Optimal. Leaf size=209 \[ \frac {a^2 (c+d x)^3}{3 d}+\frac {2 a b d (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {2 a b (c+d x)^3}{3 d}-\frac {a b d^2 \text {Li}_3\left (e^{2 (e+f x)}\right )}{f^3}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac {b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^2 (c+d x)^2}{f}+\frac {b^2 (c+d x)^3}{3 d}+\frac {b^2 d^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^3} \]
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Rubi [A] time = 0.40, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3722, 3716, 2190, 2531, 2282, 6589, 3720, 2279, 2391, 32} \[ \frac {2 a b d (c+d x) \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {a b d^2 \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3}+\frac {b^2 d^2 \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {a^2 (c+d x)^3}{3 d}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {2 a b (c+d x)^3}{3 d}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac {b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^2 (c+d x)^2}{f}+\frac {b^2 (c+d x)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2190
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 3716
Rule 3720
Rule 3722
Rule 6589
Rubi steps
\begin {align*} \int (c+d x)^2 (a+b \coth (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^2+2 a b (c+d x)^2 \coth (e+f x)+b^2 (c+d x)^2 \coth ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^3}{3 d}+(2 a b) \int (c+d x)^2 \coth (e+f x) \, dx+b^2 \int (c+d x)^2 \coth ^2(e+f x) \, dx\\ &=\frac {a^2 (c+d x)^3}{3 d}-\frac {2 a b (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \coth (e+f x)}{f}-(4 a b) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1-e^{2 (e+f x)}} \, dx+b^2 \int (c+d x)^2 \, dx+\frac {\left (2 b^2 d\right ) \int (c+d x) \coth (e+f x) \, dx}{f}\\ &=-\frac {b^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {2 a b (c+d x)^3}{3 d}+\frac {b^2 (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \coth (e+f x)}{f}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {(4 a b d) \int (c+d x) \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}-\frac {\left (4 b^2 d\right ) \int \frac {e^{2 (e+f x)} (c+d x)}{1-e^{2 (e+f x)}} \, dx}{f}\\ &=-\frac {b^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {2 a b (c+d x)^3}{3 d}+\frac {b^2 (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \coth (e+f x)}{f}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {2 a b d (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^2}-\frac {\left (2 a b d^2\right ) \int \text {Li}_2\left (e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac {\left (2 b^2 d^2\right ) \int \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac {b^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {2 a b (c+d x)^3}{3 d}+\frac {b^2 (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \coth (e+f x)}{f}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {2 a b d (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^2}-\frac {\left (a b d^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{f^3}-\frac {\left (b^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{f^3}\\ &=-\frac {b^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {2 a b (c+d x)^3}{3 d}+\frac {b^2 (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \coth (e+f x)}{f}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^2 d^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac {2 a b d (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^2}-\frac {a b d^2 \text {Li}_3\left (e^{2 (e+f x)}\right )}{f^3}\\ \end {align*}
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Mathematica [B] time = 10.33, size = 478, normalized size = 2.29 \[ \frac {1}{3} x \text {csch}(e) \left (3 c^2+3 c d x+d^2 x^2\right ) \left (a^2 \sinh (e)+2 a b \cosh (e)+b^2 \sinh (e)\right )+\frac {2}{3} b \left (-\frac {3 d \text {Li}_2\left (-e^{-e-f x}\right ) (2 a c f+b d)}{f^3}-\frac {3 d \text {Li}_2\left (e^{-e-f x}\right ) (2 a c f+b d)}{f^3}+\frac {3 d x \log \left (1-e^{-e-f x}\right ) (2 a c f+b d)}{f^2}+\frac {3 d x \log \left (e^{-e-f x}+1\right ) (2 a c f+b d)}{f^2}-\frac {3 c \left (f x-\log \left (1-e^{e+f x}\right )\right ) (a c f+b d)}{f^2}-\frac {3 c \left (f x-\log \left (e^{e+f x}+1\right )\right ) (a c f+b d)}{f^2}-\frac {3 d x^2 (2 a c f+b d)}{\left (e^{2 e}-1\right ) f}-\frac {6 c x (a c f+b d)}{\left (e^{2 e}-1\right ) f}-\frac {6 a d^2 \left (f x \text {Li}_2\left (-e^{-e-f x}\right )+\text {Li}_3\left (-e^{-e-f x}\right )\right )}{f^3}-\frac {6 a d^2 \left (f x \text {Li}_2\left (e^{-e-f x}\right )+\text {Li}_3\left (e^{-e-f x}\right )\right )}{f^3}+\frac {3 a d^2 x^2 \log \left (1-e^{-e-f x}\right )}{f}+\frac {3 a d^2 x^2 \log \left (e^{-e-f x}+1\right )}{f}-\frac {2 a d^2 x^3}{e^{2 e}-1}\right )+\frac {\text {csch}(e) \text {csch}(e+f x) \left (b^2 c^2 \sinh (f x)+2 b^2 c d x \sinh (f x)+b^2 d^2 x^2 \sinh (f x)\right )}{f} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.45, size = 1854, normalized size = 8.87 \[ \text {result too large to display} \]
