Optimal. Leaf size=209 \[ -\frac {b^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {2 a b (c+d x)^3}{3 d}+\frac {b^2 (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \coth (e+f x)}{f}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^2 d^2 \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {2 a b d (c+d x) \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {a b d^2 \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.28, 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 = {3803, 3797,
2221, 2611, 2320, 6724, 3801, 2317, 2438, 32} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3797
Rule 3801
Rule 3803
Rule 6724
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 ) \text {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 ) \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 3.31, size = 203, normalized size = 0.97 \begin {gather*} \frac {1}{3} \left (x \left (3 c^2+3 c d x+d^2 x^2\right ) \left (a^2+b^2+2 a b \coth (e)\right )+\frac {b \left (-\frac {2 e^{2 e} f^2 x \left (3 b d (2 c+d x)+2 a f \left (3 c^2+3 c d x+d^2 x^2\right )\right )}{-1+e^{2 e}}+6 f (c+d x) (b d+a f (c+d x)) \log \left (1-e^{2 (e+f x)}\right )+3 d (b d+2 a f (c+d x)) \text {PolyLog}\left (2,e^{2 (e+f x)}\right )-3 a d^2 \text {PolyLog}\left (3,e^{2 (e+f x)}\right )\right )}{f^3}+\frac {3 b^2 (c+d x)^2 \text {csch}(e) \text {csch}(e+f x) \sinh (f x)}{f}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(824\) vs.
\(2(203)=406\).
time = 2.85, size = 825, normalized size = 3.95
method | result | size |
risch | \(\frac {4 b \ln \left (1-{\mathrm e}^{f x +e}\right ) a c d x}{f}-\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 e}{f^{2}}-\frac {4 b a \,d^{2} \polylog \left (3, -{\mathrm e}^{f x +e}\right )}{f^{3}}+b^{2} c^{2} x +d \,b^{2} c \,x^{2}+\frac {8 b a c d e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {4 b \ln \left ({\mathrm e}^{f x +e}+1\right ) a c d x}{f}+\frac {d^{2} b^{2} x^{3}}{3}+\frac {a^{2} d^{2} x^{3}}{3}+\frac {4 b a \,d^{2} \polylog \left (2, -{\mathrm e}^{f x +e}\right ) x}{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 )}+a^{2} c d \,x^{2}+\frac {c^{3} a^{2}}{3 d}+\frac {b^{2} c^{3}}{3 d}-\frac {8 b a c d e x}{f}-\frac {2 a b \,d^{2} x^{3}}{3}+2 a b \,c^{2} x -2 a b c d \,x^{2}-\frac {4 b a \,d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {2 b a \,d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}-1\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 (1-{\mathrm e}^{f x +e}\right ) e^{2}}{f^{3}}+\frac {2 b a \,d^{2} \ln \left ({\mathrm e}^{f x +e}+1\right ) x^{2}}{f}+\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}}+a^{2} c^{2} x +\frac {2 b^{2} c d \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}+\frac {4 b^{2} d^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {2 b^{2} c d \ln \left ({\mathrm e}^{f x +e}+1\right )}{f^{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 {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^{2} d^{2} e x}{f^{2}}+\frac {8 b \,e^{3} a \,d^{2}}{3 f^{3}}-\frac {2 b^{2} d^{2} e \ln \left ({\mathrm e}^{f x +e}-1\right )}{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} \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 {4 b a \,d^{2} \polylog \left (3, {\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {4 b \,e^{2} a \,d^{2} x}{f^{2}}-\frac {4 b a c d \,e^{2}}{f^{2}}-\frac {2 b^{2} d^{2} x^{2}}{f}-\frac {2 b^{2} d^{2} e^{2}}{f^{3}}+\frac {2 c^{3} a b}{3 d}+\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}}\) | \(825\) |
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. 503 vs.
\(2 (207) = 414\).
time = 0.35, size = 503, normalized size = 2.41 \begin {gather*} \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 + b^{2} d^{2} f\right )} x^{3} e^{\left (2 \, e\right )} + 3 \, {\left (2 \, a b c d f + b^{2} c d f\right )} x^{2} e^{\left (2 \, e\right )}\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}} \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. 2688 vs.
\(2 (207) = 414\).
time = 0.40, size = 2688, normalized size = 12.86 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \coth {\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________