Optimal. Leaf size=293 \[ -\frac {1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {17 d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{12 c^2}-\frac {4 i b d^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^2}+\frac {2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{6} b c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2}{3} i b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac {3 a b d^2 x}{2 c}+\frac {2 b^2 d^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{3 c^2}+\frac {5 b^2 d^2 \log \left (c^2 x^2+1\right )}{6 c^2}-\frac {2 i b^2 d^2 \tan ^{-1}(c x)}{3 c^2}+\frac {2 i b^2 d^2 x}{3 c}-\frac {3 b^2 d^2 x \tan ^{-1}(c x)}{2 c}-\frac {1}{12} b^2 d^2 x^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.62, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 14, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {4876, 4852, 4916, 4846, 260, 4884, 321, 203, 4920, 4854, 2402, 2315, 266, 43} \[ \frac {2 b^2 d^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^2}-\frac {1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {17 d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{12 c^2}-\frac {4 i b d^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^2}+\frac {2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{6} b c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2}{3} i b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac {3 a b d^2 x}{2 c}+\frac {5 b^2 d^2 \log \left (c^2 x^2+1\right )}{6 c^2}-\frac {2 i b^2 d^2 \tan ^{-1}(c x)}{3 c^2}+\frac {2 i b^2 d^2 x}{3 c}-\frac {3 b^2 d^2 x \tan ^{-1}(c x)}{2 c}-\frac {1}{12} b^2 d^2 x^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 203
Rule 260
Rule 266
Rule 321
Rule 2315
Rule 2402
Rule 4846
Rule 4852
Rule 4854
Rule 4876
Rule 4884
Rule 4916
Rule 4920
Rubi steps
\begin {align*} \int x (d+i c d x)^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x \left (a+b \tan ^{-1}(c x)\right )^2+2 i c d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-c^2 d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+\left (2 i c d^2\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (c^2 d^2\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\left (b c d^2\right ) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {1}{3} \left (4 i b c^2 d^2\right ) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac {1}{2} \left (b c^3 d^2\right ) \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac {1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{3} \left (4 i b d^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac {1}{3} \left (4 i b d^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {\left (b d^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c}+\frac {\left (b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c}+\frac {1}{2} \left (b c d^2\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx-\frac {1}{2} \left (b c d^2\right ) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-\frac {a b d^2 x}{c}-\frac {2}{3} i b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} b c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {7 d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^2}+\frac {1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (4 i b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c}-\frac {\left (b d^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c}+\frac {\left (b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 c}-\frac {\left (b^2 d^2\right ) \int \tan ^{-1}(c x) \, dx}{c}+\frac {1}{3} \left (2 i b^2 c d^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {1}{6} \left (b^2 c^2 d^2\right ) \int \frac {x^3}{1+c^2 x^2} \, dx\\ &=-\frac {3 a b d^2 x}{2 c}+\frac {2 i b^2 d^2 x}{3 c}-\frac {b^2 d^2 x \tan ^{-1}(c