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 \[ \int {\left (d x + c\right )}^{2} {\left (b \coth \left (f x + e\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.59, size = 793, normalized size = 3.79 \[ -\frac {8 b a c d e x}{f}+b^{2} c d \,x^{2}+a^{2} c d \,x^{2}+\frac {2 b a \,c^{2} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f}+\frac {2 b a \,c^{2} \ln \left ({\mathrm e}^{f x +e}+1\right )}{f}+\frac {2 b^{2} c d \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}+\frac {2 b^{2} c d \ln \left ({\mathrm e}^{f x +e}+1\right )}{f^{2}}+\frac {2 b^{2} d^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) x}{f^{2}}+\frac {2 b^{2} d^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) e}{f^{3}}+\frac {2 b^{2} d^{2} \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{f^{2}}-\frac {2 b^{2} d^{2} e \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{3}}+\frac {4 b^{2} d^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {4 b a \,c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f}-\frac {4 b^{2} c d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {4 b a \,d^{2} \polylog \left (3, {\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {4 b a \,d^{2} \polylog \left (3, -{\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {4 b \ln \left (1-{\mathrm e}^{f x +e}\right ) a c d e}{f^{2}}-\frac {4 b a c d e \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}+\frac {4 b \ln \left (1-{\mathrm e}^{f x +e}\right ) a c d x}{f}+\frac {4 b \ln \left ({\mathrm e}^{f x +e}+1\right ) a c d x}{f}+\frac {8 b a c d e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {2 b^{2} \left (d^{2} x^{2}+2 c d x +c^{2}\right )}{f \left ({\mathrm e}^{2 f x +2 e}-1\right )}+\frac {a^{2} d^{2} x^{3}}{3}+\frac {b^{2} d^{2} x^{3}}{3}+c^{2} a^{2} x +c^{2} b^{2} x -\frac {2 b^{2} d^{2} x^{2}}{f}-\frac {2 b^{2} d^{2} e^{2}}{f^{3}}+\frac {2 b^{2} d^{2} \polylog \left (2, {\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {2 b^{2} d^{2} \polylog \left (2, -{\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {2 a b \,d^{2} x^{3}}{3}+2 c^{2} a b x -2 a b c d \,x^{2}-\frac {4 b^{2} d^{2} e x}{f^{2}}+\frac {8 b a \,d^{2} e^{3}}{3 f^{3}}+\frac {4 b a \,d^{2} e^{2} x}{f^{2}}-\frac {4 b a c d \,e^{2}}{f^{2}}-\frac {2 b a \,d^{2} e^{2} \ln \left (1-{\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {2 b a \,d^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) x^{2}}{f}+\frac {2 b a \,d^{2} \ln \left ({\mathrm e}^{f x +e}+1\right ) x^{2}}{f}+\frac {2 b a \,d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{3}}-\frac {4 b a \,d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {4 b a c d \polylog \left (2, {\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {4 b a c d \polylog \left (2, -{\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {4 b a \,d^{2} \polylog \left (2, {\mathrm e}^{f x +e}\right ) x}{f^{2}}+\frac {4 b a \,d^{2} \polylog \left (2, -{\mathrm e}^{f x +e}\right ) x}{f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 494, normalized size = 2.36 \[ \frac {1}{3} \, a^{2} d^{2} x^{3} + a^{2} c d x^{2} + a^{2} c^{2} x - \frac {4 \, b^{2} c d x}{f} + \frac {2 \, a b c^{2} \log \left (\sinh \left (f x + e\right )\right )}{f} + \frac {2 \, b^{2} c d \log \left (e^{\left (f x + e\right )} + 1\right )}{f^{2}} + \frac {2 \, b^{2} c d \log \left (e^{\left (f x + e\right )} - 1\right )}{f^{2}} + \frac {2 \, {\left (f^{2} x^{2} \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (f x + e\right )})\right )} a b d^{2}}{f^{3}} + \frac {2 \, {\left (f^{2} x^{2} \log \left (-e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (f x + e\right )})\right )} a b d^{2}}{f^{3}} - \frac {6 \, b^{2} c^{2} + 3 \, {\left (c^{2} f + 4 \, c d\right )} b^{2} x + {\left (2 \, a b d^{2} f + b^{2} d^{2} f\right )} x^{3} + 3 \, {\left (2 \, a b c d f + {\left (c d f + 2 \, d^{2}\right )} b^{2}\right )} x^{2} - {\left (3 \, b^{2} c^{2} f x e^{\left (2 \, e\right )} + {\left (2 \, a b d^{2} f e^{\left (2 \, e\right )} + b^{2} d^{2} f e^{\left (2 \, e\right )}\right )} x^{3} + 3 \, {\left (2 \, a b c d f e^{\left (2 \, e\right )} + b^{2} c d f e^{\left (2 \, e\right )}\right )} x^{2}\right )} e^{\left (2 \, f x\right )}}{3 \, {\left (f e^{\left (2 \, f x + 2 \, e\right )} - f\right )}} + \frac {2 \, {\left (2 \, a b c d f + b^{2} d^{2}\right )} {\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )}}{f^{3}} + \frac {2 \, {\left (2 \, a b c d f + b^{2} d^{2}\right )} {\left (f x \log \left (-e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (f x + e\right )}\right )\right )}}{f^{3}} - \frac {2 \, {\left (2 \, a b d^{2} f^{3} x^{3} + 3 \, {\left (2 \, a b c d f + b^{2} d^{2}\right )} f^{2} x^{2}\right )}}{3 \, f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2 \,d x \]
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 \[ \int \left (a + b \coth {\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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