Optimal. Leaf size=147 \[ \frac {(2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \text {ArcTan}(\sinh (c+d x))}{16 d}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{48 d}+\frac {5 b (2 a+b) \text {sech}^3(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac {b \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d} \]
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Rubi [A]
time = 0.10, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4232, 424, 540,
393, 209} \begin {gather*} \frac {(2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \text {ArcTan}(\sinh (c+d x))}{16 d}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \tanh (c+d x) \text {sech}(c+d x)}{48 d}+\frac {b \tanh (c+d x) \text {sech}^5(c+d x) \left (a \sinh ^2(c+d x)+a+b\right )^2}{6 d}+\frac {5 b (2 a+b) \tanh (c+d x) \text {sech}^3(c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{24 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 393
Rule 424
Rule 540
Rule 4232
Rubi steps
\begin {align*} \int \text {sech}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b+a x^2\right )^3}{\left (1+x^2\right )^4} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {b \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d}+\frac {\text {Subst}\left (\int \frac {\left (a+b+a x^2\right ) \left ((a+b) (6 a+5 b)+a (6 a+b) x^2\right )}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{6 d}\\ &=\frac {5 b (2 a+b) \text {sech}^3(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac {b \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d}-\frac {\text {Subst}\left (\int \frac {-(a+b) \left (24 a^2+34 a b+15 b^2\right )-a \left (24 a^2+14 a b+5 b^2\right ) x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{24 d}\\ &=\frac {b \left (44 a^2+44 a b+15 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{48 d}+\frac {5 b (2 a+b) \text {sech}^3(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac {b \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d}+\frac {\left ((2 a+b) \left (8 a^2+8 a b+5 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{16 d}\\ &=\frac {(2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{48 d}+\frac {5 b (2 a+b) \text {sech}^3(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac {b \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 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 = 9.05, size = 1430, normalized size = 9.73 \begin {gather*} \frac {\coth ^6(c+d x) \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \left (117228825 (a+b)^3 \sinh ^2(c+d x)+274542345 a (a+b)^2 \sinh ^4(c+d x)+70189350 (a+b)^3 \sinh ^4(c+d x)+215549775 a^2 (a+b) \sinh ^6(c+d x)+168951510 a (a+b)^2 \sinh ^6(c+d x)+4093425 (a+b)^3 \sinh ^6(c+d x)+58009455 a^3 \sinh ^8(c+d x)+135323370 a^2 (a+b) \sinh ^8(c+d x)+9514449 a (a+b)^2 \sinh ^8(c+d x)+36772890 a^3 \sinh ^{10}(c+d x)+7808535 a^2 (a+b) \sinh ^{10}(c+d x)-75520 (a+b)^3 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{10}(c+d x)-13824 (a+b)^3 \, _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 ^{10}(c+d x)-1024 (a+b)^3 \, _7F_6\left (\frac {3}{2},2,2,2,2,2,2;1,1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{10}(c+d x)+2160711 a^3 \sinh ^{12}(c+d x)-189696 a (a+b)^2 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{12}(c+d x)-38400 a (a+b)^2 \, _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 ^{12}(c+d x)-3072 a (a+b)^2 \, _7F_6\left (\frac {3}{2},2,2,2,2,2,2;1,1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{12}(c+d x)-158976 a^2 (a+b) \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{14}(c+d x)-35328 a^2 (a+b) \, _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 ^{14}(c+d x)-3072 a^2 (a+b) \, _7F_6\left (\frac {3}{2},2,2,2,2,2,2;1,1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{14}(c+d x)-44800 a^3 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{16}(c+d x)-10752 a^3 \, _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 ^{16}(c+d x)-1024 a^3 \, _7F_6\left (\frac {3}{2},2,2,2,2,2,2;1,1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{16}(c+d x)+117228825 (a+b)^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sqrt {-\sinh ^2(c+d x)}+215549775 a^2 (a+b) \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x) \sqrt {-\sinh ^2(c+d x)}+260465625 a (a+b)^2 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x) \sqrt {-\sinh ^2(c+d x)}+17069535 (a+b)^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x) \sqrt {-\sinh ^2(c+d x)}+58009455 a^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^6(c+d x) \sqrt {-\sinh ^2(c+d x)}+207173295 a^2 (a+b) \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^6(c+d x) \sqrt {-\sinh ^2(c+d x)}+41427855 a (a+b)^2 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^6(c+d x) \sqrt {-\sinh ^2(c+d x)}+56109375 a^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^8(c+d x) \sqrt {-\sinh ^2(c+d x)}+33756345 a^2 (a+b) \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^8(c+d x) \sqrt {-\sinh ^2(c+d x)}+210735 a (a+b)^2 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^8(c+d x) \sqrt {-\sinh ^2(c+d x)}+9261945 a^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^{10}(c+d x) \sqrt {-\sinh ^2(c+d x)}+174825 a^2 (a+b) \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^{10}(c+d x) \sqrt {-\sinh ^2(c+d x)}+48825 a^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^{12}(c+d x) \sqrt {-\sinh ^2(c+d x)}-274542345 a (a+b)^2 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (-\sinh ^2(c+d x)\right )^{3/2}-109265625 (a+b)^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (-\sinh ^2(c+d x)\right )^{3/2}+142065 (a+b)^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2}\right )}{90720 d (a+2 b+a \cosh (2 c+2 d x))^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains complex when optimal does not.
