Optimal. Leaf size=348 \[ \frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a \sqrt{a^2 c x^2+c}}-\frac{5 c^2 \sqrt{a^2 c x^2+c}}{16 a}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{5 c \left (a^2 c x^2+c\right )^{3/2}}{72 a}-\frac{\left (a^2 c x^2+c\right )^{5/2}}{30 a}+\frac{5}{24} c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x) \]
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Rubi [A] time = 0.194221, antiderivative size = 348, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4878, 4890, 4886} \[ \frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{16 a \sqrt{a^2 c x^2+c}}-\frac{5 c^2 \sqrt{a^2 c x^2+c}}{16 a}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{5 c \left (a^2 c x^2+c\right )^{3/2}}{72 a}-\frac{\left (a^2 c x^2+c\right )^{5/2}}{30 a}+\frac{5}{24} c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4878
Rule 4890
Rule 4886
Rubi steps
\begin{align*} \int \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x) \, dx &=-\frac{\left (c+a^2 c x^2\right )^{5/2}}{30 a}+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)+\frac{1}{6} (5 c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx\\ &=-\frac{5 c \left (c+a^2 c x^2\right )^{3/2}}{72 a}-\frac{\left (c+a^2 c x^2\right )^{5/2}}{30 a}+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)+\frac{1}{8} \left (5 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=-\frac{5 c^2 \sqrt{c+a^2 c x^2}}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2}}{72 a}-\frac{\left (c+a^2 c x^2\right )^{5/2}}{30 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)+\frac{1}{16} \left (5 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{5 c^2 \sqrt{c+a^2 c x^2}}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2}}{72 a}-\frac{\left (c+a^2 c x^2\right )^{5/2}}{30 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)+\frac{\left (5 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{16 \sqrt{c+a^2 c x^2}}\\ &=-\frac{5 c^2 \sqrt{c+a^2 c x^2}}{16 a}-\frac{5 c \left (c+a^2 c x^2\right )^{3/2}}{72 a}-\frac{\left (c+a^2 c x^2\right )^{5/2}}{30 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)-\frac{5 i 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 )}{8 a \sqrt{c+a^2 c x^2}}+\frac{5 i 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 )}{16 a \sqrt{c+a^2 c x^2}}-\frac{5 i 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 )}{16 a \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 6.14915, size = 643, normalized size = 1.85 \[ \frac{c^2 \sqrt{a^2 c x^2+c} \left (450 i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-450 i \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+\frac{3}{4} \left (a^2 x^2+1\right )^{5/2}+720 \sqrt{a^2 x^2+1} \left (a x \tan ^{-1}(a x)-1\right )+\frac{55}{8} \left (a^2 x^2+1\right )^3 \cos \left (3 \tan ^{-1}(a x)\right )-\frac{45}{8} \left (a^2 x^2+1\right )^3 \cos \left (5 \tan ^{-1}(a x)\right )+\frac{15}{16} \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x) \left (\frac{156 a x}{\sqrt{a^2 x^2+1}}+30 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-30 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-94 \sin \left (3 \tan ^{-1}(a x)\right )+6 \sin \left (5 \tan ^{-1}(a x)\right )+3 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cos \left (6 \tan ^{-1}(a x)\right )+45 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (2 \tan ^{-1}(a x)\right )+18 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (4 \tan ^{-1}(a x)\right )-3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cos \left (6 \tan ^{-1}(a x)\right )\right )-15 \left (a^2 x^2+1\right )^2 \left (-\frac{2}{\sqrt{a^2 x^2+1}}+3 \tan ^{-1}(a x) \left (-\frac{14 a x}{\sqrt{a^2 x^2+1}}+3 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+2 \sin \left (3 \tan ^{-1}(a x)\right )+4 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (2 \tan ^{-1}(a x)\right )+\left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (4 \tan ^{-1}(a x)\right )\right )-6 \cos \left (3 \tan ^{-1}(a x)\right )\right )+720 \tan ^{-1}(a x) \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right )\right )}{1440 a \sqrt{a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.315, size = 225, normalized size = 0.7 \begin{align*}{\frac{{c}^{2} \left ( 120\,\arctan \left ( ax \right ){x}^{5}{a}^{5}-24\,{a}^{4}{x}^{4}+390\,\arctan \left ( ax \right ){x}^{3}{a}^{3}-98\,{a}^{2}{x}^{2}+495\,\arctan \left ( ax \right ) xa-299 \right ) }{720\,a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{5\,{c}^{2}}{16\,a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( \arctan \left ( ax \right ) \ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -\arctan \left ( ax \right ) \ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i{\it dilog} \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +i{\it dilog} \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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