Optimal. Leaf size=77 \[ -\frac {3 a^2 (a+b) \text {csch}^2(c+d x)}{2 d}-\frac {3 a (a+b)^2 \text {csch}^4(c+d x)}{4 d}-\frac {(a+b)^3 \text {csch}^6(c+d x)}{6 d}+\frac {a^3 \log (\sinh (c+d x))}{d} \]
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Rubi [A]
time = 0.09, antiderivative size = 77, 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, 455, 45}
\begin {gather*} \frac {a^3 \log (\sinh (c+d x))}{d}-\frac {3 a^2 (a+b) \text {csch}^2(c+d x)}{2 d}-\frac {(a+b)^3 \text {csch}^6(c+d x)}{6 d}-\frac {3 a (a+b)^2 \text {csch}^4(c+d x)}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 455
Rule 4223
Rubi steps
\begin {align*} \int \coth ^7(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {x \left (b+a x^2\right )^3}{\left (1-x^2\right )^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {(b+a x)^3}{(1-x)^4} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {(a+b)^3}{(-1+x)^4}+\frac {3 a (a+b)^2}{(-1+x)^3}+\frac {3 a^2 (a+b)}{(-1+x)^2}+\frac {a^3}{-1+x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {3 a^2 (a+b) \text {csch}^2(c+d x)}{2 d}-\frac {3 a (a+b)^2 \text {csch}^4(c+d x)}{4 d}-\frac {(a+b)^3 \text {csch}^6(c+d x)}{6 d}+\frac {a^3 \log (\sinh (c+d x))}{d}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 98, normalized size = 1.27 \begin {gather*} -\frac {2 \left (b+a \cosh ^2(c+d x)\right )^3 \left (18 a^2 (a+b) \text {csch}^2(c+d x)+9 a (a+b)^2 \text {csch}^4(c+d x)+2 (a+b)^3 \text {csch}^6(c+d x)-12 a^3 \log (\sinh (c+d x))\right )}{3 d (a+2 b+a \cosh (2 (c+d x)))^3} \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. \(148\) vs.
\(2(71)=142\).
time = 2.13, size = 149, normalized size = 1.94
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\coth ^{6}\left (d x +c \right )\right )}{6}\right )+3 a^{2} b \left (-\frac {\cosh ^{4}\left (d x +c \right )}{2 \sinh \left (d x +c \right )^{6}}+\frac {\cosh ^{2}\left (d x +c \right )}{2 \sinh \left (d x +c \right )^{6}}-\frac {1}{6 \sinh \left (d x +c \right )^{6}}\right )+3 a \,b^{2} \left (-\frac {\cosh ^{2}\left (d x +c \right )}{4 \sinh \left (d x +c \right )^{6}}+\frac {1}{12 \sinh \left (d x +c \right )^{6}}\right )-\frac {b^{3}}{6 \sinh \left (d x +c \right )^{6}}}{d}\) | \(149\) |
default | \(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\coth ^{6}\left (d x +c \right )\right )}{6}\right )+3 a^{2} b \left (-\frac {\cosh ^{4}\left (d x +c \right )}{2 \sinh \left (d x +c \right )^{6}}+\frac {\cosh ^{2}\left (d x +c \right )}{2 \sinh \left (d x +c \right )^{6}}-\frac {1}{6 \sinh \left (d x +c \right )^{6}}\right )+3 a \,b^{2} \left (-\frac {\cosh ^{2}\left (d x +c \right )}{4 \sinh \left (d x +c \right )^{6}}+\frac {1}{12 \sinh \left (d x +c \right )^{6}}\right )-\frac {b^{3}}{6 \sinh \left (d x +c \right )^{6}}}{d}\) | \(149\) |
risch | \(-a^{3} x -\frac {2 a^{3} c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (9 a^{3} {\mathrm e}^{8 d x +8 c}+9 a^{2} b \,{\mathrm e}^{8 d x +8 c}-18 a^{3} {\mathrm e}^{6 d x +6 c}+18 a \,b^{2} {\mathrm e}^{6 d x +6 c}+34 a^{3} {\mathrm e}^{4 d x +4 c}+30 a^{2} b \,{\mathrm e}^{4 d x +4 c}+12 a \,b^{2} {\mathrm e}^{4 d x +4 c}+16 b^{3} {\mathrm e}^{4 d x +4 c}-18 a^{3} {\mathrm e}^{2 d x +2 c}+18 a \,b^{2} {\mathrm e}^{2 d x +2 c}+9 a^{3}+9 a^{2} b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{6}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{3}}{d}\) | \(220\) |
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. 727 vs.
\(2 (71) = 142\).
time = 0.27, size = 727, normalized size = 9.44 \begin {gather*} \frac {1}{3} \, a^{3} {\left (3 \, x + \frac {3 \, c}{d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 18 \, e^{\left (-4 \, d x - 4 \, c\right )} + 34 \, e^{\left (-6 \, d x - 6 \, c\right )} - 18 \, e^{\left (-8 \, d x - 8 \, c\right )} + 9 \, e^{\left (-10 \, d x - 10 \, c\right )}\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + 2 \, a^{2} b {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} + \frac {10 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} + \frac {3 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + 4 \, a b^{2} {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} + \frac {2 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} + \frac {3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} - \frac {32 \, b^{3}}{3 \, d {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{6}} \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. 2632 vs.
\(2 (71) = 142\).
time = 0.48, size = 2632, normalized size = 34.18 \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. 242 vs.
\(2 (71) = 142\).
time = 0.55, size = 242, normalized size = 3.14 \begin {gather*} -\frac {60 \, {\left (d x + c\right )} a^{3} - 60 \, a^{3} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {147 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} - 522 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 360 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 1485 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 720 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 1580 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1200 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 480 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 640 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1485 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 720 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 522 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 360 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 147 \, a^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{6}}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.61, size = 411, normalized size = 5.34 \begin {gather*} \frac {a^3\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )}{d}-\frac {32\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )}-\frac {32\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{3\,d\,\left (15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}-\frac {6\,\left (a^3+b\,a^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {6\,\left (3\,a^3+5\,a^2\,b+2\,a\,b^2\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,\left (13\,a^3+30\,a^2\,b+21\,a\,b^2+4\,b^3\right )}{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 {4\,\left (11\,a^3+30\,a^2\,b+27\,a\,b^2+8\,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 )}-a^3\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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