Optimal. Leaf size=97 \[ \frac {b f x}{2 d}+\frac {(e+f x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 f}+\frac {b (d e+f-c f)^2 \log (1-c-d x)}{4 d^2 f}-\frac {b (d e-(1+c) f)^2 \log (1+c+d x)}{4 d^2 f} \]
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Rubi [A]
time = 0.12, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6246, 6063,
716, 647, 31} \begin {gather*} \frac {(e+f x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 f}+\frac {b (-c f+d e+f)^2 \log (-c-d x+1)}{4 d^2 f}-\frac {b (d e-(c+1) f)^2 \log (c+d x+1)}{4 d^2 f}+\frac {b f x}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 647
Rule 716
Rule 6063
Rule 6246
Rubi steps
\begin {align*} \int (e+f x) \left (a+b \tanh ^{-1}(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right ) \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 f}-\frac {b \text {Subst}\left (\int \frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^2}{1-x^2} \, dx,x,c+d x\right )}{2 f}\\ &=\frac {(e+f x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 f}-\frac {b \text {Subst}\left (\int \left (-\frac {f^2}{d^2}+\frac {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2+2 f (d e-c f) x}{d^2 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{2 f}\\ &=\frac {b f x}{2 d}+\frac {(e+f x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 f}-\frac {b \text {Subst}\left (\int \frac {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2+2 f (d e-c f) x}{1-x^2} \, dx,x,c+d x\right )}{2 d^2 f}\\ &=\frac {b f x}{2 d}+\frac {(e+f x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 f}-\frac {\left (b (d e+f-c f)^2\right ) \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,c+d x\right )}{4 d^2 f}+\frac {\left (b (d e-(1+c) f)^2\right ) \text {Subst}\left (\int \frac {1}{-1-x} \, dx,x,c+d x\right )}{4 d^2 f}\\ &=\frac {b f x}{2 d}+\frac {(e+f x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 f}+\frac {b (d e+f-c f)^2 \log (1-c-d x)}{4 d^2 f}-\frac {b (d e-(1+c) f)^2 \log (1+c+d x)}{4 d^2 f}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 138, normalized size = 1.42 \begin {gather*} a e x+\frac {b f x}{2 d}+\frac {1}{2} a f x^2+b e x \tanh ^{-1}(c+d x)+\frac {1}{2} b f x^2 \tanh ^{-1}(c+d x)+\frac {b \left (1-2 c+c^2\right ) f \log (1-c-d x)}{4 d^2}+\frac {b \left (-1-2 c-c^2\right ) f \log (1+c+d x)}{4 d^2}+\frac {b e (-((-1+c) \log (1-c-d x))+(1+c) \log (1+c+d x))}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(184\) vs.
\(2(89)=178\).
time = 0.19, size = 185, normalized size = 1.91
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (f c \left (d x +c \right )-e \left (d x +c \right ) d -\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b \arctanh \left (d x +c \right ) f c \left (d x +c \right )}{d}+e \left (d x +c \right ) b \arctanh \left (d x +c \right )+\frac {b \arctanh \left (d x +c \right ) f \left (d x +c \right )^{2}}{2 d}+\frac {b f \left (d x +c \right )}{2 d}-\frac {b \ln \left (d x +c -1\right ) f c}{2 d}+\frac {b e \ln \left (d x +c -1\right )}{2}+\frac {b \ln \left (d x +c -1\right ) f}{4 d}-\frac {b \ln \left (d x +c +1\right ) f c}{2 d}+\frac {b e \ln \left (d x +c +1\right )}{2}-\frac {b \ln \left (d x +c +1\right ) f}{4 d}}{d}\) | \(185\) |
default | \(\frac {-\frac {a \left (f c \left (d x +c \right )-e \left (d x +c \right ) d -\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b \arctanh \left (d x +c \right ) f c \left (d x +c \right )}{d}+e \left (d x +c \right ) b \arctanh \left (d x +c \right )+\frac {b \arctanh \left (d x +c \right ) f \left (d x +c \right )^{2}}{2 d}+\frac {b f \left (d x +c \right )}{2 d}-\frac {b \ln \left (d x +c -1\right ) f c}{2 d}+\frac {b e \ln \left (d x +c -1\right )}{2}+\frac {b \ln \left (d x +c -1\right ) f}{4 d}-\frac {b \ln \left (d x +c +1\right ) f c}{2 d}+\frac {b e \ln \left (d x +c +1\right )}{2}-\frac {b \ln \left (d x +c +1\right ) f}{4 d}}{d}\) | \(185\) |
