Optimal. Leaf size=70 \[ -\frac {b^2 (a+b) \text {sech}^3(c+d x)}{d}+\frac {(a+b)^3 \cosh (c+d x)}{d}+\frac {3 b (a+b)^2 \text {sech}(c+d x)}{d}+\frac {b^3 \text {sech}^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.07, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3664, 270} \[ -\frac {b^2 (a+b) \text {sech}^3(c+d x)}{d}+\frac {(a+b)^3 \cosh (c+d x)}{d}+\frac {3 b (a+b)^2 \text {sech}(c+d x)}{d}+\frac {b^3 \text {sech}^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 270
Rule 3664
Rubi steps
\begin {align*} \int \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (a+b-b x^2\right )^3}{x^2} \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-3 b (a+b)^2+\frac {(a+b)^3}{x^2}+3 b^2 (a+b) x^2-b^3 x^4\right ) \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=\frac {(a+b)^3 \cosh (c+d x)}{d}+\frac {3 b (a+b)^2 \text {sech}(c+d x)}{d}-\frac {b^2 (a+b) \text {sech}^3(c+d x)}{d}+\frac {b^3 \text {sech}^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.88, size = 63, normalized size = 0.90 \[ \frac {b \text {sech}(c+d x) \left (-5 b (a+b) \text {sech}^2(c+d x)+15 (a+b)^2+b^2 \text {sech}^4(c+d x)\right )+5 (a+b)^3 \cosh (c+d x)}{5 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 383, normalized size = 5.47 \[ \frac {5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{6} + 5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{6} + 30 \, {\left (a^{3} + 5 \, a^{2} b + 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 15 \, {\left (2 \, a^{3} + 10 \, a^{2} b + 14 \, a b^{2} + 6 \, b^{3} + 5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} + 50 \, a^{3} + 330 \, a^{2} b + 430 \, a b^{2} + 182 \, b^{3} + 5 \, {\left (15 \, a^{3} + 93 \, a^{2} b + 125 \, a b^{2} + 47 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 5 \, {\left (15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{4} + 15 \, a^{3} + 93 \, a^{2} b + 125 \, a b^{2} + 47 \, b^{3} + 36 \, {\left (a^{3} + 5 \, a^{2} b + 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2}}{10 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 322, normalized size = 4.60 \[ \frac {5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-d x - c\right )} + 5 \, {\left (a^{3} e^{\left (d x + 14 \, c\right )} + 3 \, a^{2} b e^{\left (d x + 14 \, c\right )} + 3 \, a b^{2} e^{\left (d x + 14 \, c\right )} + b^{3} e^{\left (d x + 14 \, c\right )}\right )} e^{\left (-13 \, c\right )} + \frac {4 \, {\left (15 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 30 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 15 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} + 60 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} + 100 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 40 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} + 90 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 140 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 66 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 60 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 100 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 40 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 15 \, a^{2} b e^{\left (d x + c\right )} + 30 \, a b^{2} e^{\left (d x + c\right )} + 15 \, b^{3} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.23, size = 170, normalized size = 2.43 \[ \frac {a^{3} \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {\sinh ^{2}\left (d x +c \right )}{\cosh \left (d x +c \right )}+\frac {2}{\cosh \left (d x +c \right )}\right )+3 a \,b^{2} \left (\frac {\sinh ^{4}\left (d x +c \right )}{\cosh \left (d x +c \right )^{3}}+\frac {4 \left (\sinh ^{2}\left (d x +c \right )\right )}{\cosh \left (d x +c \right )^{3}}+\frac {8}{3 \cosh \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sinh ^{6}\left (d x +c \right )}{\cosh \left (d x +c \right )^{5}}+\frac {6 \left (\sinh ^{4}\left (d x +c \right )\right )}{\cosh \left (d x +c \right )^{5}}+\frac {8 \left (\sinh ^{2}\left (d x +c \right )\right )}{\cosh \left (d x +c \right )^{5}}+\frac {16}{5 \cosh \left (d x +c \right )^{5}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 321, normalized size = 4.59 \[ \frac {1}{10} \, b^{3} {\left (\frac {5 \, e^{\left (-d x - c\right )}}{d} + \frac {85 \, e^{\left (-2 \, d x - 2 \, c\right )} + 210 \, e^{\left (-4 \, d x - 4 \, c\right )} + 314 \, e^{\left (-6 \, d x - 6 \, c\right )} + 185 \, e^{\left (-8 \, d x - 8 \, c\right )} + 65 \, e^{\left (-10 \, d x - 10 \, c\right )} + 5}{d {\left (e^{\left (-d x - c\right )} + 5 \, e^{\left (-3 \, d x - 3 \, c\right )} + 10 \, e^{\left (-5 \, d x - 5 \, c\right )} + 10 \, e^{\left (-7 \, d x - 7 \, c\right )} + 5 \, e^{\left (-9 \, d x - 9 \, c\right )} + e^{\left (-11 \, d x - 11 \, c\right )}\right )}}\right )} + \frac {1}{2} \, a b^{2} {\left (\frac {3 \, e^{\left (-d x - c\right )}}{d} + \frac {33 \, e^{\left (-2 \, d x - 2 \, c\right )} + 41 \, e^{\left (-4 \, d x - 4 \, c\right )} + 27 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3}{d {\left (e^{\left (-d x - c\right )} + 3 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )}\right )}}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {e^{\left (-d x - c\right )}}{d} + \frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}\right )} + \frac {a^{3} \cosh \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 308, normalized size = 4.40 \[ \frac {{\mathrm {e}}^{c+d\,x}\,{\left (a+b\right )}^3}{2\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,{\left (a+b\right )}^3}{2\,d}+\frac {6\,{\mathrm {e}}^{c+d\,x}\,\left (a^2\,b+2\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {64\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (9\,b^3+5\,a\,b^2\right )}{5\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (b^3+a\,b^2\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \sinh {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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