Optimal. Leaf size=263 \[ \frac {a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^2}+\frac {a x}{2 c^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}-\frac {\sqrt {c+a^2 c x^2}}{2 a c^3 x \text {ArcTan}(a x)^2}+\frac {3}{2 c \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)}-\frac {1}{2 c^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {Si}(\text {ArcTan}(a x))}{8 c^2 \sqrt {c+a^2 c x^2}}+\frac {9 \sqrt {1+a^2 x^2} \text {Si}(3 \text {ArcTan}(a x))}{8 c^2 \sqrt {c+a^2 c x^2}}-\frac {\text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2},x\right )}{2 a c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.19, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \text {ArcTan}(a x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx &=-\left (a^2 \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx\right )+\frac {\int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx}{c}\\ &=\frac {a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2} a \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx+a^3 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx+\frac {\int \frac {1}{x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx}{c^2}-\frac {a^2 \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx}{c}\\ &=\frac {a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac {a x}{2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {\sqrt {c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac {1}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-a \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx+\frac {1}{2} \left (3 a^2\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx-\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a c^2}-\frac {a \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{2 c}+\frac {a \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{c}\\ &=\frac {a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac {a x}{2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {\sqrt {c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac {3}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac {1}{2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}+\left (3 a^2\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx-\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a c^2}+\frac {a^2 \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{2 c}-\frac {a^2 \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c}+\frac {\left (3 a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{2 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac {a x}{2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {\sqrt {c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac {3}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac {1}{2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a c^2}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac {a x}{2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {\sqrt {c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac {3}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac {1}{2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a c^2}+\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\sin (x)}{4 x}+\frac {\sin (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac {a x}{2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {\sqrt {c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac {3}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac {1}{2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}\left (\tan ^{-1}(a x)\right )}{2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a c^2}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\sin (x)}{4 x}+\frac {\sin (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac {a x}{2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {\sqrt {c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac {3}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac {1}{2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}\left (\tan ^{-1}(a x)\right )}{8 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \text {Si}\left (3 \tan ^{-1}(a x)\right )}{8 c^2 \sqrt {c+a^2 c x^2}}-\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a c^2}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac {a x}{2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {\sqrt {c+a^2 c x^2}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac {3}{2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac {1}{2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {Si}\left (\tan ^{-1}(a x)\right )}{8 c^2 \sqrt {c+a^2 c x^2}}+\frac {9 \sqrt {1+a^2 x^2} \text {Si}\left (3 \tan ^{-1}(a x)\right )}{8 c^2 \sqrt {c+a^2 c x^2}}-\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{2 a c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 2.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \text {ArcTan}(a x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.56, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
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]
________________________________________________________________________________________
Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________