Optimal. Leaf size=50 \[ -\frac {\text {Ci}\left (2 \tan ^{-1}(a x)\right )}{2 a^5 c^3}+\frac {\text {Ci}\left (4 \tan ^{-1}(a x)\right )}{8 a^5 c^3}+\frac {3 \log \left (\tan ^{-1}(a x)\right )}{8 a^5 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4970, 3312, 3302} \[ -\frac {\text {CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{2 a^5 c^3}+\frac {\text {CosIntegral}\left (4 \tan ^{-1}(a x)\right )}{8 a^5 c^3}+\frac {3 \log \left (\tan ^{-1}(a x)\right )}{8 a^5 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3302
Rule 3312
Rule 4970
Rubi steps
\begin {align*} \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin ^4(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {3}{8 x}-\frac {\cos (2 x)}{2 x}+\frac {\cos (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^3}\\ &=\frac {3 \log \left (\tan ^{-1}(a x)\right )}{8 a^5 c^3}+\frac {\operatorname {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^5 c^3}-\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^5 c^3}\\ &=-\frac {\text {Ci}\left (2 \tan ^{-1}(a x)\right )}{2 a^5 c^3}+\frac {\text {Ci}\left (4 \tan ^{-1}(a x)\right )}{8 a^5 c^3}+\frac {3 \log \left (\tan ^{-1}(a x)\right )}{8 a^5 c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 34, normalized size = 0.68 \[ \frac {-4 \text {Ci}\left (2 \tan ^{-1}(a x)\right )+\text {Ci}\left (4 \tan ^{-1}(a x)\right )+3 \log \left (\tan ^{-1}(a x)\right )}{8 a^5 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.56, size = 174, normalized size = 3.48 \[ \frac {6 \, \log \left (\arctan \left (a x\right )\right ) + \operatorname {log\_integral}\left (\frac {a^{4} x^{4} + 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) + \operatorname {log\_integral}\left (\frac {a^{4} x^{4} - 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) - 4 \, \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 4 \, \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{16 \, a^{5} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.31, size = 45, normalized size = 0.90 \[ -\frac {\Ci \left (2 \arctan \left (a x \right )\right )}{2 a^{5} c^{3}}+\frac {\Ci \left (4 \arctan \left (a x \right )\right )}{8 a^{5} c^{3}}+\frac {3 \ln \left (\arctan \left (a x \right )\right )}{8 a^{5} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^4}{\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{4}}{a^{6} x^{6} \operatorname {atan}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}{\left (a x \right )} + \operatorname {atan}{\left (a x \right )}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________