Optimal. Leaf size=103 \[ \frac {3 (a-b) \left (4 b^2+(a+b)^2\right ) \text {ArcTan}(\sinh (c+d x))}{8 d}+\frac {b^3 \sinh (c+d x)}{d}+\frac {3 (a-b)^2 (a+3 b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {(a-b)^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3269, 398,
1171, 393, 209} \begin {gather*} \frac {3 (a-b) \left ((a+b)^2+4 b^2\right ) \text {ArcTan}(\sinh (c+d x))}{8 d}+\frac {(a-b)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}+\frac {3 (a-b)^2 (a+3 b) \tanh (c+d x) \text {sech}(c+d x)}{8 d}+\frac {b^3 \sinh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 393
Rule 398
Rule 1171
Rule 3269
Rubi steps
\begin {align*} \int \text {sech}^5(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 )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (b^3+\frac {a^3-b^3+3 b \left (a^2-b^2\right ) x^2+3 (a-b) b^2 x^4}{\left (1+x^2\right )^3}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {b^3 \sinh (c+d x)}{d}+\frac {\text {Subst}\left (\int \frac {a^3-b^3+3 b \left (a^2-b^2\right ) x^2+3 (a-b) b^2 x^4}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {b^3 \sinh (c+d x)}{d}+\frac {(a-b)^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 d}-\frac {\text {Subst}\left (\int \frac {-3 (a-b) (a+b)^2-12 (a-b) b^2 x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 d}\\ &=\frac {b^3 \sinh (c+d x)}{d}+\frac {3 (a-b)^2 (a+3 b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {(a-b)^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac {\left (3 (a-b) \left (4 b^2+(a+b)^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=\frac {3 (a-b) \left (4 b^2+(a+b)^2\right ) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac {b^3 \sinh (c+d x)}{d}+\frac {3 (a-b)^2 (a+3 b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {(a-b)^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 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 = 9.39, size = 472, normalized size = 4.58 \begin {gather*} -\frac {\text {csch}^5(c+d x) \left (256 \, _6F_5\left (\frac {3}{2},2,2,2,2,2;1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^8(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3+384 \, _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 )^2 \left (7 a+5 b \sinh ^2(c+d x)\right )-21 \left (15 a b^2 \sinh ^4(c+d x) \left (36015+21529 \sinh ^2(c+d x)+1128 \sinh ^4(c+d x)\right )+9 a^2 b \sinh ^2(c+d x) \left (72030+41615 \sinh ^2(c+d x)+2131 \sinh ^4(c+d x)\right )+b^3 \sinh ^6(c+d x) \left (149460+90805 \sinh ^2(c+d x)+4887 \sinh ^4(c+d x)\right )+a^3 \left (252105+140965 \sinh ^2(c+d x)+8226 \sinh ^4(c+d x)\right )\right )+\frac {315 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (a^3 \left (16807+15000 \sinh ^2(c+d x)+2187 \sinh ^4(c+d x)-62 \sinh ^6(c+d x)\right )+9 a^2 b \sinh ^2(c+d x) \left (4802+4375 \sinh ^2(c+d x)+640 \sinh ^4(c+d x)+3 \sinh ^6(c+d x)\right )+b^3 \sinh ^6(c+d x) \left (9964+9375 \sinh ^2(c+d x)+1458 \sinh ^4(c+d x)+7 \sinh ^6(c+d x)\right )+3 a b^2 \sinh ^4(c+d x) \left (12005+11178 \sinh ^2(c+d x)+1701 \sinh ^4(c+d x)+8 \sinh ^6(c+d x)\right )\right )}{\sqrt {-\sinh ^2(c+d x)}}\right )}{60480 d} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains complex when optimal does not.