x)}{c}-\frac {2}{3} i b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} b c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {17 d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{12 c^2}+\frac {1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {4 i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^2}+\left (b^2 d^2\right ) \int \frac {x}{1+c^2 x^2} \, dx-\frac {\left (2 i b^2 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c}+\frac {\left (4 i b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c}-\frac {\left (b^2 d^2\right ) \int \tan ^{-1}(c x) \, dx}{2 c}-\frac {1}{12} \left (b^2 c^2 d^2\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac {3 a b d^2 x}{2 c}+\frac {2 i b^2 d^2 x}{3 c}-\frac {2 i b^2 d^2 \tan ^{-1}(c x)}{3 c^2}-\frac {3 b^2 d^2 x \tan ^{-1}(c x)}{2 c}-\frac {2}{3} i b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} b c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {17 d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{12 c^2}+\frac {1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {4 i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^2}+\frac {b^2 d^2 \log \left (1+c^2 x^2\right )}{2 c^2}+\frac {1}{2} \left (b^2 d^2\right ) \int \frac {x}{1+c^2 x^2} \, dx+\frac {\left (4 b^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{3 c^2}-\frac {1}{12} \left (b^2 c^2 d^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {3 a b d^2 x}{2 c}+\frac {2 i b^2 d^2 x}{3 c}-\frac {1}{12} b^2 d^2 x^2-\frac {2 i b^2 d^2 \tan ^{-1}(c x)}{3 c^2}-\frac {3 b^2 d^2 x \tan ^{-1}(c x)}{2 c}-\frac {2}{3} i b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} b c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {17 d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{12 c^2}+\frac {1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {4 i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^2}+\frac {5 b^2 d^2 \log \left (1+c^2 x^2\right )}{6 c^2}+\frac {2 b^2 d^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{3 c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.82, size = 257, normalized size = 0.88 \[ -\frac {d^2 \left (3 a^2 c^4 x^4-8 i a^2 c^3 x^3-6 a^2 c^2 x^2-2 a b c^3 x^3+8 i a b c^2 x^2-8 i a b \log \left (c^2 x^2+1\right )+2 b \tan ^{-1}(c x) \left (a \left (3 c^4 x^4-8 i c^3 x^3-6 c^2 x^2-9\right )+b \left (-c^3 x^3+4 i c^2 x^2+9 c x+4 i\right )+8 i b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )+18 a b c x+b^2 c^2 x^2-10 b^2 \log \left (c^2 x^2+1\right )+8 b^2 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )-8 i b^2 c x+b^2 (c x-i)^3 (3 c x+i) \tan ^{-1}(c x)^2+b^2\right )}{12 c^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ \frac {1}{48} \, {\left (3 \, b^{2} c^{2} d^{2} x^{4} - 8 i \, b^{2} c d^{2} x^{3} - 6 \, b^{2} d^{2} x^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} + {\rm integral}\left (-\frac {12 \, a^{2} c^{4} d^{2} x^{5} - 24 i \, a^{2} c^{3} d^{2} x^{4} - 24 i \, a^{2} c d^{2} x^{2} - 12 \, a^{2} d^{2} x - {\left (-12 i \, a b c^{4} d^{2} x^{5} - {\left (24 \, a b - 3 i \, b^{2}\right )} c^{3} d^{2} x^{4} + 8 \, b^{2} c^{2} d^{2} x^{3} - {\left (24 \, a b + 6 i \, b^{2}\right )} c d^{2} x^{2} + 12 i \, a b d^{2} x\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{12 \, {\left (c^{2} x^{2} + 1\right )}}, x\right ) \]
