Optimal. Leaf size=438 \[ \frac {5 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{6 a}+\frac {3 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}+\frac {3 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a \sqrt {a^2 c x^2+c}}+\frac {1}{12} c x \sqrt {a^2 c x^2+c}+\frac {3}{8} c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {3 c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{4 a}+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac {\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{6 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.31, 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 = {4880, 4890, 4888, 4181, 2531, 2282, 6589, 217, 206, 195} \[ \frac {3 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}+\frac {3 c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a \sqrt {a^2 c x^2+c}}+\frac {5 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{6 a}+\frac {1}{12} c x \sqrt {a^2 c x^2+c}+\frac {3}{8} c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {3 c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{4 a}+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac {\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{6 a} \]
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 6589
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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 0.98, size = 439, normalized size = 1.00 \[ \frac {c \sqrt {a^2 c x^2+c} \left (2 a^4 x^4 \sin \left (3 \tan ^{-1}(a x)\right )-3 a^4 x^4 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+6 a^4 x^4 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )+2 a x \sqrt {a^2 x^2+1}+2 a^2 x^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)+69 a x \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2-94 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)+80 \tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )+4 a^2 x^2 \sin \left (3 \tan ^{-1}(a x)\right )-6 a^2 x^2 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+12 a^2 x^2 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )+2 a^3 x^3 \sqrt {a^2 x^2+1}+21 a^3 x^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2+72 i \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )-72 i \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )-72 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )+72 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )-72 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+2 \sin \left (3 \tan ^{-1}(a x)\right )+6 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )\right )}{96 a \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{2}, x\right ) \]
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 = 0.71, size = 304, normalized size = 0.69 \[ \frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (6 \arctan \left (a x \right )^{2} x^{3} a^{3}-4 \arctan \left (a x \right ) a^{2} x^{2}+15 \arctan \left (a x \right )^{2} x a +2 a x -22 \arctan \left (a x \right )\right )}{24 a}-\frac {i c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (9 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-9 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 i \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 i \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+40 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{24 a \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________