Optimal. Leaf size=117 \[ -\frac {12 d^2 (c+d x) \text {Li}_2\left (-e^{e+f x}\right )}{a f^3}-\frac {6 d (c+d x)^2 \log \left (e^{e+f x}+1\right )}{a f^2}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {(c+d x)^3}{a f}+\frac {12 d^3 \text {Li}_3\left (-e^{e+f x}\right )}{a f^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3318, 4184, 3718, 2190, 2531, 2282, 6589} \[ -\frac {12 d^2 (c+d x) \text {PolyLog}\left (2,-e^{e+f x}\right )}{a f^3}+\frac {12 d^3 \text {PolyLog}\left (3,-e^{e+f x}\right )}{a f^4}-\frac {6 d (c+d x)^2 \log \left (e^{e+f x}+1\right )}{a f^2}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {(c+d x)^3}{a f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2282
Rule 2531
Rule 3318
Rule 3718
Rule 4184
Rule 6589
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{a+a \cosh (e+f x)} \, dx &=\frac {\int (c+d x)^3 \csc ^2\left (\frac {1}{2} (i e+\pi )+\frac {i f x}{2}\right ) \, dx}{2 a}\\ &=\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {(3 d) \int (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a f}\\ &=\frac {(c+d x)^3}{a f}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {(6 d) \int \frac {e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1+e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a f}\\ &=\frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a f^2}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {\left (12 d^2\right ) \int (c+d x) \log \left (1+e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^2}\\ &=\frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a f^2}-\frac {12 d^2 (c+d x) \text {Li}_2\left (-e^{e+f x}\right )}{a f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {\left (12 d^3\right ) \int \text {Li}_2\left (-e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^3}\\ &=\frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a f^2}-\frac {12 d^2 (c+d x) \text {Li}_2\left (-e^{e+f x}\right )}{a f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {\left (12 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a f^4}\\ &=\frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a f^2}-\frac {12 d^2 (c+d x) \text {Li}_2\left (-e^{e+f x}\right )}{a f^3}+\frac {12 d^3 \text {Li}_3\left (-e^{e+f x}\right )}{a f^4}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.30, size = 154, normalized size = 1.32 \[ \frac {2 \cosh \left (\frac {1}{2} (e+f x)\right ) \left (2 \cosh \left (\frac {1}{2} (e+f x)\right ) \left (6 d^2 \left (f (c+d x) \text {Li}_2\left (-e^{-e-f x}\right )+d \text {Li}_3\left (-e^{-e-f x}\right )\right )-\frac {f^3 (c+d x)^3}{e^e+1}-3 d f^2 (c+d x)^2 \log \left (e^{-e-f x}+1\right )\right )+f^3 \text {sech}\left (\frac {e}{2}\right ) (c+d x)^3 \sinh \left (\frac {f x}{2}\right )\right )}{a f^4 (\cosh (e+f x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.64, size = 438, normalized size = 3.74 \[ \frac {2 \, {\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - c^{3} f^{3} + {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2}\right )} \cosh \left (f x + e\right ) - 6 \, {\left (d^{3} f x + c d^{2} f + {\left (d^{3} f x + c d^{2} f\right )} \cosh \left (f x + e\right ) + {\left (d^{3} f x + c d^{2} f\right )} \sinh \left (f x + e\right )\right )} {\rm Li}_2\left (-\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right ) - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2}\right )} \cosh \left (f x + e\right ) + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2}\right )} \sinh \left (f x + e\right )\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1\right ) + 6 \, {\left (d^{3} \cosh \left (f x + e\right ) + d^{3} \sinh \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, -\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right ) + {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2}\right )} \sinh \left (f x + e\right )\right )}}{a f^{4} \cosh \left (f x + e\right ) + a f^{4} \sinh \left (f x + e\right ) + a f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{3}}{a \cosh \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.24, size = 325, normalized size = 2.78 \[ -\frac {2 \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{f a \left ({\mathrm e}^{f x +e}+1\right )}-\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{f x +e}+1\right )}{a \,f^{2}}+\frac {6 d \ln \left ({\mathrm e}^{f x +e}\right ) c^{2}}{a \,f^{2}}+\frac {6 d^{3} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{a \,f^{4}}+\frac {2 d^{3} x^{3}}{a f}-\frac {6 d^{3} e^{2} x}{a \,f^{3}}-\frac {4 d^{3} e^{3}}{a \,f^{4}}-\frac {6 d^{3} \ln \left ({\mathrm e}^{f x +e}+1\right ) x^{2}}{a \,f^{2}}-\frac {12 d^{3} \polylog \left (2, -{\mathrm e}^{f x +e}\right ) x}{a \,f^{3}}+\frac {12 d^{3} \polylog \left (3, -{\mathrm e}^{f x +e}\right )}{a \,f^{4}}-\frac {12 d^{2} c e \ln \left ({\mathrm e}^{f x +e}\right )}{a \,f^{3}}+\frac {6 d^{2} c \,x^{2}}{a f}+\frac {12 d^{2} c e x}{a \,f^{2}}+\frac {6 d^{2} c \,e^{2}}{a \,f^{3}}-\frac {12 d^{2} c \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{a \,f^{2}}-\frac {12 d^{2} c \polylog \left (2, -{\mathrm e}^{f x +e}\right )}{a \,f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.69, size = 228, normalized size = 1.95 \[ 6 \, c^{2} d {\left (\frac {x e^{\left (f x + e\right )}}{a f e^{\left (f x + e\right )} + a f} - \frac {\log \left ({\left (e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a f^{2}}\right )} + \frac {2 \, c^{3}}{{\left (a e^{\left (-f x - e\right )} + a\right )} f} - \frac {2 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2}\right )}}{a f e^{\left (f x + e\right )} + a f} - \frac {12 \, {\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )} c d^{2}}{a f^{3}} - \frac {6 \, {\left (f^{2} x^{2} \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (f x + e\right )})\right )} d^{3}}{a f^{4}} + \frac {2 \, {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2}\right )}}{a f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^3}{a+a\,\mathrm {cosh}\left (e+f\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {c^{3}}{\cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{3} x^{3}}{\cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c d^{2} x^{2}}{\cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c^{2} d x}{\cosh {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________