Optimal. Leaf size=127 \[ \frac {a^2 (c+d x)^2}{2 d}+\frac {2 a b (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {a b (c+d x)^2}{d}+\frac {a b d \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {b^2 (c+d x) \tanh (e+f x)}{f}+b^2 c x+\frac {b^2 d \log (\cosh (e+f x))}{f^2}+\frac {1}{2} b^2 d x^2 \]
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Rubi [A] time = 0.18, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3722, 3718, 2190, 2279, 2391, 3720, 3475} \[ \frac {a b d \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}+\frac {a^2 (c+d x)^2}{2 d}+\frac {2 a b (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {a b (c+d x)^2}{d}-\frac {b^2 (c+d x) \tanh (e+f x)}{f}+b^2 c x+\frac {b^2 d \log (\cosh (e+f x))}{f^2}+\frac {1}{2} b^2 d x^2 \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3475
Rule 3718
Rule 3720
Rule 3722
Rubi steps
\begin {align*} \int (c+d x) (a+b \tanh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)+2 a b (c+d x) \tanh (e+f x)+b^2 (c+d x) \tanh ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^2}{2 d}+(2 a b) \int (c+d x) \tanh (e+f x) \, dx+b^2 \int (c+d x) \tanh ^2(e+f x) \, dx\\ &=\frac {a^2 (c+d x)^2}{2 d}-\frac {a b (c+d x)^2}{d}-\frac {b^2 (c+d x) \tanh (e+f x)}{f}+(4 a b) \int \frac {e^{2 (e+f x)} (c+d x)}{1+e^{2 (e+f x)}} \, dx+b^2 \int (c+d x) \, dx+\frac {\left (b^2 d\right ) \int \tanh (e+f x) \, dx}{f}\\ &=b^2 c x+\frac {1}{2} b^2 d x^2+\frac {a^2 (c+d x)^2}{2 d}-\frac {a b (c+d x)^2}{d}+\frac {2 a b (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^2 d \log (\cosh (e+f x))}{f^2}-\frac {b^2 (c+d x) \tanh (e+f x)}{f}-\frac {(2 a b d) \int \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f}\\ &=b^2 c x+\frac {1}{2} b^2 d x^2+\frac {a^2 (c+d x)^2}{2 d}-\frac {a b (c+d x)^2}{d}+\frac {2 a b (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^2 d \log (\cosh (e+f x))}{f^2}-\frac {b^2 (c+d x) \tanh (e+f x)}{f}-\frac {(a b d) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{f^2}\\ &=b^2 c x+\frac {1}{2} b^2 d x^2+\frac {a^2 (c+d x)^2}{2 d}-\frac {a b (c+d x)^2}{d}+\frac {2 a b (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^2 d \log (\cosh (e+f x))}{f^2}+\frac {a b d \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {b^2 (c+d x) \tanh (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 2.45, size = 192, normalized size = 1.51 \[ \frac {\cosh (e+f x) (a+b \tanh (e+f x))^2 \left (\cosh (e+f x) \left (-\left ((e+f x) \left (a^2 (-2 c f+d e-d f x)-2 a b d (e+f x)+b^2 (-2 c f+d e-d f x)\right )\right )+2 b \log (\cosh (e+f x)) (2 a c f-2 a d e+b d)+4 a b d (e+f x) \log \left (e^{-2 (e+f x)}+1\right )\right )-2 a b d \text {Li}_2\left (-e^{-2 (e+f x)}\right ) \cosh (e+f x)-2 b^2 f (c+d x) \sinh (e+f x)\right )}{2 f^2 (a \cosh (e+f x)+b \sinh (e+f x))^2} \]
Antiderivative was successfully verified.
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fricas [C] time = 1.00, size = 944, normalized size = 7.43 \[ \text {result too large to display} \]
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 \[ \int {\left (d x + c\right )} {\left (b \tanh \left (f x + e\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 221, normalized size = 1.74 \[ \frac {a^{2} d \,x^{2}}{2}-a b d \,x^{2}+\frac {b^{2} d \,x^{2}}{2}+a^{2} c x +2 c a b x +b^{2} c x +\frac {2 b^{2} \left (d x +c \right )}{f \left ({\mathrm e}^{2 f x +2 e}+1\right )}+\frac {b^{2} d \ln \left ({\mathrm e}^{2 f x +2 e}+1\right )}{f^{2}}-\frac {2 b^{2} d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {4 b a c \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {2 b a c \ln \left ({\mathrm e}^{2 f x +2 e}+1\right )}{f}+\frac {4 b d a e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {4 b a d e x}{f}-\frac {2 b a d \,e^{2}}{f^{2}}+\frac {2 b \ln \left ({\mathrm e}^{2 f x +2 e}+1\right ) a d x}{f}+\frac {a b d \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a^{2} d x^{2} + {\left (x^{2} - 4 \, \int \frac {x}{e^{\left (2 \, f x + 2 \, e\right )} + 1}\,{d x}\right )} a b d + b^{2} c {\left (x + \frac {e}{f} - \frac {2}{f {\left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}}\right )} + a^{2} c x + \frac {1}{2} \, b^{2} d {\left (\frac {f x^{2} + {\left (f x^{2} e^{\left (2 \, e\right )} - 4 \, x e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f e^{\left (2 \, f x + 2 \, e\right )} + f} + \frac {2 \, \log \left ({\left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )} e^{\left (-2 \, e\right )}\right )}{f^{2}}\right )} + \frac {2 \, a b c \log \left (\cosh \left (f x + e\right )\right )}{f} \]
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 \[ \int {\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^2\,\left (c+d\,x\right ) \,d x \]
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 \[ \int \left (a + b \tanh {\left (e + f x \right )}\right )^{2} \left (c + d x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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