Optimal. Leaf size=164 \[ -\frac {3 b^2 e \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )}{d}-\frac {3 i b e \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}-\frac {3 b e (c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}+\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {e \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}-\frac {3 i b^3 e \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right )}{2 d} \]
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Rubi [A] time = 0.24, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5043, 12, 4852, 4916, 4846, 4920, 4854, 2402, 2315, 4884} \[ -\frac {3 i b^3 e \text {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d}-\frac {3 b^2 e \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )}{d}-\frac {3 i b e \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}-\frac {3 b e (c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}+\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {e \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2315
Rule 2402
Rule 4846
Rule 4852
Rule 4854
Rule 4884
Rule 4916
Rule 4920
Rule 5043
Rubi steps
\begin {align*} \int (c e+d e x) \left (a+b \tan ^{-1}(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int e x \left (a+b \tan ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \left (a+b \tan ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}-\frac {(3 b e) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \tan ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}-\frac {(3 b e) \operatorname {Subst}\left (\int \left (a+b \tan ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{2 d}+\frac {(3 b e) \operatorname {Subst}\left (\int \frac {\left (a+b \tan ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac {3 b e (c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}+\frac {e \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {\left (3 b^2 e\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \tan ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {3 i b e \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}-\frac {3 b e (c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}+\frac {e \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}-\frac {\left (3 b^2 e\right ) \operatorname {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{i-x} \, dx,x,c+d x\right )}{d}\\ &=-\frac {3 i b e \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}-\frac {3 b e (c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}+\frac {e \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}-\frac {3 b^2 e \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {\left (3 b^3 e\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {3 i b e \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}-\frac {3 b e (c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}+\frac {e \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}-\frac {3 b^2 e \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {\left (3 i b^3 e\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\right )}{d}\\ &=-\frac {3 i b e \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}-\frac {3 b e (c+d x) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d}+\frac {e \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}+\frac {e (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^3}{2 d}-\frac {3 b^2 e \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {3 i b^3 e \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 196, normalized size = 1.20 \[ \frac {e \left (3 b \tan ^{-1}(c+d x) \left (a \left (a \left (c^2+2 c d x+d^2 x^2+1\right )-2 b (c+d x)\right )-2 b^2 \log \left (1+e^{2 i \tan ^{-1}(c+d x)}\right )\right )+a \left (a (c+d x) (a c+a d x-3 b)-6 b^2 \log \left (\frac {1}{\sqrt {(c+d x)^2+1}}\right )\right )+3 b^2 (c+d x-i) \tan ^{-1}(c+d x)^2 (-b+a (c+d x+i))+b^3 \left (c^2+2 c d x+d^2 x^2+1\right ) \tan ^{-1}(c+d x)^3+3 i b^3 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c+d x)}\right )\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{3} d e x + a^{3} c e + {\left (b^{3} d e x + b^{3} c e\right )} \arctan \left (d x + c\right )^{3} + 3 \, {\left (a b^{2} d e x + a b^{2} c e\right )} \arctan \left (d x + c\right )^{2} + 3 \, {\left (a^{2} b d e x + a^{2} b c e\right )} \arctan \left (d x + c\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [B] time = 0.14, size = 567, normalized size = 3.46 \[ \frac {3 i e \,b^{3} \dilog \left (\frac {i \left (d x +c -i\right )}{2}\right )}{4 d}-\frac {3 i e \,b^{3} \ln \left (d x +c -i\right )^{2}}{8 d}+\frac {3 i e \,b^{3} \ln \left (d x +c +i\right )^{2}}{8 d}-\frac {3 i e \,b^{3} \dilog \left (-\frac {i \left (d x +c +i\right )}{2}\right )}{4 d}+\frac {3 e a \,b^{2} \ln \left (1+\left (d x +c \right )^{2}\right )}{2 d}-3 \arctan \left (d x +c \right ) x a \,b^{2} e +\arctan \left (d x +c \right )^{3} x \,b^{3} c e +\frac {\arctan \left (d x +c \right )^{3} b^{3} c^{2} e}{2 d}-\frac {3 \arctan \left (d x +c \right )^{2} b^{3} c e}{2 d}+\frac {3 e \,a^{2} b \arctan \left (d x +c \right )}{2 d}+\frac {3 e \,b^{3} \arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2 d}+\frac {3 e a \,b^{2} \arctan \left (d x +c \right )^{2}}{2 d}-\frac {3 a^{2} b c e}{2 d}+\frac {3 i e \,b^{3} \ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{4 d}+\frac {3 i e \,b^{3} \ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )}{4 d}-\frac {3 i e \,b^{3} \ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{4 d}-\frac {3 i e \,b^{3} \ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )}{4 d}+\frac {d \arctan \left (d x +c \right )^{3} x^{2} b^{3} e}{2}+3 \arctan \left (d x +c \right )^{2} x a \,b^{2} c e +\frac {3 \arctan \left (d x +c \right )^{2} a \,b^{2} c^{2} e}{2 d}+\frac {a^{3} c^{2} e}{2 d}-\frac {3 e x \,a^{2} b}{2}+x \,a^{3} c e +\frac {d \,x^{2} a^{3} e}{2}-\frac {3 \arctan \left (d x +c \right )^{2} x \,b^{3} e}{2}+\frac {e \,b^{3} \arctan \left (d x +c \right )^{3}}{2 d}+3 \arctan \left (d x +c \right ) x \,a^{2} b c e +\frac {3 \arctan \left (d x +c \right ) a^{2} b \,c^{2} e}{2 d}-\frac {3 \arctan \left (d x +c \right ) a \,b^{2} c e}{d}+\frac {3 d \arctan \left (d x +c \right )^{2} x^{2} a \,b^{2} e}{2}+\frac {3 d \arctan \left (d x +c \right ) x^{2} a^{2} b e}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e \left (\int a^{3} c\, dx + \int a^{3} d x\, dx + \int b^{3} c \operatorname {atan}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} c \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b c \operatorname {atan}{\left (c + d x \right )}\, dx + \int b^{3} d x \operatorname {atan}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} d x \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b d x \operatorname {atan}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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