Optimal. Leaf size=169 \[ \frac {3 d^3 x}{8 a f^3}+\frac {3 d (c+d x)^2}{8 a f^2}+\frac {(c+d x)^3}{4 a f}+\frac {(c+d x)^4}{8 a d}-\frac {3 d^3}{8 f^4 (a+a \coth (e+f x))}-\frac {3 d^2 (c+d x)}{4 f^3 (a+a \coth (e+f x))}-\frac {3 d (c+d x)^2}{4 f^2 (a+a \coth (e+f x))}-\frac {(c+d x)^3}{2 f (a+a \coth (e+f x))} \]
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Rubi [A]
time = 0.13, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3804, 3560, 8}
\begin {gather*} -\frac {3 d^2 (c+d x)}{4 f^3 (a \coth (e+f x)+a)}-\frac {3 d (c+d x)^2}{4 f^2 (a \coth (e+f x)+a)}-\frac {(c+d x)^3}{2 f (a \coth (e+f x)+a)}+\frac {3 d (c+d x)^2}{8 a f^2}+\frac {(c+d x)^3}{4 a f}+\frac {(c+d x)^4}{8 a d}-\frac {3 d^3}{8 f^4 (a \coth (e+f x)+a)}+\frac {3 d^3 x}{8 a f^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3560
Rule 3804
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{a+a \coth (e+f x)} \, dx &=\frac {(c+d x)^4}{8 a d}-\frac {(c+d x)^3}{2 f (a+a \coth (e+f x))}+\frac {(3 d) \int \frac {(c+d x)^2}{a+a \coth (e+f x)} \, dx}{2 f}\\ &=\frac {(c+d x)^3}{4 a f}+\frac {(c+d x)^4}{8 a d}-\frac {3 d (c+d x)^2}{4 f^2 (a+a \coth (e+f x))}-\frac {(c+d x)^3}{2 f (a+a \coth (e+f x))}+\frac {\left (3 d^2\right ) \int \frac {c+d x}{a+a \coth (e+f x)} \, dx}{2 f^2}\\ &=\frac {3 d (c+d x)^2}{8 a f^2}+\frac {(c+d x)^3}{4 a f}+\frac {(c+d x)^4}{8 a d}-\frac {3 d^2 (c+d x)}{4 f^3 (a+a \coth (e+f x))}-\frac {3 d (c+d x)^2}{4 f^2 (a+a \coth (e+f x))}-\frac {(c+d x)^3}{2 f (a+a \coth (e+f x))}+\frac {\left (3 d^3\right ) \int \frac {1}{a+a \coth (e+f x)} \, dx}{4 f^3}\\ &=\frac {3 d (c+d x)^2}{8 a f^2}+\frac {(c+d x)^3}{4 a f}+\frac {(c+d x)^4}{8 a d}-\frac {3 d^3}{8 f^4 (a+a \coth (e+f x))}-\frac {3 d^2 (c+d x)}{4 f^3 (a+a \coth (e+f x))}-\frac {3 d (c+d x)^2}{4 f^2 (a+a \coth (e+f x))}-\frac {(c+d x)^3}{2 f (a+a \coth (e+f x))}+\frac {\left (3 d^3\right ) \int 1 \, dx}{8 a f^3}\\ &=\frac {3 d^3 x}{8 a f^3}+\frac {3 d (c+d x)^2}{8 a f^2}+\frac {(c+d x)^3}{4 a f}+\frac {(c+d x)^4}{8 a d}-\frac {3 d^3}{8 f^4 (a+a \coth (e+f x))}-\frac {3 d^2 (c+d x)}{4 f^3 (a+a \coth (e+f x))}-\frac {3 d (c+d x)^2}{4 f^2 (a+a \coth (e+f x))}-\frac {(c+d x)^3}{2 f (a+a \coth (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 244, normalized size = 1.44 \begin {gather*} \frac {\text {csch}(e+f x) (\cosh (f x)+\sinh (f x)) \left (\left (4 c^3 f^3+6 c^2 d f^2 (1+2 f x)+6 c d^2 f \left (1+2 f x+2 f^2 x^2\right )+d^3 \left (3+6 f x+6 f^2 x^2+4 f^3 x^3\right )\right ) \cosh (2 f x) (\cosh (e)-\sinh (e))+2 f^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) (\cosh (e)+\sinh (e))+\left (4 c^3 f^3+6 c^2 d f^2 (1+2 f x)+6 c d^2 f \left (1+2 f x+2 f^2 x^2\right )+d^3 \left (3+6 f x+6 f^2 x^2+4 f^3 x^3\right )\right ) (-\cosh (e)+\sinh (e)) \sinh (2 f x)\right )}{16 a f^4 (1+\coth (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(928\) vs.
\(2(153)=306\).
