Optimal. Leaf size=48 \[ \frac {b e x}{2}-\frac {b e \tanh ^{-1}(c+d x)}{2 d}+\frac {e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6242, 12, 6037,
327, 212} \begin {gather*} \frac {e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}-\frac {b e \tanh ^{-1}(c+d x)}{2 d}+\frac {b e x}{2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 212
Rule 327
Rule 6037
Rule 6242
Rubi steps
\begin {align*} \int (c e+d e x) \left (a+b \tanh ^{-1}(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int e x \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int x \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {b e x}{2}+\frac {e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {b e x}{2}-\frac {b e \tanh ^{-1}(c+d x)}{2 d}+\frac {e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 77, normalized size = 1.60 \begin {gather*} \frac {e \left (2 b c+2 a c^2+2 b d x+4 a c d x+2 a d^2 x^2+2 b (c+d x)^2 \tanh ^{-1}(c+d x)+b \log (1-c-d x)-b \log (1+c+d x)\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.04, size = 65, normalized size = 1.35
method | result | size |
derivativedivides | \(\frac {\frac {e \left (d x +c \right )^{2} a}{2}+\frac {b e \left (d x +c \right )^{2} \arctanh \left (d x +c \right )}{2}+\frac {e \left (d x +c \right ) b}{2}+\frac {b e \ln \left (d x +c -1\right )}{4}-\frac {b e \ln \left (d x +c +1\right )}{4}}{d}\) | \(65\) |
default | \(\frac {\frac {e \left (d x +c \right )^{2} a}{2}+\frac {b e \left (d x +c \right )^{2} \arctanh \left (d x +c \right )}{2}+\frac {e \left (d x +c \right ) b}{2}+\frac {b e \ln \left (d x +c -1\right )}{4}-\frac {b e \ln \left (d x +c +1\right )}{4}}{d}\) | \(65\) |
risch | \(\frac {e b x \left (d x +2 c \right ) \ln \left (d x +c +1\right )}{4}-\frac {e d b \,x^{2} \ln \left (-d x -c +1\right )}{4}-\frac {e b x \ln \left (-d x -c +1\right ) c}{2}+\frac {a d e \,x^{2}}{2}+\frac {e \ln \left (-d x -c -1\right ) b \,c^{2}}{4 d}-\frac {e \ln \left (d x +c -1\right ) b \,c^{2}}{4 d}+a c e x +\frac {b e x}{2}-\frac {e b \ln \left (-d x -c -1\right )}{4 d}+\frac {e b \ln \left (d x +c -1\right )}{4 d}\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs.
\(2 (45) = 90\).
time = 0.27, size = 117, normalized size = 2.44 \begin {gather*} \frac {1}{2} \, a d x^{2} e + \frac {1}{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 d e + a c x e + \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 e}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs.
\(2 (45) = 90\).
time = 0.34, size = 121, normalized size = 2.52 \begin {gather*} \frac {2 \, {\left (a d^{2} x^{2} + {\left (2 \, a c + b\right )} d x\right )} \cosh \left (1\right ) + {\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} - b\right )} \cosh \left (1\right ) + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} - b\right )} \sinh \left (1\right )\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right ) + 2 \, {\left (a d^{2} x^{2} + {\left (2 \, a c + b\right )} d x\right )} \sinh \left (1\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs.
\(2 (41) = 82\).
time = 0.68, size = 95, normalized size = 1.98 \begin {gather*} \begin {cases} a c e x + \frac {a d e x^{2}}{2} + \frac {b c^{2} e \operatorname {atanh}{\left (c + d x \right )}}{2 d} + b c e x \operatorname {atanh}{\left (c + d x \right )} + \frac {b d e x^{2} \operatorname {atanh}{\left (c + d x \right )}}{2} + \frac {b e x}{2} - \frac {b e \operatorname {atanh}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {atanh}{\left (c \right )}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 180 vs.
\(2 (42) = 84\).
time = 0.42, size = 180, normalized size = 3.75 \begin {gather*} \frac {1}{2} \, {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )} {\left (\frac {{\left (d x + c + 1\right )} b e \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{{\left (\frac {{\left (d x + c + 1\right )}^{2} d^{2}}{{\left (d x + c - 1\right )}^{2}} - \frac {2 \, {\left (d x + c + 1\right )} d^{2}}{d x + c - 1} + d^{2}\right )} {\left (d x + c - 1\right )}} + \frac {\frac {2 \, {\left (d x + c + 1\right )} a e}{d x + c - 1} + \frac {{\left (d x + c + 1\right )} b e}{d x + c - 1} - b e}{\frac {{\left (d x + c + 1\right )}^{2} d^{2}}{{\left (d x + c - 1\right )}^{2}} - \frac {2 \, {\left (d x + c + 1\right )} d^{2}}{d x + c - 1} + d^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.84, size = 73, normalized size = 1.52 \begin {gather*} \frac {b\,e\,x}{2}+a\,c\,e\,x-\frac {b\,e\,\mathrm {atanh}\left (c+d\,x\right )}{2\,d}+\frac {a\,d\,e\,x^2}{2}+\frac {b\,c^2\,e\,\mathrm {atanh}\left (c+d\,x\right )}{2\,d}+b\,c\,e\,x\,\mathrm {atanh}\left (c+d\,x\right )+\frac {b\,d\,e\,x^2\,\mathrm {atanh}\left (c+d\,x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________