Optimal. Leaf size=91 \[ \frac {(a-b)^2 (a+5 b) \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {(3 a-2 b) b^2 \sinh (c+d x)}{d}+\frac {b^3 \sinh ^3(c+d x)}{3 d}+\frac {(a-b)^3 \text {sech}(c+d x) \tanh (c+d x)}{2 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3269, 398, 393,
209} \begin {gather*} \frac {(a+5 b) (a-b)^2 \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {b^2 (3 a-2 b) \sinh (c+d x)}{d}+\frac {(a-b)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}+\frac {b^3 \sinh ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 393
Rule 398
Rule 3269
Rubi steps
\begin {align*} \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^3}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left ((3 a-2 b) b^2+b^3 x^2+\frac {(a-b)^2 (a+2 b)+3 (a-b)^2 b x^2}{\left (1+x^2\right )^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {(3 a-2 b) b^2 \sinh (c+d x)}{d}+\frac {b^3 \sinh ^3(c+d x)}{3 d}+\frac {\text {Subst}\left (\int \frac {(a-b)^2 (a+2 b)+3 (a-b)^2 b x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {(3 a-2 b) b^2 \sinh (c+d x)}{d}+\frac {b^3 \sinh ^3(c+d x)}{3 d}+\frac {(a-b)^3 \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {\left ((a-b)^2 (a+5 b)\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}\\ &=\frac {(a-b)^2 (a+5 b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {(3 a-2 b) b^2 \sinh (c+d x)}{d}+\frac {b^3 \sinh ^3(c+d x)}{3 d}+\frac {(a-b)^3 \text {sech}(c+d x) \tanh (c+d x)}{2 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 5.73, size = 347, normalized size = 3.81 \begin {gather*} \frac {\text {csch}^5(c+d x) \left (-256 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^8(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3+21 \left (36015 a^3+5 a^2 (3224 a+21609 b) \sinh ^2(c+d x)+3 a \left (491 a^2+16120 a b+36015 b^2\right ) \sinh ^4(c+d x)+3 b \left (753 a^2+18280 a b+10805 b^2\right ) \sinh ^6(c+d x)+b^2 (2259 a+17320 b) \sinh ^8(c+d x)+753 b^3 \sinh ^{10}(c+d x)\right )-\frac {315 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (2401 a^3+3 a^2 (625 a+2401 b) \sinh ^2(c+d x)+3 a \left (81 a^2+1875 a b+2401 b^2\right ) \sinh ^4(c+d x)+\left (-47 a^3+585 a^2 b+6057 a b^2+2161 b^3\right ) \sinh ^6(c+d x)+3 b \left (a^2+243 a b+625 b^2\right ) \sinh ^8(c+d x)+3 b^2 (a+81 b) \sinh ^{10}(c+d x)+b^3 \sinh ^{12}(c+d x)\right )}{\sqrt {-\sinh ^2(c+d x)}}\right )}{30240 d} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains complex when optimal does not.
time = 1.50, size = 311, normalized size = 3.42
method | result | size |
risch | \(\frac {{\mathrm e}^{3 d x +3 c} b^{3}}{24 d}+\frac {3 a \,{\mathrm e}^{d x +c} b^{2}}{2 d}-\frac {9 b^{3} {\mathrm e}^{d x +c}}{8 d}-\frac {3 a \,{\mathrm e}^{-d x -c} b^{2}}{2 d}+\frac {9 b^{3} {\mathrm e}^{-d x -c}}{8 d}-\frac {b^{3} {\mathrm e}^{-3 d x -3 c}}{24 d}+\frac {{\mathrm e}^{d x +c} \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{2 d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2} b}{2 d}-\frac {9 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a \,b^{2}}{2 d}+\frac {5 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{2 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{2 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2} b}{2 d}+\frac {9 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a \,b^{2}}{2 d}-\frac {5 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{3}}{2 d}\) | \(311\) |
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. 357 vs.
\(2 (85) = 170\).
time = 0.51, size = 357, normalized size = 3.92 \begin {gather*} \frac {1}{24} \, b^{3} {\left (\frac {27 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} - \frac {120 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {25 \, e^{\left (-2 \, d x - 2 \, c\right )} + 77 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 2 \, e^{\left (-5 \, d x - 5 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )}\right )}}\right )} + \frac {3}{2} \, a b^{2} {\left (\frac {6 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )}}{d} + \frac {4 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} - 3 \, a^{2} b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} - a^{3} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \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. 1679 vs.
\(2 (85) = 170\).
time = 0.42, size = 1679, normalized size = 18.45 \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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. 247 vs.
\(2 (85) = 170\).
time = 0.44, size = 247, normalized size = 2.71 \begin {gather*} \frac {b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 36 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 24 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 6 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (a^{3} + 3 \, a^{2} b - 9 \, a b^{2} + 5 \, b^{3}\right )} + \frac {24 \, {\left (a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 3 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 3 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.28, size = 308, normalized size = 3.38 \begin {gather*} \frac {b^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {b^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}+\frac {3\,b^2\,{\mathrm {e}}^{c+d\,x}\,\left (4\,a-3\,b\right )}{8\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+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 {3\,b^2\,{\mathrm {e}}^{-c-d\,x}\,\left (4\,a-3\,b\right )}{8\,d}+\frac {\ln \left (-{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3+3\,a^2\,b-9\,a\,b^2+5\,b^3\right )-{\left (a-b\right )}^2\,\left (a+5\,b\right )\,1{}\mathrm {i}\right )\,{\left (a-b\right )}^2\,\left (a+5\,b\right )\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (-{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3+3\,a^2\,b-9\,a\,b^2+5\,b^3\right )+{\left (a-b\right )}^2\,\left (a+5\,b\right )\,1{}\mathrm {i}\right )\,{\left (a-b\right )}^2\,\left (a+5\,b\right )\,1{}\mathrm {i}}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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