Optimal. Leaf size=72 \[ \frac{e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )}{4 d}+\frac{b e^3 (c+d x)^3}{12 d}-\frac{b e^3 \tanh ^{-1}(c+d x)}{4 d}+\frac{1}{4} b e^3 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0666851, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {6107, 12, 5916, 302, 206} \[ \frac{e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )}{4 d}+\frac{b e^3 (c+d x)^3}{12 d}-\frac{b e^3 \tanh ^{-1}(c+d x)}{4 d}+\frac{1}{4} b e^3 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6107
Rule 12
Rule 5916
Rule 302
Rule 206
Rubi steps
\begin{align*} \int (c e+d e x)^3 \left (a+b \tanh ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int e^3 x^3 \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int x^3 \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )}{4 d}-\frac{\left (b e^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{1-x^2} \, dx,x,c+d x\right )}{4 d}\\ &=\frac{e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )}{4 d}-\frac{\left (b e^3\right ) \operatorname{Subst}\left (\int \left (-1-x^2+\frac{1}{1-x^2}\right ) \, dx,x,c+d x\right )}{4 d}\\ &=\frac{1}{4} b e^3 x+\frac{b e^3 (c+d x)^3}{12 d}+\frac{e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )}{4 d}-\frac{\left (b e^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,c+d x\right )}{4 d}\\ &=\frac{1}{4} b e^3 x+\frac{b e^3 (c+d x)^3}{12 d}-\frac{b e^3 \tanh ^{-1}(c+d x)}{4 d}+\frac{e^3 (c+d x)^4 \left (a+b \tanh ^{-1}(c+d x)\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 0.10449, size = 78, normalized size = 1.08 \[ \frac{e^3 \left (6 a (c+d x)^4+2 b (c+d x)^3+6 b (c+d x)+3 b \log (-c-d x+1)-3 b \log (c+d x+1)+6 b (c+d x)^4 \tanh ^{-1}(c+d x)\right )}{24 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.035, size = 242, normalized size = 3.4 \begin{align*}{\frac{{d}^{3}{x}^{4}a{e}^{3}}{4}}+{d}^{2}{x}^{3}ac{e}^{3}+{\frac{3\,d{x}^{2}a{c}^{2}{e}^{3}}{2}}+xa{c}^{3}{e}^{3}+{\frac{a{c}^{4}{e}^{3}}{4\,d}}+{\frac{{d}^{3}{\it Artanh} \left ( dx+c \right ){x}^{4}b{e}^{3}}{4}}+{d}^{2}{\it Artanh} \left ( dx+c \right ){x}^{3}bc{e}^{3}+{\frac{3\,d{\it Artanh} \left ( dx+c \right ){x}^{2}b{c}^{2}{e}^{3}}{2}}+{\it Artanh} \left ( dx+c \right ) xb{c}^{3}{e}^{3}+{\frac{{\it Artanh} \left ( dx+c \right ) b{c}^{4}{e}^{3}}{4\,d}}+{\frac{{d}^{2}{x}^{3}b{e}^{3}}{12}}+{\frac{d{x}^{2}bc{e}^{3}}{4}}+{\frac{xb{c}^{2}{e}^{3}}{4}}+{\frac{b{c}^{3}{e}^{3}}{12\,d}}+{\frac{b{e}^{3}x}{4}}+{\frac{bc{e}^{3}}{4\,d}}+{\frac{{e}^{3}b\ln \left ( dx+c-1 \right ) }{8\,d}}-{\frac{{e}^{3}b\ln \left ( dx+c+1 \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.995547, size = 482, normalized size = 6.69 \begin{align*} \frac{1}{4} \, a d^{3} e^{3} x^{4} + a c d^{2} e^{3} x^{3} + \frac{3}{2} \, a c^{2} d e^{3} x^{2} + \frac{3}{4} \,{\left (2 \, x^{2} \operatorname{artanh}\left (d x + c\right ) + d{\left (\frac{2 \, x}{d^{2}} - \frac{{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac{{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} b c^{2} d e^{3} + \frac{1}{2} \,{\left (2 \, x^{3} \operatorname{artanh}\left (d x + c\right ) + d{\left (\frac{d x^{2} - 4 \, c x}{d^{3}} + \frac{{\left (c^{3} + 3 \, c^{2} + 3 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{4}} - \frac{{\left (c^{3} - 3 \, c^{2} + 3 \, c - 1\right )} \log \left (d x + c - 1\right )}{d^{4}}\right )}\right )} b c d^{2} e^{3} + \frac{1}{24} \,{\left (6 \, x^{4} \operatorname{artanh}\left (d x + c\right ) + d{\left (\frac{2 \,{\left (d^{2} x^{3} - 3 \, c d x^{2} + 3 \,{\left (3 \, c^{2} + 1\right )} x\right )}}{d^{4}} - \frac{3 \,{\left (c^{4} + 4 \, c^{3} + 6 \, c^{2} + 4 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{5}} + \frac{3 \,{\left (c^{4} - 4 \, c^{3} + 6 \, c^{2} - 4 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{5}}\right )}\right )} b d^{3} e^{3} + a c^{3} e^{3} x + \frac{{\left (2 \,{\left (d x + c\right )} \operatorname{artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b c^{3} e^{3}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.23478, size = 342, normalized size = 4.75 \begin{align*} \frac{6 \, a d^{4} e^{3} x^{4} + 2 \,{\left (12 \, a c + b\right )} d^{3} e^{3} x^{3} + 6 \,{\left (6 \, a c^{2} + b c\right )} d^{2} e^{3} x^{2} + 6 \,{\left (4 \, a c^{3} + b c^{2} + b\right )} d e^{3} x + 3 \,{\left (b d^{4} e^{3} x^{4} + 4 \, b c d^{3} e^{3} x^{3} + 6 \, b c^{2} d^{2} e^{3} x^{2} + 4 \, b c^{3} d e^{3} x +{\left (b c^{4} - b\right )} e^{3}\right )} \log \left (-\frac{d x + c + 1}{d x + c - 1}\right )}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 26.819, size = 231, normalized size = 3.21 \begin{align*} \begin{cases} a c^{3} e^{3} x + \frac{3 a c^{2} d e^{3} x^{2}}{2} + a c d^{2} e^{3} x^{3} + \frac{a d^{3} e^{3} x^{4}}{4} + \frac{b c^{4} e^{3} \operatorname{atanh}{\left (c + d x \right )}}{4 d} + b c^{3} e^{3} x \operatorname{atanh}{\left (c + d x \right )} + \frac{3 b c^{2} d e^{3} x^{2} \operatorname{atanh}{\left (c + d x \right )}}{2} + \frac{b c^{2} e^{3} x}{4} + b c d^{2} e^{3} x^{3} \operatorname{atanh}{\left (c + d x \right )} + \frac{b c d e^{3} x^{2}}{4} + \frac{b d^{3} e^{3} x^{4} \operatorname{atanh}{\left (c + d x \right )}}{4} + \frac{b d^{2} e^{3} x^{3}}{12} + \frac{b e^{3} x}{4} - \frac{b e^{3} \operatorname{atanh}{\left (c + d x \right )}}{4 d} & \text{for}\: d \neq 0 \\c^{3} e^{3} x \left (a + b \operatorname{atanh}{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.29449, size = 362, normalized size = 5.03 \begin{align*} \frac{3 \, b d^{4} x^{4} e^{3} \log \left (-\frac{d x + c + 1}{d x + c - 1}\right ) + 6 \, a d^{4} x^{4} e^{3} + 12 \, b c d^{3} x^{3} e^{3} \log \left (-\frac{d x + c + 1}{d x + c - 1}\right ) + 24 \, a c d^{3} x^{3} e^{3} + 18 \, b c^{2} d^{2} x^{2} e^{3} \log \left (-\frac{d x + c + 1}{d x + c - 1}\right ) + 36 \, a c^{2} d^{2} x^{2} e^{3} + 2 \, b d^{3} x^{3} e^{3} + 12 \, b c^{3} d x e^{3} \log \left (-\frac{d x + c + 1}{d x + c - 1}\right ) + 24 \, a c^{3} d x e^{3} + 6 \, b c d^{2} x^{2} e^{3} + 3 \, b c^{4} e^{3} \log \left (d x + c + 1\right ) - 3 \, b c^{4} e^{3} \log \left (-d x - c + 1\right ) + 6 \, b c^{2} d x e^{3} + 6 \, b d x e^{3} - 3 \, b e^{3} \log \left (d x + c + 1\right ) + 3 \, b e^{3} \log \left (-d x - c + 1\right )}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]