Optimal. Leaf size=103 \[ \frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b d (c+d x) \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {b d^2 \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3803, 3799,
2221, 2611, 2320, 6724} \begin {gather*} \frac {a (c+d x)^3}{3 d}+\frac {b d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}+\frac {b (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {b (c+d x)^3}{3 d}-\frac {b d^2 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 3803
Rule 6724
Rubi steps
\begin {align*} \int (c+d x)^2 (a+b \tanh (e+f x)) \, dx &=\int \left (a (c+d x)^2+b (c+d x)^2 \tanh (e+f x)\right ) \, dx\\ &=\frac {a (c+d x)^3}{3 d}+b \int (c+d x)^2 \tanh (e+f x) \, dx\\ &=\frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+(2 b) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1+e^{2 (e+f x)}} \, dx\\ &=\frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac {(2 b d) \int (c+d x) \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f}\\ &=\frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {\left (b d^2\right ) \int \text {Li}_2\left (-e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=\frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^3}\\ &=\frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {b d^2 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}\\ \end {align*}
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Mathematica [A]
time = 1.40, size = 104, normalized size = 1.01 \begin {gather*} \frac {2 f^2 \left ((a-b) f x \left (3 c^2+3 c d x+d^2 x^2\right )+3 b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )\right )+6 b d f (c+d x) \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )-3 b d^2 \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{6 f^3} \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. \(289\) vs.
\(2(97)=194\).
time = 2.04, size = 290, normalized size = 2.82
method | result | size |
risch | \(\frac {d^{2} a \,x^{3}}{3}-\frac {b \,d^{2} x^{3}}{3}+d a c \,x^{2}-b c d \,x^{2}+a \,c^{2} x +b \,c^{2} x +\frac {c^{3} a}{3 d}+\frac {b \,c^{3}}{3 d}-\frac {2 b \,c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {b \,c^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}-\frac {2 b \,d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {2 b \,d^{2} e^{2} x}{f^{2}}+\frac {4 b \,d^{2} e^{3}}{3 f^{3}}+\frac {b \,d^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x^{2}}{f}+\frac {b \,d^{2} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right ) x}{f^{2}}-\frac {b \,d^{2} \polylog \left (3, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{3}}+\frac {4 b d e c \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {4 b c d e x}{f}-\frac {2 b c d \,e^{2}}{f^{2}}+\frac {2 b c d \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f}+\frac {b c d \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{f^{2}}\) | \(290\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 185, normalized size = 1.80 \begin {gather*} \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{3} \, b d^{2} x^{3} + a c d x^{2} + b c d x^{2} + a c^{2} x + \frac {b c^{2} \log \left (\cosh \left (f x + e\right )\right )}{f} + \frac {{\left (2 \, f x \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right )\right )} b c d}{f^{2}} + \frac {{\left (2 \, f^{2} x^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )})\right )} b d^{2}}{2 \, f^{3}} - \frac {2 \, {\left (b d^{2} f^{3} x^{3} + 3 \, b c d f^{3} x^{2}\right )}}{3 \, f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.45, size = 530, normalized size = 5.15 \begin {gather*} \frac {{\left (a - b\right )} d^{2} f^{3} x^{3} + 3 \, {\left (a - b\right )} c d f^{3} x^{2} + 3 \, {\left (a - b\right )} c^{2} f^{3} x - 6 \, b d^{2} {\rm polylog}\left (3, i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right ) - 6 \, b d^{2} {\rm polylog}\left (3, -i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right ) + 6 \, {\left (b d^{2} f x + b c d f\right )} {\rm Li}_2\left (i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right ) + 6 \, {\left (b d^{2} f x + b c d f\right )} {\rm Li}_2\left (-i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right ) + 3 \, {\left (b c^{2} f^{2} - 2 \, b c d f \cosh \left (1\right ) + b d^{2} \cosh \left (1\right )^{2} + b d^{2} \sinh \left (1\right )^{2} - 2 \, {\left (b c d f - b d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + i\right ) + 3 \, {\left (b c^{2} f^{2} - 2 \, b c d f \cosh \left (1\right ) + b d^{2} \cosh \left (1\right )^{2} + b d^{2} \sinh \left (1\right )^{2} - 2 \, {\left (b c d f - b d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - i\right ) + 3 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + 2 \, b c d f \cosh \left (1\right ) - b d^{2} \cosh \left (1\right )^{2} - b d^{2} \sinh \left (1\right )^{2} + 2 \, {\left (b c d f - b d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 1\right ) + 3 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + 2 \, b c d f \cosh \left (1\right ) - b d^{2} \cosh \left (1\right )^{2} - b d^{2} \sinh \left (1\right )^{2} + 2 \, {\left (b c d f - b d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (-i \, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - i \, \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 1\right )}{3 \, f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tanh {\left (e + f x \right )}\right ) \left (c + d x\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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