Optimal. Leaf size=127 \[ b^2 c x+\frac {1}{2} b^2 d x^2+\frac {a^2 (c+d x)^2}{2 d}-\frac {a b (c+d x)^2}{d}-\frac {b^2 (c+d x) \coth (e+f x)}{f}+\frac {2 a b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^2 d \log (\sinh (e+f x))}{f^2}+\frac {a b d \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, 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 = {3803, 3797,
2221, 2317, 2438, 3801, 3556} \begin {gather*} \frac {a^2 (c+d x)^2}{2 d}+\frac {2 a b (c+d x) \log \left (1-e^{2 (e+f x)}\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) \coth (e+f x)}{f}+b^2 c x+\frac {b^2 d \log (\sinh (e+f x))}{f^2}+\frac {1}{2} b^2 d x^2 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2317
Rule 2438
Rule 3556
Rule 3797
Rule 3801
Rule 3803
Rubi steps
\begin {align*} \int (c+d x) (a+b \coth (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)+2 a b (c+d x) \coth (e+f x)+b^2 (c+d x) \coth ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^2}{2 d}+(2 a b) \int (c+d x) \coth (e+f x) \, dx+b^2 \int (c+d x) \coth ^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) \coth (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 \coth (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 {b^2 (c+d x) \coth (e+f x)}{f}+\frac {2 a b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^2 d \log (\sinh (e+f x))}{f^2}-\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 {b^2 (c+d x) \coth (e+f x)}{f}+\frac {2 a b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^2 d \log (\sinh (e+f x))}{f^2}-\frac {(a b d) \text {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 {b^2 (c+d x) \coth (e+f x)}{f}+\frac {2 a b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^2 d \log (\sinh (e+f x))}{f^2}+\frac {a b d \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.12, size = 192, normalized size = 1.51 \begin {gather*} \frac {(a+b \coth (e+f x))^2 \sinh (e+f x) \left (-2 b^2 f (c+d x) \cosh (e+f x)+\left (-\left ((e+f x) \left (-2 a b d (e+f x)+a^2 (d e-2 c f-d f x)+b^2 (d e-2 c f-d f x)\right )\right )+4 a b d (e+f x) \log \left (1-e^{-2 (e+f x)}\right )+2 b (b d-2 a d e+2 a c f) \log (\sinh (e+f x))\right ) \sinh (e+f x)-2 a b d \text {PolyLog}\left (2,e^{-2 (e+f x)}\right ) \sinh (e+f x)\right )}{2 f^2 (b \cosh (e+f x)+a \sinh (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(317\) vs.
\(2(123)=246\).
time = 2.50, size = 318, normalized size = 2.50
method | result | size |
risch | \(\frac {d \,a^{2} x^{2}}{2}-a b d \,x^{2}+\frac {b^{2} d \,x^{2}}{2}+a^{2} c x +2 a b c x +b^{2} c x -\frac {2 \left (d x +c \right ) b^{2}}{f \left ({\mathrm e}^{2 f x +2 e}-1\right )}-\frac {2 b^{2} d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {b^{2} d \ln \left ({\mathrm e}^{f x +e}+1\right )}{f^{2}}+\frac {b^{2} d \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}-\frac {4 b a c \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {2 b a c \ln \left ({\mathrm e}^{f x +e}+1\right )}{f}+\frac {2 b a c \ln \left ({\mathrm e}^{f x +e}-1\right )}{f}+\frac {4 b a d e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {2 b a d e \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}+\frac {2 b \ln \left (1-{\mathrm e}^{f x +e}\right ) a d x}{f}+\frac {2 b \ln \left (1-{\mathrm e}^{f x +e}\right ) a d e}{f^{2}}+\frac {2 b \ln \left ({\mathrm e}^{f x +e}+1\right ) a d x}{f}-\frac {4 b a d e x}{f}-\frac {2 b a d \,e^{2}}{f^{2}}+\frac {2 b a d \polylog \left (2, -{\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {2 b a d \polylog \left (2, {\mathrm e}^{f x +e}\right )}{f^{2}}\) | \(318\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.35, size = 250, normalized size = 1.97 \begin {gather*} \frac {1}{2} \, a^{2} d x^{2} - 2 \, a b d x^{2} + a^{2} c x - \frac {2 \, b^{2} d x}{f} + \frac {2 \, a b c \log \left (\sinh \left (f x + e\right )\right )}{f} + \frac {2 \, {\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 )} a b d}{f^{2}} + \frac {2 \, {\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 )} a b d}{f^{2}} + \frac {b^{2} d \log \left (e^{\left (f x + e\right )} + 1\right )}{f^{2}} + \frac {b^{2} d \log \left (e^{\left (f x + e\right )} - 1\right )}{f^{2}} - \frac {2 \, {\left (c f + 2 \, d\right )} b^{2} x + 4 \, b^{2} c + {\left (2 \, a b d f + b^{2} d f\right )} x^{2} - {\left (2 \, b^{2} c f x e^{\left (2 \, e\right )} + {\left (2 \, a b d f + b^{2} d f\right )} x^{2} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{2 \, {\left (f e^{\left (2 \, f x + 2 \, e\right )} - f\right )}} \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. 1159 vs.
\(2 (126) = 252\).
time = 0.39, size = 1159, normalized size = 9.13 \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 )\, 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 = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^2\,\left (c+d\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________