time = 1.72, size = 409, normalized size = 3.97
method | result | size |
risch | \(\frac {b^{3} {\mathrm e}^{d x +c}}{2 d}-\frac {b^{3} {\mathrm e}^{-d x -c}}{2 d}+\frac {{\mathrm e}^{d x +c} \left (3 a^{3} {\mathrm e}^{6 d x +6 c}+3 a^{2} b \,{\mathrm e}^{6 d x +6 c}-15 a \,b^{2} {\mathrm e}^{6 d x +6 c}+9 b^{3} {\mathrm e}^{6 d x +6 c}+11 a^{3} {\mathrm e}^{4 d x +4 c}-21 a^{2} b \,{\mathrm e}^{4 d x +4 c}+9 a \,b^{2} {\mathrm e}^{4 d x +4 c}+b^{3} {\mathrm e}^{4 d x +4 c}-11 a^{3} {\mathrm e}^{2 d x +2 c}+21 a^{2} b \,{\mathrm e}^{2 d x +2 c}-9 a \,b^{2} {\mathrm e}^{2 d x +2 c}-b^{3} {\mathrm e}^{2 d x +2 c}-3 a^{3}-3 a^{2} b +15 a \,b^{2}-9 b^{3}\right )}{4 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{4}}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{8 d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2} b}{8 d}+\frac {9 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a \,b^{2}}{8 d}-\frac {15 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{8 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{8 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2} b}{8 d}-\frac {9 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a \,b^{2}}{8 d}+\frac {15 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{3}}{8 d}\) | \(409\) |
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. 489 vs.
\(2 (97) = 194\).
time = 0.48, size = 489, normalized size = 4.75 \begin {gather*} \frac {1}{4} \, b^{3} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {2 \, e^{\left (-d x - c\right )}}{d} + \frac {17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 13 \, e^{\left (-4 \, d x - 4 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} - 7 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2}{d {\left (e^{\left (-d x - c\right )} + 4 \, e^{\left (-3 \, d x - 3 \, c\right )} + 6 \, e^{\left (-5 \, d x - 5 \, c\right )} + 4 \, e^{\left (-7 \, d x - 7 \, c\right )} + e^{\left (-9 \, d x - 9 \, c\right )}\right )}}\right )} - \frac {3}{4} \, a b^{2} {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {5 \, e^{\left (-d x - c\right )} - 3 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, 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 {1}{4} \, a^{3} {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, 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 {3}{4} \, a^{2} b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - 7 \, e^{\left (-3 \, d x - 3 \, c\right )} + 7 \, e^{\left (-5 \, d x - 5 \, c\right )} - e^{\left (-7 \, d x - 7 \, 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 )} \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. 2245 vs.
\(2 (97) = 194\).
time = 0.43, size = 2245, normalized size = 21.80 \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. 301 vs.
\(2 (97) = 194\).
time = 0.46, size = 301, normalized size = 2.92 \begin {gather*} \frac {8 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 3 \, {\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} + a^{2} b + 3 \, a b^{2} - 5 \, b^{3}\right )} + \frac {4 \, {\left (3 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 3 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 15 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 9 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 20 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 12 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 36 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 28 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{2}}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 430, normalized size = 4.17 \begin {gather*} \frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3\,\sqrt {d^2}-5\,b^3\,\sqrt {d^2}+3\,a\,b^2\,\sqrt {d^2}+a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {a^6+2\,a^5\,b+7\,a^4\,b^2-4\,a^3\,b^3-a^2\,b^4-30\,a\,b^5+25\,b^6}}\right )\,\sqrt {a^6+2\,a^5\,b+7\,a^4\,b^2-4\,a^3\,b^3-a^2\,b^4-30\,a\,b^5+25\,b^6}}{4\,\sqrt {d^2}}+\frac {b^3\,{\mathrm {e}}^{c+d\,x}}{2\,d}-\frac {b^3\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {3\,{\mathrm {e}}^{c+d\,x}\,\left (a^3+a^2\,b-5\,a\,b^2+3\,b^3\right )}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3-15\,a^2\,b+27\,a\,b^2-13\,b^3\right )}{2\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {6\,{\mathrm {e}}^{c+d\,x}\,\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\,{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{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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