risch | \(\frac {b x \left (f x +2 e \right ) \ln \left (d x +c +1\right )}{4}-\frac {b f \,x^{2} \ln \left (-d x -c +1\right )}{4}-\frac {b e x \ln \left (-d x -c +1\right )}{2}+\frac {a f \,x^{2}}{2}-\frac {\ln \left (d x +c +1\right ) b \,c^{2} f}{4 d^{2}}+\frac {\ln \left (d x +c +1\right ) b c e}{2 d}+\frac {\ln \left (-d x -c +1\right ) b \,c^{2} f}{4 d^{2}}-\frac {\ln \left (-d x -c +1\right ) b c e}{2 d}+a e x -\frac {\ln \left (d x +c +1\right ) b c f}{2 d^{2}}+\frac {\ln \left (d x +c +1\right ) b e}{2 d}-\frac {\ln \left (-d x -c +1\right ) b c f}{2 d^{2}}+\frac {\ln \left (-d x -c +1\right ) b e}{2 d}+\frac {b f x}{2 d}-\frac {\ln \left (d x +c +1\right ) b f}{4 d^{2}}+\frac {\ln \left (-d x -c +1\right ) b f}{4 d^{2}}\) | \(236\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 111, normalized size = 1.14 \begin {gather*} \frac {1}{2} \, a f x^{2} + \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 f + a 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 e}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 177, normalized size = 1.82 \begin {gather*} \frac {2 \, a d^{2} f x^{2} + 4 \, a d^{2} x \cosh \left (1\right ) + 4 \, a d^{2} x \sinh \left (1\right ) + 2 \, b d f x + {\left (2 \, {\left (b c + b\right )} d \cosh \left (1\right ) + 2 \, {\left (b c + b\right )} d \sinh \left (1\right ) - {\left (b c^{2} + 2 \, b c + b\right )} f\right )} \log \left (d x + c + 1\right ) - {\left (2 \, {\left (b c - b\right )} d \cosh \left (1\right ) + 2 \, {\left (b c - b\right )} d \sinh \left (1\right ) - {\left (b c^{2} - 2 \, b c + b\right )} f\right )} \log \left (d x + c - 1\right ) + {\left (b d^{2} f x^{2} + 2 \, b d^{2} x \cosh \left (1\right ) + 2 \, b d^{2} x \sinh \left (1\right )\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{4 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 173 vs.
\(2 (82) = 164\).
time = 1.04, size = 173, normalized size = 1.78 \begin {gather*} \begin {cases} a e x + \frac {a f x^{2}}{2} - \frac {b c^{2} f \operatorname {atanh}{\left (c + d x \right )}}{2 d^{2}} + \frac {b c e \operatorname {atanh}{\left (c + d x \right )}}{d} - \frac {b c f \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{2}} + \frac {b c f \operatorname {atanh}{\left (c + d x \right )}}{d^{2}} + b e x \operatorname {atanh}{\left (c + d x \right )} + \frac {b f x^{2} \operatorname {atanh}{\left (c + d x \right )}}{2} + \frac {b e \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d} - \frac {b e \operatorname {atanh}{\left (c + d x \right )}}{d} + \frac {b f x}{2 d} - \frac {b f \operatorname {atanh}{\left (c + d x \right )}}{2 d^{2}} & \text {for}\: d \neq 0 \\\left (a + b \operatorname {atanh}{\left (c \right )}\right ) \left (e x + \frac {f x^{2}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 341 vs.
\(2 (89) = 178\).
time = 0.43, size = 341, normalized size = 3.52 \begin {gather*} \frac {1}{2} \, {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )} {\left (\frac {{\left (\frac {{\left (d x + c + 1\right )} b d e}{d x + c - 1} - b d e - \frac {{\left (d x + c + 1\right )} b c f}{d x + c - 1} + b c f + \frac {{\left (d x + c + 1\right )} b f}{d x + c - 1}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{\frac {{\left (d x + c + 1\right )}^{2} d^{3}}{{\left (d x + c - 1\right )}^{2}} - \frac {2 \, {\left (d x + c + 1\right )} d^{3}}{d x + c - 1} + d^{3}} + \frac {\frac {2 \, {\left (d x + c + 1\right )} a d e}{d x + c - 1} - 2 \, a d e - \frac {2 \, {\left (d x + c + 1\right )} a c f}{d x + c - 1} + 2 \, a c f + \frac {2 \, {\left (d x + c + 1\right )} a f}{d x + c - 1} + \frac {{\left (d x + c + 1\right )} b f}{d x + c - 1} - b f}{\frac {{\left (d x + c + 1\right )}^{2} d^{3}}{{\left (d x + c - 1\right )}^{2}} - \frac {2 \, {\left (d x + c + 1\right )} d^{3}}{d x + c - 1} + d^{3}} - \frac {{\left (b d e - b c f\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1} + 1\right )}{d^{3}} + \frac {{\left (b d e - b c f\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{d^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.34, size = 136, normalized size = 1.40 \begin {gather*} a\,e\,x+\frac {a\,f\,x^2}{2}+\frac {b\,e\,\ln \left (c^2+2\,c\,d\,x+d^2\,x^2-1\right )}{2\,d}-\frac {b\,f\,\mathrm {atanh}\left (c+d\,x\right )}{2\,d^2}+\frac {b\,f\,x^2\,\mathrm {atanh}\left (c+d\,x\right )}{2}+\frac {b\,f\,x}{2\,d}+b\,e\,x\,\mathrm {atanh}\left (c+d\,x\right )-\frac {b\,c^2\,f\,\mathrm {atanh}\left (c+d\,x\right )}{2\,d^2}-\frac {b\,c\,f\,\ln \left (c^2+2\,c\,d\,x+d^2\,x^2-1\right )}{2\,d^2}+\frac {b\,c\,e\,\mathrm {atanh}\left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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