Optimal. Leaf size=93 \[ \frac {3 b \left (8 a^2+4 a b+b^2\right ) \text {ArcTan}(\sinh (c+d x))}{8 d}+\frac {a^3 \sinh (c+d x)}{d}+\frac {3 b^2 (4 a+b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4232, 398,
1171, 393, 209} \begin {gather*} \frac {a^3 \sinh (c+d x)}{d}+\frac {3 b \left (8 a^2+4 a b+b^2\right ) \text {ArcTan}(\sinh (c+d x))}{8 d}+\frac {3 b^2 (4 a+b) \tanh (c+d x) \text {sech}(c+d x)}{8 d}+\frac {b^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 393
Rule 398
Rule 1171
Rule 4232
Rubi steps
\begin {align*} \int \cosh (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 )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (a^3+\frac {b \left (3 a^2+3 a b+b^2\right )+3 a b (2 a+b) x^2+3 a^2 b x^4}{\left (1+x^2\right )^3}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^3 \sinh (c+d x)}{d}+\frac {\text {Subst}\left (\int \frac {b \left (3 a^2+3 a b+b^2\right )+3 a b (2 a+b) x^2+3 a^2 b x^4}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^3 \sinh (c+d x)}{d}+\frac {b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 d}-\frac {\text {Subst}\left (\int \frac {-3 b (2 a+b)^2-12 a^2 b x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 d}\\ &=\frac {a^3 \sinh (c+d x)}{d}+\frac {3 b^2 (4 a+b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac {\left (3 b \left (8 a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=\frac {3 b \left (8 a^2+4 a b+b^2\right ) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac {a^3 \sinh (c+d x)}{d}+\frac {3 b^2 (4 a+b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {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 = 7.46, size = 575, normalized size = 6.18 \begin {gather*} -\frac {\cosh (c+d x) \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \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+a \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+a \sinh ^2(c+d x)\right )^2 \left (7 b+a \left (7+5 \sinh ^2(c+d x)\right )\right )+\frac {315 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (b^3 \left (16807+15000 \sinh ^2(c+d x)+2187 \sinh ^4(c+d x)-62 \sinh ^6(c+d x)\right )+a^3 \cosh ^4(c+d x) \left (16807+24604 \sinh ^2(c+d x)+11562 \sinh ^4(c+d x)+1468 \sinh ^6(c+d x)+7 \sinh ^8(c+d x)\right )+3 a b^2 \left (16807+29406 \sinh ^2(c+d x)+15312 \sinh ^4(c+d x)+1858 \sinh ^6(c+d x)+9 \sinh ^8(c+d x)\right )+3 a^2 b \left (16807+43812 \sinh ^2(c+d x)+40442 \sinh ^4(c+d x)+14956 \sinh ^6(c+d x)+1719 \sinh ^8(c+d x)+8 \sinh ^{10}(c+d x)\right )\right )}{\sqrt {-\sinh ^2(c+d x)}}-21 \left (b^3 \left (252105+140965 \sinh ^2(c+d x)+8226 \sinh ^4(c+d x)\right )+3 a b^2 \left (252105+357055 \sinh ^2(c+d x)+133071 \sinh ^4(c+d x)+6393 \sinh ^6(c+d x)\right )+3 a^2 b \left (252105+573145 \sinh ^2(c+d x)+437991 \sinh ^4(c+d x)+120431 \sinh ^6(c+d x)+5640 \sinh ^8(c+d x)\right )+a^3 \left (252105+789235 \sinh ^2(c+d x)+922986 \sinh ^4(c+d x)+491574 \sinh ^6(c+d x)+107725 \sinh ^8(c+d x)+4887 \sinh ^{10}(c+d x)\right )\right )\right )}{7560 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 = 2.30, size = 257, normalized size = 2.76
method | result | size |
risch | \(\frac {{\mathrm e}^{d x +c} a^{3}}{2 d}-\frac {{\mathrm e}^{-d x -c} a^{3}}{2 d}+\frac {b^{2} {\mathrm e}^{d x +c} \left (12 a \,{\mathrm e}^{6 d x +6 c}+3 b \,{\mathrm e}^{6 d x +6 c}+12 a \,{\mathrm e}^{4 d x +4 c}+11 b \,{\mathrm e}^{4 d x +4 c}-12 a \,{\mathrm e}^{2 d x +2 c}-11 b \,{\mathrm e}^{2 d x +2 c}-12 a -3 b \right )}{4 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{4}}+\frac {3 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{d}+\frac {3 i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{2 d}+\frac {3 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{8 d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{d}-\frac {3 i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{2 d}-\frac {3 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{8 d}\) | \(257\) |
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. 221 vs.
\(2 (87) = 174\).
time = 0.53, size = 221, normalized size = 2.38 \begin {gather*} -\frac {1}{4} \, b^{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 )} - 3 \, a b^{2} {\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 {6 \, a^{2} b \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {a^{3} \sinh \left (d x + c\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. 1992 vs.
\(2 (87) = 174\).
time = 0.39, size = 1992, normalized size = 21.42 \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} \cosh {\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. 199 vs.
\(2 (87) = 174\).
time = 0.43, size = 199, normalized size = 2.14 \begin {gather*} \frac {8 \, a^{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 (8 \, a^{2} b + 4 \, a b^{2} + b^{3}\right )} + \frac {4 \, {\left (12 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 3 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 48 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 20 \, 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 = 344, normalized size = 3.70 \begin {gather*} \frac {a^3\,{\mathrm {e}}^{c+d\,x}}{2\,d}-\frac {a^3\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (b^3\,\sqrt {d^2}+4\,a\,b^2\,\sqrt {d^2}+8\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {64\,a^4\,b^2+64\,a^3\,b^3+32\,a^2\,b^4+8\,a\,b^5+b^6}}\right )\,\sqrt {64\,a^4\,b^2+64\,a^3\,b^3+32\,a^2\,b^4+8\,a\,b^5+b^6}}{4\,\sqrt {d^2}}-\frac {6\,b^3\,{\mathrm {e}}^{c+d\,x}}{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 {{\mathrm {e}}^{c+d\,x}\,\left (12\,a\,b^2-b^3\right )}{2\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {4\,b^3\,{\mathrm {e}}^{c+d\,x}}{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 {3\,{\mathrm {e}}^{c+d\,x}\,\left (b^3+4\,a\,b^2\right )}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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