45 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {45 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {6 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {45 a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {45 a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}}-\frac {\left (9 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {\left (9 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{4} a c^2 \sqrt {c+a^2 c x^2}+\frac {1}{4} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {21}{8} a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {45 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {45 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt {c+a^2 c x^2}}-\frac {6 i a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {11 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 a c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {45 a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {45 a c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {6 a c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {45 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {45 i a c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(3267\) vs. \(2(1027)=2054\).
time = 14.28, size = 3267, normalized size = 3.18 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 2.99, size = 655, normalized size = 0.64
method | result | size |
default | \(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 \arctan \left (a x \right )^{3} a^{4} x^{4}-2 \arctan \left (a x \right )^{2} a^{3} x^{3}+9 \arctan \left (a x \right )^{3} a^{2} x^{2}+2 \arctan \left (a x \right ) a^{2} x^{2}-23 \arctan \left (a x \right )^{2} a x -8 \arctan \left (a x \right )^{3}-2 a x \right )}{8 x}-\frac {i a \,c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (15 i \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-48 i \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+24 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+90 i \arctan \left (a x \right ) \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+45 \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-45 \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-24 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-90 i \arctan \left (a x \right ) \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+44 i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-44 i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+48 \arctan \left (a x \right ) \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-48 \arctan \left (a x \right ) \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+48 i \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-15 i \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+44 \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-44 \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+90 \polylog \left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 \polylog \left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8 \sqrt {a^{2} x^{2}+1}}\) | \(655\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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