Optimal. Leaf size=58 \[ \frac {\sqrt {\text {ArcTan}(a x)}}{4 a^3 c^3}-\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{8 a^3 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5090, 4491,
3385, 3433} \begin {gather*} \frac {\sqrt {\text {ArcTan}(a x)}}{4 a^3 c^3}-\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{8 a^3 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3385
Rule 3433
Rule 4491
Rule 5090
Rubi steps
\begin {align*} \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {\cos ^2(x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{8 \sqrt {x}}-\frac {\cos (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^3 c^3}-\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 c^3}\\ &=\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^3 c^3}-\frac {\text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{4 a^3 c^3}\\ &=\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^3 c^3}-\frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{8 a^3 c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.19, size = 229, normalized size = 3.95 \begin {gather*} \frac {-2 \sqrt {2 \pi } \sqrt {\text {ArcTan}(a x)^2} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )+16 \sqrt {\pi } \sqrt {\text {ArcTan}(a x)^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcTan}(a x)}}{\sqrt {\pi }}\right )+\sqrt {\text {ArcTan}(a x)} \left (64 \sqrt {\text {ArcTan}(a x)^2}+4 \sqrt {2} \sqrt {i \text {ArcTan}(a x)} \text {Gamma}\left (\frac {1}{2},-2 i \text {ArcTan}(a x)\right )+4 \sqrt {2} \sqrt {-i \text {ArcTan}(a x)} \text {Gamma}\left (\frac {1}{2},2 i \text {ArcTan}(a x)\right )+7 \sqrt {i \text {ArcTan}(a x)} \text {Gamma}\left (\frac {1}{2},-4 i \text {ArcTan}(a x)\right )+7 \sqrt {-i \text {ArcTan}(a x)} \text {Gamma}\left (\frac {1}{2},4 i \text {ArcTan}(a x)\right )\right )}{256 a^3 c^3 \sqrt {\text {ArcTan}(a x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.21, size = 48, normalized size = 0.83
method | result | size |
default | \(\frac {\sqrt {2}\, \left (2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }-\pi \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{16 c^{3} a^{3} \sqrt {\pi }}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {x^{2}}{a^{6} x^{6} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{4} x^{4} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{2} x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}} + \sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2}{\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________