Optimal. Leaf size=222 \[ \frac{i b^2 (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{d^2}+\frac{i (d e-c f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac{2 b (d e-c f) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )}{d^2}+\frac{(e+f x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 f}-\frac{a b f x}{d}+\frac{b^2 f \log \left ((c+d x)^2+1\right )}{2 d^2}-\frac{b^2 f (c+d x) \tan ^{-1}(c+d x)}{d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.373148, antiderivative size = 222, normalized size of antiderivative = 1., 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 = {5047, 4864, 4846, 260, 4984, 4884, 4920, 4854, 2402, 2315} \[ \frac{i b^2 (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{d^2}+\frac{i (d e-c f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac{2 b (d e-c f) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )}{d^2}+\frac{(e+f x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 f}-\frac{a b f x}{d}+\frac{b^2 f \log \left ((c+d x)^2+1\right )}{2 d^2}-\frac{b^2 f (c+d x) \tan ^{-1}(c+d x)}{d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5047
Rule 4864
Rule 4846
Rule 260
Rule 4984
Rule 4884
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int (e+f x) \left (a+b \tan ^{-1}(c+d x)\right )^2 \, dx &=\frac{\operatorname{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 \operatorname{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 \operatorname{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) \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,c+d x\right )}{d^2}-\frac{(2 b (d e-c f)) \operatorname{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)) \operatorname{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)) \operatorname{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 ) \operatorname{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 ) \operatorname{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 = 0.379037, size = 264, normalized size = 1.19 \[ \frac{-2 i b^2 (d e-c f) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c+d x)}\right )-a^2 c^2 f+2 a^2 c d e+2 a^2 d^2 e x+a^2 d^2 f x^2-2 b \tan ^{-1}(c+d x) \left (a \left (c^2 f-2 c d e-2 d^2 e x-f \left (d^2 x^2+1\right )\right )-2 b (d e-c f) \log \left (1+e^{2 i \tan ^{-1}(c+d x)}\right )+b f (c+d x)\right )+4 a b d e \log \left (\frac{1}{\sqrt{(c+d x)^2+1}}\right )-4 a b c f \log \left (\frac{1}{\sqrt{(c+d x)^2+1}}\right )-2 a b c f-2 a b d f x+b^2 (c+d x-i) \tan ^{-1}(c+d x)^2 (-c f+2 d e+d f x+i f)-2 b^2 f \log \left (\frac{1}{\sqrt{(c+d x)^2+1}}\right )}{2 d^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.119, size = 748, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (a^{2} f x + a^{2} e +{\left (b^{2} f x + b^{2} e\right )} \arctan \left (d x + c\right )^{2} + 2 \,{\left (a b f x + a b e\right )} \arctan \left (d x + c\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{atan}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}{\left (b \arctan \left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]