Optimal. Leaf size=222 \[ -\frac {a b f x}{d}-\frac {b^2 f (c+d x) \text {ArcTan}(c+d x)}{d^2}+\frac {i (d e-c f) (a+b \text {ArcTan}(c+d x))^2}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) (a+b \text {ArcTan}(c+d x))^2}{2 d^2 f}+\frac {(e+f x)^2 (a+b \text {ArcTan}(c+d x))^2}{2 f}+\frac {2 b (d e-c f) (a+b \text {ArcTan}(c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {b^2 f \log \left (1+(c+d x)^2\right )}{2 d^2}+\frac {i b^2 (d e-c f) \text {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.28, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5155, 4974,
4930, 266, 5104, 5004, 5040, 4964, 2449, 2352} \begin {gather*} \frac {i (d e-c f) (a+b \text {ArcTan}(c+d x))^2}{d^2}-\frac {(-c f+d e+f) (d e-(c+1) f) (a+b \text {ArcTan}(c+d x))^2}{2 d^2 f}+\frac {2 b (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \text {ArcTan}(c+d x))}{d^2}+\frac {(e+f x)^2 (a+b \text {ArcTan}(c+d x))^2}{2 f}-\frac {a b f x}{d}-\frac {b^2 f (c+d x) \text {ArcTan}(c+d x)}{d^2}+\frac {i b^2 (d e-c f) \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right )}{d^2}+\frac {b^2 f \log \left ((c+d x)^2+1\right )}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 2352
Rule 2449
Rule 4930
Rule 4964
Rule 4974
Rule 5004
Rule 5040
Rule 5104
Rule 5155
Rubi steps
\begin {align*} \int (e+f x) \left (a+b \tan ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right ) \left (a+b \tan ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 f}-\frac {b \text {Subst}\left (\int \left (\frac {f^2 \left (a+b \tan ^{-1}(x)\right )}{d^2}+\frac {((d e-f-c f) (d e+f-c f)+2 f (d e-c f) x) \left (a+b \tan ^{-1}(x)\right )}{d^2 \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f}\\ &=\frac {(e+f x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 f}-\frac {b \text {Subst}\left (\int \frac {((d e-f-c f) (d e+f-c f)+2 f (d e-c f) x) \left (a+b \tan ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2 f}-\frac {(b f) \text {Subst}\left (\int \left (a+b \tan ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^2}\\ &=-\frac {a b f x}{d}+\frac {(e+f x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 f}-\frac {b \text {Subst}\left (\int \left (\frac {(d e+f-c f) (d e-(1+c) f) \left (a+b \tan ^{-1}(x)\right )}{1+x^2}-\frac {2 f (-d e+c f) x \left (a+b \tan ^{-1}(x)\right )}{1+x^2}\right ) \, dx,x,c+d x\right )}{d^2 f}-\frac {\left (b^2 f\right ) \text {Subst}\left (\int \tan ^{-1}(x) \, dx,x,c+d x\right )}{d^2}\\ &=-\frac {a b f x}{d}-\frac {b^2 f (c+d x) \tan ^{-1}(c+d x)}{d^2}+\frac {(e+f x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 f}+\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{d^2}-\frac {(2 b (d e-c f)) \text {Subst}\left (\int \frac {x \left (a+b \tan ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2}-\frac {(b (d e+f-c f) (d e-(1+c) f)) \text {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{d^2 f}\\ &=-\frac {a b f x}{d}-\frac {b^2 f (c+d x) \tan ^{-1}(c+d x)}{d^2}+\frac {i (d e-c f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 f}+\frac {b^2 f \log \left (1+(c+d x)^2\right )}{2 d^2}+\frac {(2 b (d e-c f)) \text {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{i-x} \, dx,x,c+d x\right )}{d^2}\\ &=-\frac {a b f x}{d}-\frac {b^2 f (c+d x) \tan ^{-1}(c+d x)}{d^2}+\frac {i (d e-c f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 f}+\frac {2 b (d e-c f) \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {b^2 f \log \left (1+(c+d