Optimal. Leaf size=438 \[ \frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{4 a \sqrt {c+a^2 c x^2}}+\frac {5 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {3 c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.22, antiderivative size = 438, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5000, 5010,
5008, 4266, 2611, 2320, 6724, 223, 212, 201} \begin {gather*} \frac {3 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (-i e^{i \text {ArcTan}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (i e^{i \text {ArcTan}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \text {ArcTan}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}+\frac {3 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \text {ArcTan}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{4 a \sqrt {a^2 c x^2+c}}+\frac {3}{8} c x \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}-\frac {3 c \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{4 a}+\frac {1}{4} x \text {ArcTan}(a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\text {ArcTan}(a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {5 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{6 a}+\frac {1}{12} c x \sqrt {a^2 c x^2+c} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 2320
Rule 2611
Rule 4266
Rule 5000
Rule 5008
Rule 5010
Rule 6724
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx &=-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{6} c \int \sqrt {c+a^2 c x^2} \, dx+\frac {1}{4} (3 c) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=\frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{12} c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{4} \left (3 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{12} c^2 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {1}{4} \left (3 c^2\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{8 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {5 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a \sqrt {c+a^2 c x^2}}+\frac {5 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}-\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a \sqrt {c+a^2 c x^2}}+\frac {5 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {\left (3 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {\left (3 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a \sqrt {c+a^2 c x^2}}+\frac {5 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a \sqrt {c+a^2 c x^2}}+\frac {5 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {3 c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.42, size = 811, normalized size = 1.85 \begin {gather*} \frac {c \sqrt {c+a^2 c x^2} \left (2 a x \sqrt {1+a^2 x^2}+2 a^3 x^3 \sqrt {1+a^2 x^2}-94 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)+2 a^2 x^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)+69 a x \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2+21 a^3 x^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2-96 i \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2+96 \tanh ^{-1}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+6 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x))+12 a^2 x^2 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x))+6 a^4 x^4 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x))-12 \text {ArcTan}(a x)^2 \log \left (1-i e^{i \text {ArcTan}(a x)}\right )-12 \pi \text {ArcTan}(a x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (1-i e^{i \text {ArcTan}(a x)}\right )\right )+12 \text {ArcTan}(a x)^2 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )+12 \text {ArcTan}(a x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )-12 \pi \text {ArcTan}(a x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )-12 \text {ArcTan}(a x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left ((1+i)+(1-i) e^{i \text {ArcTan}(a x)}\right )\right )+12 \pi \text {ArcTan}(a x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )+16 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-12 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-16 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+12 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+12 \pi \text {ArcTan}(a x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )+72 i \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )-72 i \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )-72 \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )+72 \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )+2 \sin (3 \text {ArcTan}(a x))+4 a^2 x^2 \sin (3 \text {ArcTan}(a x))+2 a^4 x^4 \sin (3 \text {ArcTan}(a x))-3 \text {ArcTan}(a x)^2 \sin (3 \text {ArcTan}(a x))-6 a^2 x^2 \text {ArcTan}(a x)^2 \sin (3 \text {ArcTan}(a x))-3 a^4 x^4 \text {ArcTan}(a x)^2 \sin (3 \text {ArcTan}(a x))\right )}{96 a \sqrt {1+a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.35, size = 304, normalized size = 0.69
method | result | size |
default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (6 \arctan \left (a x \right )^{2} a^{3} x^{3}-4 \arctan \left (a x \right ) a^{2} x^{2}+15 \arctan \left (a x \right )^{2} a x +2 a x -22 \arctan \left (a x \right )\right )}{24 a}+\frac {i c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (9 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-9 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 i \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 i \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-40 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{24 a \sqrt {a^{2} x^{2}+1}}\) | \(304\) |
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 \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, 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 {\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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