time = 1.46, size = 929, normalized size = 5.50
method | result | size |
risch | \(\frac {d^{3} x^{4}}{8 a}+\frac {d^{2} c \,x^{3}}{2 a}+\frac {3 c^{2} d \,x^{2}}{4 a}+\frac {c^{3} x}{2 a}+\frac {c^{4}}{8 a d}+\frac {\left (4 d^{3} x^{3} f^{3}+12 c \,d^{2} f^{3} x^{2}+12 c^{2} d \,f^{3} x +6 d^{3} f^{2} x^{2}+4 c^{3} f^{3}+12 c \,d^{2} f^{2} x +6 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +3 d^{3}\right ) {\mathrm e}^{-2 f x -2 e}}{16 a \,f^{4}}\) | \(165\) |
derivativedivides | \(\text {Expression too large to display}\) | \(929\) |
default | \(\text {Expression too large to display}\) | \(929\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 191, normalized size = 1.13 \begin {gather*} \frac {1}{4} \, c^{3} {\left (\frac {2 \, {\left (f x + e\right )}}{a f} + \frac {e^{\left (-2 \, f x - 2 \, e\right )}}{a f}\right )} + \frac {3 \, {\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} + {\left (2 \, f x + 1\right )} e^{\left (-2 \, f x\right )}\right )} c^{2} d e^{\left (-2 \, e\right )}}{8 \, a f^{2}} + \frac {{\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} + 3 \, {\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x\right )}\right )} c d^{2} e^{\left (-2 \, e\right )}}{8 \, a f^{3}} + \frac {{\left (2 \, f^{4} x^{4} e^{\left (2 \, e\right )} + {\left (4 \, f^{3} x^{3} + 6 \, f^{2} x^{2} + 6 \, f x + 3\right )} e^{\left (-2 \, f x\right )}\right )} d^{3} e^{\left (-2 \, e\right )}}{16 \, a f^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 316 vs.
\(2 (157) = 314\).
time = 0.41, size = 316, normalized size = 1.87 \begin {gather*} \frac {{\left (2 \, d^{3} f^{4} x^{4} + 4 \, c^{3} f^{3} + 6 \, c^{2} d f^{2} + 6 \, c d^{2} f + 4 \, {\left (2 \, c d^{2} f^{4} + d^{3} f^{3}\right )} x^{3} + 3 \, d^{3} + 6 \, {\left (2 \, c^{2} d f^{4} + 2 \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} + 2 \, {\left (4 \, c^{3} f^{4} + 6 \, c^{2} d f^{3} + 6 \, c d^{2} f^{2} + 3 \, d^{3} f\right )} x\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (2 \, d^{3} f^{4} x^{4} - 4 \, c^{3} f^{3} - 6 \, c^{2} d f^{2} - 6 \, c d^{2} f + 4 \, {\left (2 \, c d^{2} f^{4} - d^{3} f^{3}\right )} x^{3} - 3 \, d^{3} + 6 \, {\left (2 \, c^{2} d f^{4} - 2 \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 2 \, {\left (4 \, c^{3} f^{4} - 6 \, c^{2} d f^{3} - 6 \, c d^{2} f^{2} - 3 \, d^{3} f\right )} x\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )}{16 \, {\left (a f^{4} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a f^{4} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 864 vs.
\(2 (144) = 288\).
time = 0.84, size = 864, normalized size = 5.11 \begin {gather*} \begin {cases} \frac {4 c^{3} f^{4} x \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {4 c^{3} f^{4} x}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {4 c^{3} f^{3}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c^{2} d f^{4} x^{2} \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c^{2} d f^{4} x^{2}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} - \frac {6 c^{2} d f^{3} x \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c^{2} d f^{3} x}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c^{2} d f^{2}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {4 c d^{2} f^{4} x^{3} \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {4 c d^{2} f^{4} x^{3}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} - \frac {6 c d^{2} f^{3} x^{2} \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c d^{2} f^{3} x^{2}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} - \frac {6 c d^{2} f^{2} x \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c d^{2} f^{2} x}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c d^{2} f}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {d^{3} f^{4} x^{4} \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {d^{3} f^{4} x^{4}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} - \frac {2 d^{3} f^{3} x^{3} \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {2 d^{3} f^{3} x^{3}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} - \frac {3 d^{3} f^{2} x^{2} \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {3 d^{3} f^{2} x^{2}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} - \frac {3 d^{3} f x \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {3 d^{3} f x}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {3 d^{3}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} & \text {for}\: f \neq 0 \\\frac {c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}}{a \coth {\left (e \right )} + a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 188, normalized size = 1.11 \begin {gather*} \frac {{\left (2 \, d^{3} f^{4} x^{4} e^{\left (2 \, f x + 2 \, e\right )} + 8 \, c d^{2} f^{4} x^{3} e^{\left (2 \, f x + 2 \, e\right )} + 12 \, c^{2} d f^{4} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 4 \, d^{3} f^{3} x^{3} + 8 \, c^{3} f^{4} x e^{\left (2 \, f x + 2 \, e\right )} + 12 \, c d^{2} f^{3} x^{2} + 12 \, c^{2} d f^{3} x + 6 \, d^{3} f^{2} x^{2} + 4 \, c^{3} f^{3} + 12 \, c d^{2} f^{2} x + 6 \, c^{2} d f^{2} + 6 \, d^{3} f x + 6 \, c d^{2} f + 3 \, d^{3}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, a f^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.45, size = 223, normalized size = 1.32 \begin {gather*} \frac {4\,c^3\,f^4\,x+6\,c^2\,d\,f^4\,x^2+6\,c^2\,d\,f^3\,x+4\,c\,d^2\,f^4\,x^3+6\,c\,d^2\,f^3\,x^2+6\,c\,d^2\,f^2\,x+d^3\,f^4\,x^4+2\,d^3\,f^3\,x^3+3\,d^3\,f^2\,x^2+3\,d^3\,f\,x}{8\,a\,f^4}-\frac {4\,c^3\,f^3+12\,c^2\,d\,f^3\,x+6\,c^2\,d\,f^2+12\,c\,d^2\,f^3\,x^2+12\,c\,d^2\,f^2\,x+6\,c\,d^2\,f+4\,d^3\,f^3\,x^3+6\,d^3\,f^2\,x^2+6\,d^3\,f\,x+3\,d^3}{8\,a\,f^4\,\left (\mathrm {coth}\left (e+f\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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