Optimal. Leaf size=81 \[ -\frac {(2 a-b) (a+b)^2 \text {csch}^2(c+d x)}{2 d}-\frac {(a+b)^3 \text {csch}^4(c+d x)}{4 d}-\frac {b^3 \log (\cosh (c+d x))}{d}+\frac {\left (a^3+b^3\right ) \log (\sinh (c+d x))}{d} \]
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Rubi [A]
time = 0.09, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4223, 457, 90}
\begin {gather*} \frac {\left (a^3+b^3\right ) \log (\sinh (c+d x))}{d}-\frac {(a+b)^3 \text {csch}^4(c+d x)}{4 d}-\frac {(2 a-b) (a+b)^2 \text {csch}^2(c+d x)}{2 d}-\frac {b^3 \log (\cosh (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 457
Rule 4223
Rubi steps
\begin {align*} \int \coth ^5(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 \left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {(b+a x)^3}{(1-x)^3 x} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {(a+b)^3}{(-1+x)^3}-\frac {(2 a-b) (a+b)^2}{(-1+x)^2}+\frac {-a^3-b^3}{-1+x}+\frac {b^3}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {(2 a-b) (a+b)^2 \text {csch}^2(c+d x)}{2 d}-\frac {(a+b)^3 \text {csch}^4(c+d x)}{4 d}-\frac {b^3 \log (\cosh (c+d x))}{d}+\frac {\left (a^3+b^3\right ) \log (\sinh (c+d x))}{d}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 101, normalized size = 1.25 \begin {gather*} -\frac {2 \left (b+a \cosh ^2(c+d x)\right )^3 \left (2 (2 a-b) (a+b)^2 \text {csch}^2(c+d x)+(a+b)^3 \text {csch}^4(c+d x)+4 b^3 \log (\cosh (c+d x))-4 \left (a^3+b^3\right ) \log (\sinh (c+d x))\right )}{d (a+2 b+a \cosh (2 (c+d x)))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.98, size = 119, normalized size = 1.47
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}\right )+3 a^{2} b \left (-\frac {\cosh ^{2}\left (d x +c \right )}{2 \sinh \left (d x +c \right )^{4}}+\frac {1}{4 \sinh \left (d x +c \right )^{4}}\right )-\frac {3 a \,b^{2}}{4 \sinh \left (d x +c \right )^{4}}+b^{3} \left (-\frac {1}{4 \sinh \left (d x +c \right )^{4}}+\frac {1}{2 \sinh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\) | \(119\) |
default | \(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}\right )+3 a^{2} b \left (-\frac {\cosh ^{2}\left (d x +c \right )}{2 \sinh \left (d x +c \right )^{4}}+\frac {1}{4 \sinh \left (d x +c \right )^{4}}\right )-\frac {3 a \,b^{2}}{4 \sinh \left (d x +c \right )^{4}}+b^{3} \left (-\frac {1}{4 \sinh \left (d x +c \right )^{4}}+\frac {1}{2 \sinh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\) | \(119\) |
risch | \(-a^{3} x -\frac {2 a^{3} c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (2 a^{3} {\mathrm e}^{4 d x +4 c}+3 a^{2} b \,{\mathrm e}^{4 d x +4 c}-b^{3} {\mathrm e}^{4 d x +4 c}-2 a^{3} {\mathrm e}^{2 d x +2 c}+6 a \,b^{2} {\mathrm e}^{2 d x +2 c}+4 b^{3} {\mathrm e}^{2 d x +2 c}+2 a^{3}+3 a^{2} b -b^{3}\right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{3}}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{3}}{d}-\frac {b^{3} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d}\) | \(205\) |
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. 422 vs.
\(2 (77) = 154\).
time = 0.49, size = 422, normalized size = 5.21 \begin {gather*} a^{3} {\left (x + \frac {c}{d} + \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 {4 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 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 )} + 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 )} + 6 \, a^{2} b {\left (\frac {e^{\left (-2 \, d x - 2 \, c\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 )}} + \frac {e^{\left (-6 \, d x - 6 \, c\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 )} - \frac {12 \, a b^{2}}{d {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{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. 1830 vs.
\(2 (77) = 154\).
time = 0.47, size = 1830, normalized size = 22.59 \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. 227 vs.
\(2 (77) = 154\).
time = 0.52, size = 227, normalized size = 2.80 \begin {gather*} -\frac {2 \, b^{3} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right ) - 2 \, {\left (a^{3} + b^{3}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right ) + \frac {3 \, a^{3} {\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} + 4 \, a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 24 \, a^{2} b {\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 )} - 4 \, a^{3} + 48 \, 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 = 384, normalized size = 4.74 \begin {gather*} -a^3\,x-\frac {2\,\left (4\,a^3+9\,a^2\,b+6\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )\,\left (b^3\,d-d\,\left (a^3+b^3\right )\right )}{2\,d^2}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (a^6\,\sqrt {-d^2}+4\,b^6\,\sqrt {-d^2}+4\,a^3\,b^3\,\sqrt {-d^2}\right )}{a^3\,d\,\sqrt {a^6+4\,a^3\,b^3+4\,b^6}+2\,b^3\,d\,\sqrt {a^6+4\,a^3\,b^3+4\,b^6}}\right )\,\sqrt {a^6+4\,a^3\,b^3+4\,b^6}}{\sqrt {-d^2}}-\frac {2\,\left (2\,a^3+3\,a^2\,b-b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {8\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{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 {4\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,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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