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 [B] time = 0.11, size = 556, normalized size = 1.90 \[ \frac {d^{2} b^{2} \arctan \left (c x \right )^{2} x^{2}}{2}-\frac {c^{2} d^{2} a^{2} x^{4}}{4}+\frac {3 d^{2} b^{2} \arctan \left (c x \right )^{2}}{4 c^{2}}+\frac {2 i d^{2} a b \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}+\frac {2 i c \,d^{2} b^{2} \arctan \left (c x \right )^{2} x^{3}}{3}+\frac {2 i d^{2} b^{2} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}-\frac {3 a b \,d^{2} x}{2 c}-\frac {3 b^{2} d^{2} x \arctan \left (c x \right )}{2 c}+\frac {5 b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{6 c^{2}}-\frac {c^{2} d^{2} a b \arctan \left (c x \right ) x^{4}}{2}-\frac {d^{2} b^{2} \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}-\frac {d^{2} b^{2} \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{3 c^{2}}-\frac {2 i d^{2} b^{2} \arctan \left (c x \right ) x^{2}}{3}+\frac {2 i c \,d^{2} a^{2} x^{3}}{3}-\frac {2 i b^{2} d^{2} \arctan \left (c x \right )}{3 c^{2}}+\frac {2 i b^{2} d^{2} x}{3 c}-\frac {2 i d^{2} a b \,x^{2}}{3}+\frac {d^{2} b^{2} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{3 c^{2}}+\frac {d^{2} b^{2} \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}+\frac {c \,d^{2} a b \,x^{3}}{6}-\frac {c^{2} d^{2} b^{2} \arctan \left (c x \right )^{2} x^{4}}{4}+d^{2} a b \arctan \left (c x \right ) x^{2}+\frac {3 d^{2} a b \arctan \left (c x \right )}{2 c^{2}}+\frac {c \,d^{2} b^{2} \arctan \left (c x \right ) x^{3}}{6}-\frac {d^{2} b^{2} \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{3 c^{2}}+\frac {d^{2} b^{2} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{3 c^{2}}+\frac {d^{2} b^{2} \ln \left (c x -i\right )^{2}}{6 c^{2}}-\frac {d^{2} b^{2} \ln \left (c x +i\right )^{2}}{6 c^{2}}-\frac {b^{2} d^{2} x^{2}}{12}+\frac {d^{2} a^{2} x^{2}}{2}+\frac {4 i c \,d^{2} a b \arctan \left (c x \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{4} \, a^{2} c^{2} d^{2} x^{4} + \frac {2}{3} i \, a^{2} c d^{2} x^{3} + \frac {1}{2} \, b^{2} d^{2} x^{2} \arctan \left (c x\right )^{2} - \frac {1}{6} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} a b c^{2} d^{2} + \frac {2}{3} i \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} a b c d^{2} + \frac {1}{2} \, a^{2} d^{2} x^{2} + {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} a b d^{2} - \frac {1}{2} \, {\left (2 \, c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )} \arctan \left (c x\right ) + \frac {\arctan \left (c x\right )^{2} - \log \left (c^{2} x^{2} + 1\right )}{c^{2}}\right )} b^{2} d^{2} - \frac {1}{192} \, {\left (12 \, b^{2} c^{2} d^{2} x^{4} - 32 i \, b^{2} c d^{2} x^{3}\right )} \arctan \left (c x\right )^{2} + \frac {1}{48} \, {\left (-3 i \, b^{2} c^{2} d^{2} x^{4} - 8 \, b^{2} c d^{2} x^{3}\right )} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right ) + \frac {1}{192} \, {\left (3 \, b^{2} c^{2} d^{2} x^{4} - 8 i \, b^{2} c d^{2} x^{3}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - \int -\frac {22 \, b^{2} c^{3} d^{2} x^{4} \arctan \left (c x\right ) - 36 \, {\left (b^{2} c^{4} d^{2} x^{5} + b^{2} c^{2} d^{2} x^{3}\right )} \arctan \left (c x\right )^{2} - 3 \, {\left (b^{2} c^{4} d^{2} x^{5} + b^{2} c^{2} d^{2} x^{3}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - {\left (3 \, b^{2} c^{4} d^{2} x^{5} - 8 \, b^{2} c^{2} d^{2} x^{3} - 24 \, {\left (b^{2} c^{3} d^{2} x^{4} + b^{2} c d^{2} x^{2}\right )} \arctan \left (c x\right )\right )} \log \left (c^{2} x^{2} + 1\right )}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + i \, \int \frac {72 \, {\left (b^{2} c^{3} d^{2} x^{4} + b^{2} c d^{2} x^{2}\right )} \arctan \left (c x\right )^{2} + 6 \, {\left (b^{2} c^{3} d^{2} x^{4} + b^{2} c d^{2} x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} + 2 \, {\left (3 \, b^{2} c^{4} d^{2} x^{5} - 8 \, b^{2} c^{2} d^{2} x^{3}\right )} \arctan \left (c x\right ) + {\left (11 \, b^{2} c^{3} d^{2} x^{4} + 12 \, {\left (b^{2} c^{4} d^{2} x^{5} + b^{2} c^{2} d^{2} x^{3}\right )} \arctan \left (c x\right )\right )} \log \left (c^{2} x^{2} + 1\right )}{48 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________