Optimal. Leaf size=84 \[ -\frac {b \left (3 a^2+3 a b+b^2\right ) \log (\cosh (c+d x))}{d}+\frac {(a+b)^3 \log (\sinh (c+d x))}{d}+\frac {b^2 (3 a+b) \text {sech}^2(c+d x)}{2 d}+\frac {b^3 \text {sech}^4(c+d x)}{4 d} \]
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Rubi [A]
time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4223, 457, 90}
\begin {gather*} -\frac {b \left (3 a^2+3 a b+b^2\right ) \log (\cosh (c+d x))}{d}+\frac {b^2 (3 a+b) \text {sech}^2(c+d x)}{2 d}+\frac {(a+b)^3 \log (\sinh (c+d x))}{d}+\frac {b^3 \text {sech}^4(c+d x)}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 457
Rule 4223
Rubi steps
\begin {align*} \int \coth (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (b+a x^2\right )^3}{x^5 \left (1-x^2\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {(b+a x)^3}{(1-x) x^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {(a+b)^3}{-1+x}+\frac {b^3}{x^3}+\frac {b^2 (3 a+b)}{x^2}+\frac {b \left (3 a^2+3 a b+b^2\right )}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {b \left (3 a^2+3 a b+b^2\right ) \log (\cosh (c+d x))}{d}+\frac {(a+b)^3 \log (\sinh (c+d x))}{d}+\frac {b^2 (3 a+b) \text {sech}^2(c+d x)}{2 d}+\frac {b^3 \text {sech}^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 114, normalized size = 1.36 \begin {gather*} -\frac {2 \cosh ^6(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \left (4 b \left (3 a^2+3 a b+b^2\right ) \log (\cosh (c+d x))-4 (a+b)^3 \log (\sinh (c+d x))-2 b^2 (3 a+b) \text {sech}^2(c+d x)-b^3 \text {sech}^4(c+d x)\right )}{d (a+2 b+a \cosh (2 c+2 d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.83, size = 86, normalized size = 1.02
method | result | size |
derivativedivides | \(\frac {a^{3} \ln \left (\sinh \left (d x +c \right )\right )+3 a^{2} b \ln \left (\tanh \left (d x +c \right )\right )+3 a \,b^{2} \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )+b^{3} \left (\frac {1}{4 \cosh \left (d x +c \right )^{4}}+\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\) | \(86\) |
default | \(\frac {a^{3} \ln \left (\sinh \left (d x +c \right )\right )+3 a^{2} b \ln \left (\tanh \left (d x +c \right )\right )+3 a \,b^{2} \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )+b^{3} \left (\frac {1}{4 \cosh \left (d x +c \right )^{4}}+\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\) | \(86\) |
risch | \(-a^{3} x -\frac {2 a^{3} c}{d}+\frac {2 b^{2} {\mathrm e}^{2 d x +2 c} \left (3 a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+6 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+3 a +b \right )}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{3}}{d}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{2} b}{d}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{3}}{d}-\frac {3 b \ln \left (1+{\mathrm e}^{2 d x +2 c}\right ) a^{2}}{d}-\frac {3 b^{2} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right ) a}{d}-\frac {b^{3} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d}\) | \(241\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 300 vs.
\(2 (80) = 160\).
time = 0.49, size = 300, normalized size = 3.57 \begin {gather*} b^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 4 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} + 3 \, a b^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + 3 \, a^{2} b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d}\right )} + \frac {a^{3} \log \left (\sinh \left (d x + c\right )\right )}{d} \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. 2376 vs.
\(2 (80) = 160\).
time = 0.38, size = 2376, normalized size = 28.29 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 283 vs.
\(2 (80) = 160\).
time = 0.42, size = 283, normalized size = 3.37 \begin {gather*} -\frac {2 \, {\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right ) - 2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right ) - \frac {9 \, a^{2} b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 9 \, a b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 3 \, b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 36 \, a^{2} b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 60 \, a b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 20 \, b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 36 \, a^{2} b + 84 \, a b^{2} + 44 \, b^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right )}^{2}}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.64, size = 360, normalized size = 4.29 \begin {gather*} \frac {2\,\left (b^3+3\,a\,b^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-a^3\,x-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (a^3\,\sqrt {-d^2}+2\,b^3\,\sqrt {-d^2}+6\,a\,b^2\,\sqrt {-d^2}+6\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^6+12\,a^5\,b+48\,a^4\,b^2+76\,a^3\,b^3+60\,a^2\,b^4+24\,a\,b^5+4\,b^6}}\right )\,\sqrt {a^6+12\,a^5\,b+48\,a^4\,b^2+76\,a^3\,b^3+60\,a^2\,b^4+24\,a\,b^5+4\,b^6}}{\sqrt {-d^2}}+\frac {a^3\,\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )}{2\,d}-\frac {8\,b^3}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {2\,\left (3\,a\,b^2-b^3\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {4\,b^3}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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