Optimal. Leaf size=75 \[ \frac {a (c+d x)^2}{2 d}+\frac {b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b (c+d x)^2}{2 d}+\frac {b d \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3722, 3716, 2190, 2279, 2391} \[ \frac {b d \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac {a (c+d x)^2}{2 d}+\frac {b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b (c+d x)^2}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 3722
Rubi steps
\begin {align*} \int (c+d x) (a+b \coth (e+f x)) \, dx &=\int (a (c+d x)+b (c+d x) \coth (e+f x)) \, dx\\ &=\frac {a (c+d x)^2}{2 d}+b \int (c+d x) \coth (e+f x) \, dx\\ &=\frac {a (c+d x)^2}{2 d}-\frac {b (c+d x)^2}{2 d}-(2 b) \int \frac {e^{2 (e+f x)} (c+d x)}{1-e^{2 (e+f x)}} \, dx\\ &=\frac {a (c+d x)^2}{2 d}-\frac {b (c+d x)^2}{2 d}+\frac {b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {(b d) \int \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}\\ &=\frac {a (c+d x)^2}{2 d}-\frac {b (c+d x)^2}{2 d}+\frac {b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {(b d) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^2}\\ &=\frac {a (c+d x)^2}{2 d}-\frac {b (c+d x)^2}{2 d}+\frac {b (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b d \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 87, normalized size = 1.16 \[ a c x+\frac {1}{2} a d x^2+\frac {b c (\log (\tanh (e+f x))+\log (\cosh (e+f x)))}{f}+\frac {b d \text {Li}_2\left (e^{2 e+2 f x}\right )}{2 f^2}+\frac {b d x \log \left (1-e^{2 e+2 f x}\right )}{f}-\frac {1}{2} b d x^2 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.41, size = 156, normalized size = 2.08 \[ \frac {{\left (a - b\right )} d f^{2} x^{2} + 2 \, {\left (a - b\right )} c f^{2} x + 2 \, b d {\rm Li}_2\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) + 2 \, b d {\rm Li}_2\left (-\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right ) + 2 \, {\left (b d f x + b c f\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1\right ) - 2 \, {\left (b d e - b c f\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - 1\right ) + 2 \, {\left (b d f x + b d e\right )} \log \left (-\cosh \left (f x + e\right ) - \sinh \left (f x + e\right ) + 1\right )}{2 \, f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} {\left (b \coth \left (f x + e\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.31, size = 201, normalized size = 2.68 \[ \frac {a d \,x^{2}}{2}-\frac {b d \,x^{2}}{2}+a c x +b c x -\frac {2 b c \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {b c \ln \left ({\mathrm e}^{f x +e}-1\right )}{f}+\frac {b c \ln \left ({\mathrm e}^{f x +e}+1\right )}{f}-\frac {2 b d e x}{f}-\frac {b d \,e^{2}}{f^{2}}+\frac {b d \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{f}+\frac {b d \polylog \left (2, -{\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {b d \ln \left (1-{\mathrm e}^{f x +e}\right ) x}{f}+\frac {b d \ln \left (1-{\mathrm e}^{f x +e}\right ) e}{f^{2}}+\frac {b d \polylog \left (2, {\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {2 b d e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {b d e \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a d x^{2} + \frac {1}{2} \, {\left (x^{2} - 2 \, \int \frac {x}{e^{\left (f x + e\right )} + 1}\,{d x} + 2 \, \int \frac {x}{e^{\left (f x + e\right )} - 1}\,{d x}\right )} b d + a c x + \frac {b c \log \left (\sinh \left (f x + e\right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )\,\left (c+d\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \coth {\left (e + f x \right )}\right ) \left (c + d x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________