time = 1.82, size = 403, normalized size = 2.74
method | result | size |
risch | \(\frac {b \,{\mathrm e}^{d x +c} \left (72 a^{2} {\mathrm e}^{10 d x +10 c}+54 a b \,{\mathrm e}^{10 d x +10 c}+15 b^{2} {\mathrm e}^{10 d x +10 c}+216 a^{2} {\mathrm e}^{8 d x +8 c}+306 a b \,{\mathrm e}^{8 d x +8 c}+85 b^{2} {\mathrm e}^{8 d x +8 c}+144 a^{2} {\mathrm e}^{6 d x +6 c}+252 a b \,{\mathrm e}^{6 d x +6 c}+198 b^{2} {\mathrm e}^{6 d x +6 c}-144 a^{2} {\mathrm e}^{4 d x +4 c}-252 a b \,{\mathrm e}^{4 d x +4 c}-198 b^{2} {\mathrm e}^{4 d x +4 c}-216 a^{2} {\mathrm e}^{2 d x +2 c}-306 a b \,{\mathrm e}^{2 d x +2 c}-85 b^{2} {\mathrm e}^{2 d x +2 c}-72 a^{2}-54 a b -15 b^{2}\right )}{24 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{6}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{d}+\frac {3 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{2 d}+\frac {9 i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{8 d}+\frac {5 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{16 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 d}-\frac {9 i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{8 d}-\frac {5 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{16 d}\) | \(403\) |
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. 365 vs.
\(2 (139) = 278\).
time = 0.50, size = 365, normalized size = 2.48 \begin {gather*} -\frac {1}{24} \, b^{3} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {15 \, e^{\left (-d x - c\right )} + 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} - 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} - 15 \, e^{\left (-11 \, d x - 11 \, 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 {3}{4} \, a b^{2} {\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 )} - 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 )} + \frac {a^{3} \arctan \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. 3465 vs.
\(2 (139) = 278\).
time = 0.38, size = 3465, normalized size = 23.57 \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3} \operatorname {sech}{\left (c + d x \right )}\, dx \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. 310 vs.
\(2 (139) = 278\).
time = 0.42, size = 310, normalized size = 2.11 \begin {gather*} \frac {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 (16 \, a^{3} + 24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} + \frac {4 \, {\left (72 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 54 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 15 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 576 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 576 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 160 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 1152 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 1440 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 528 \, 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 )}^{3}}}{96 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.47, size = 535, normalized size = 3.64 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (16\,a^3\,\sqrt {d^2}+5\,b^3\,\sqrt {d^2}+18\,a\,b^2\,\sqrt {d^2}+24\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {256\,a^6+768\,a^5\,b+1152\,a^4\,b^2+1024\,a^3\,b^3+564\,a^2\,b^4+180\,a\,b^5+25\,b^6}}\right )\,\sqrt {256\,a^6+768\,a^5\,b+1152\,a^4\,b^2+1024\,a^3\,b^3+564\,a^2\,b^4+180\,a\,b^5+25\,b^6}}{8\,\sqrt {d^2}}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (54\,a\,b^2-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 {80\,b^3\,{\mathrm {e}}^{c+d\,x}}{3\,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 {6\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a\,b^2-3\,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 )}-\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,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 {{\mathrm {e}}^{c+d\,x}\,\left (24\,a^2\,b+18\,a\,b^2+5\,b^3\right )}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (-72\,a^2\,b+18\,a\,b^2+5\,b^3\right )}{12\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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