Optimal. Leaf size=81 \[ \frac {a^3 \sinh ^3(c+d x)}{3 d}+\frac {a^2 (a+3 b) \sinh (c+d x)}{d}+\frac {b^2 (6 a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {b^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d} \]
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Rubi [A] time = 0.09, antiderivative size = 81, 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 = {4147, 390, 385, 203} \[ \frac {a^2 (a+3 b) \sinh (c+d x)}{d}+\frac {a^3 \sinh ^3(c+d x)}{3 d}+\frac {b^2 (6 a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {b^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 385
Rule 390
Rule 4147
Rubi steps
\begin {align*} \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b+a x^2\right )^3}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 (a+3 b)+a^3 x^2+\frac {b^2 (3 a+b)+3 a b^2 x^2}{\left (1+x^2\right )^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^2 (a+3 b) \sinh (c+d x)}{d}+\frac {a^3 \sinh ^3(c+d x)}{3 d}+\frac {\operatorname {Subst}\left (\int \frac {b^2 (3 a+b)+3 a b^2 x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^2 (a+3 b) \sinh (c+d x)}{d}+\frac {a^3 \sinh ^3(c+d x)}{3 d}+\frac {b^3 \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {\left (b^2 (6 a+b)\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}\\ &=\frac {b^2 (6 a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {a^2 (a+3 b) \sinh (c+d x)}{d}+\frac {a^3 \sinh ^3(c+d x)}{3 d}+\frac {b^3 \text {sech}(c+d x) \tanh (c+d x)}{2 d}\\ \end {align*}
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Mathematica [C] time = 6.88, size = 483, normalized size = 5.96 \[ \frac {\coth ^3(c+d x) \text {csch}^2(c+d x) (a \cosh (c+d x)+b \text {sech}(c+d x))^3 \left (-256 \sinh ^8(c+d x) \left (a \sinh ^2(c+d x)+a+b\right )^3 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right )+21 \left (a^3 \left (753 \sinh ^{10}(c+d x)+19579 \sinh ^8(c+d x)+89514 \sinh ^6(c+d x)+157878 \sinh ^4(c+d x)+124165 \sinh ^2(c+d x)+36015\right )+3 a^2 b \left (753 \sinh ^8(c+d x)+19786 \sinh ^6(c+d x)+69728 \sinh ^4(c+d x)+88150 \sinh ^2(c+d x)+36015\right )+3 a b^2 \left (753 \sinh ^6(c+d x)+17593 \sinh ^4(c+d x)+52135 \sinh ^2(c+d x)+36015\right )+b^3 \left (1473 \sinh ^4(c+d x)+16120 \sinh ^2(c+d x)+36015\right )\right )-\frac {315 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (a^3 \left (\sinh ^6(c+d x)+243 \sinh ^4(c+d x)+1875 \sinh ^2(c+d x)+2401\right ) \cosh ^6(c+d x)+3 a^2 b \left (\sinh ^6(c+d x)+243 \sinh ^4(c+d x)+1875 \sinh ^2(c+d x)+2401\right ) \cosh ^4(c+d x)+3 a b^2 \left (\sinh ^8(c+d x)+148 \sinh ^6(c+d x)+2118 \sinh ^4(c+d x)+4276 \sinh ^2(c+d x)+2401\right )+b^3 \left (-47 \sinh ^6(c+d x)+243 \sinh ^4(c+d x)+1875 \sinh ^2(c+d x)+2401\right )\right )}{\sqrt {-\sinh ^2(c+d x)}}\right )}{3780 d (a \cosh (2 c+2 d x)+a+2 b)^3} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.42, size = 1409, normalized size = 17.40 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 163, normalized size = 2.01 \[ \frac {a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 12 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 36 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + \frac {24 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4} + 6 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (6 \, a b^{2} + b^{3}\right )}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 103, normalized size = 1.27 \[ \frac {2 a^{3} \sinh \left (d x +c \right )}{3 d}+\frac {a^{3} \sinh \left (d x +c \right ) \left (\cosh ^{2}\left (d x +c \right )\right )}{3 d}+\frac {3 a^{2} b \sinh \left (d x +c \right )}{d}+\frac {6 a \,b^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{d}+\frac {b^{3} \mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2 d}+\frac {b^{3} \arctan \left ({\mathrm e}^{d x +c}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 179, normalized size = 2.21 \[ -b^{3} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} - \frac {6 \, a b^{2} \arctan \left (e^{\left (-d x - c\right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 218, normalized size = 2.69 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (b^3\,\sqrt {d^2}+6\,a\,b^2\,\sqrt {d^2}\right )}{d\,\sqrt {36\,a^2\,b^4+12\,a\,b^5+b^6}}\right )\,\sqrt {36\,a^2\,b^4+12\,a\,b^5+b^6}}{\sqrt {d^2}}-\frac {a^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}+\frac {a^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {3\,a^2\,{\mathrm {e}}^{-c-d\,x}\,\left (a+4\,b\right )}{8\,d}+\frac {3\,a^2\,{\mathrm {e}}^{c+d\,x}\,\left (a+4\,b\right )}{8\,d}+\frac {b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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