x)^2\right )}{2 d^2}-\frac {\left (2 b^2 (d e-c f)\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2}\\ &=-\frac {a b f x}{d}-\frac {b^2 f (c+d x) \tan ^{-1}(c+d x)}{d^2}+\frac {i (d e-c f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 f}+\frac {2 b (d e-c f) \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {b^2 f \log \left (1+(c+d x)^2\right )}{2 d^2}+\frac {\left (2 i b^2 (d e-c f)\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\right )}{d^2}\\ &=-\frac {a b f x}{d}-\frac {b^2 f (c+d x) \tan ^{-1}(c+d x)}{d^2}+\frac {i (d e-c f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 f}+\frac {2 b (d e-c f) \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {b^2 f \log \left (1+(c+d x)^2\right )}{2 d^2}+\frac {i b^2 (d e-c f) \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.27, size = 264, normalized size = 1.19 \begin {gather*} \frac {2 a^2 c d e-2 a b c f-a^2 c^2 f+2 a^2 d^2 e x-2 a b d f x+a^2 d^2 f x^2+b^2 (-i+c+d x) (2 d e+i f-c f+d f x) \text {ArcTan}(c+d x)^2-2 b \text {ArcTan}(c+d x) \left (b f (c+d x)+a \left (-2 c d e+c^2 f-2 d^2 e x-f \left (1+d^2 x^2\right )\right )-2 b (d e-c f) \log \left (1+e^{2 i \text {ArcTan}(c+d x)}\right )\right )+4 a b d e \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )-2 b^2 f \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )-4 a b c f \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )-2 i b^2 (d e-c f) \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c+d x)}\right )}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 700 vs. \(2 (212 ) = 424\).
time = 0.20, size = 701, normalized size = 3.16
method | result | size |
derivativedivides | \(\frac {\frac {a b \arctan \left (d x +c \right ) f \left (d x +c \right )^{2}}{d}-\frac {2 a b \arctan \left (d x +c \right ) f c \left (d x +c \right )}{d}+\frac {i b^{2} \dilog \left (-\frac {i \left (d x +c +i\right )}{2}\right ) e}{2}-\frac {i b^{2} \dilog \left (\frac {i \left (d x +c -i\right )}{2}\right ) e}{2}-\frac {i b^{2} \ln \left (d x +c +i\right )^{2} e}{4}+\frac {i b^{2} \ln \left (d x +c -i\right )^{2} e}{4}+\frac {i b^{2} \ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right ) c f}{2 d}+\frac {i b^{2} \ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right ) c f}{2 d}-\frac {i b^{2} \ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right ) c f}{2 d}-\frac {i b^{2} \ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right ) c f}{2 d}+\frac {a b \ln \left (1+\left (d x +c \right )^{2}\right ) c f}{d}-\frac {b^{2} \arctan \left (d x +c \right )^{2} f c \left (d x +c \right )}{d}+\frac {b^{2} \ln \left (1+\left (d x +c \right )^{2}\right ) \arctan \left (d x +c \right ) c f}{d}-\frac {a^{2} \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}+\frac {i b^{2} \ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right ) e}{2}+\frac {i b^{2} \ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right ) e}{2}-\frac {i b^{2} \ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right ) e}{2}-\frac {i b^{2} \ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right ) e}{2}+\frac {b^{2} \arctan \left (d x +c \right )^{2} f \left (d x +c \right )^{2}}{2 d}-\frac {b^{2} \arctan \left (d x +c \right ) f \left (d x +c \right )}{d}+2 e \left (d x +c \right ) a b \arctan \left (d x +c \right )+\frac {a b f \arctan \left (d x +c \right )}{d}-\frac {a b f \left (d x +c \right )}{d}-a b \ln \left (1+\left (d x +c \right )^{2}\right ) e +b^{2} \arctan \left (d x +c \right )^{2} e \left (d x +c \right )-b^{2} \ln \left (1+\left (d x +c \right )^{2}\right ) \arctan \left (d x +c \right ) e +\frac {b^{2} \arctan \left (d x +c \right )^{2} f}{2 d}+\frac {b^{2} f \ln \left (1+\left (d x +c \right )^{2}\right )}{2 d}-\frac {i b^{2} \ln \left (d x +c -i\right )^{2} c f}{4 d}+\frac {i b^{2} \ln \left (d x +c +i\right )^{2} c f}{4 d}-\frac {i b^{2} \dilog \left (-\frac {i \left (d x +c +i\right )}{2}\right ) c f}{2 d}+\frac {i b^{2} \dilog \left (\frac {i \left (d x +c -i\right )}{2}\right ) c f}{2 d}}{d}\) | \(701\) |
default | \(\frac {\frac {a b \arctan \left (d x +c \right ) f \left (d x +c \right )^{2}}{d}-\frac {2 a b \arctan \left (d x +c \right ) f c \left (d x +c \right )}{d}+\frac {i b^{2} \dilog \left (-\frac {i \left (d x +c +i\right )}{2}\right ) e}{2}-\frac {i b^{2} \dilog \left (\frac {i \left (d x +c -i\right )}{2}\right ) e}{2}-\frac {i b^{2} \ln \left (d x +c +i\right )^{2} e}{4}+\frac {i b^{2} \ln \left (d x +c -i\right )^{2} e}{4}+\frac {i b^{2} \ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right ) c f}{2 d}+\frac {i b^{2} \ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right ) c f}{2 d}-\frac {i b^{2} \ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right ) c f}{2 d}-\frac {i b^{2} \ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right ) c f}{2 d}+\frac {a b \ln \left (1+\left (d x +c \right )^{2}\right ) c f}{d}-\frac {b^{2} \arctan \left (d x +c \right )^{2} f c \left (d x +c \right )}{d}+\frac {b^{2} \ln \left (1+\left (d x +c \right )^{2}\right ) \arctan \left (d x +c \right ) c f}{d}-\frac {a^{2} \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}+\frac {i b^{2} \ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right ) e}{2}+\frac {i b^{2} \ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right ) e}{2}-\frac {i b^{2} \ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right ) e}{2}-\frac {i b^{2} \ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right ) e}{2}+\frac {b^{2} \arctan \left (d x +c \right )^{2} f \left (d x +c \right )^{2}}{2 d}-\frac {b^{2} \arctan \left (d x +c \right ) f \left (d x +c \right )}{d}+2 e \left (d x +c \right ) a b \arctan \left (d x +c \right )+\frac {a b f \arctan \left (d x +c \right )}{d}-\frac {a b f \left (d x +c \right )}{d}-a b \ln \left (1+\left (d x +c \right )^{2}\right ) e +b^{2} \arctan \left (d x +c \right )^{2} e \left (d x +c \right )-b^{2} \ln \left (1+\left (d x +c \right )^{2}\right ) \arctan \left (d x +c \right ) e +\frac {b^{2} \arctan \left (d x +c \right )^{2} f}{2 d}+\frac {b^{2} f \ln \left (1+\left (d x +c \right )^{2}\right )}{2 d}-\frac {i b^{2} \ln \left (d x +c -i\right )^{2} c f}{4 d}+\frac {i b^{2} \ln \left (d x +c +i\right )^{2} c f}{4 d}-\frac {i b^{2} \dilog \left (-\frac {i \left (d x +c +i\right )}{2}\right ) c f}{2 d}+\frac {i b^{2} \dilog \left (\frac {i \left (d x +c -i\right )}{2}\right ) c f}{2 d}}{d}\) | \(701\) |
risch | \(\text {Expression too large to display}\) | \(1208\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
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 [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {atan}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )\, dx \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 = -0.00 \begin {gather*} \int \left (e+f\,x\right )\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________