Optimal. Leaf size=47 \[ -\frac {a^2 \coth (c+d x)}{d}+\frac {a b \tanh ^2(c+d x)}{d}+\frac {b^2 \tanh ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3663, 270} \[ -\frac {a^2 \coth (c+d x)}{d}+\frac {a b \tanh ^2(c+d x)}{d}+\frac {b^2 \tanh ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 270
Rule 3663
Rubi steps
\begin {align*} \int \text {csch}^2(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^3\right )^2}{x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{x^2}+2 a b x+b^2 x^4\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a^2 \coth (c+d x)}{d}+\frac {a b \tanh ^2(c+d x)}{d}+\frac {b^2 \tanh ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 94, normalized size = 2.00 \[ -\frac {a^2 \coth (c+d x)}{d}-\frac {a b \text {sech}^2(c+d x)}{d}+\frac {b^2 \tanh (c+d x)}{5 d}+\frac {b^2 \tanh (c+d x) \text {sech}^4(c+d x)}{5 d}-\frac {2 b^2 \tanh (c+d x) \text {sech}^2(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 518, normalized size = 11.02 \[ -\frac {4 \, {\left ({\left (5 \, a^{2} + 5 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (5 \, a^{2} + 5 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + {\left (5 \, a b + 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{5} + {\left (25 \, a^{2} + 5 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (10 \, {\left (5 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 15 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + {\left (10 \, {\left (5 \, a^{2} + 5 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (25 \, a^{2} + 5 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, {\left (5 \, a^{2} - a b\right )} \cosh \left (d x + c\right ) + {\left (5 \, {\left (5 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 9 \, {\left (5 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 10 \, a b + 10 \, b^{2}\right )} \sinh \left (d x + c\right )\right )}}{5 \, {\left (d \cosh \left (d x + c\right )^{7} + 7 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + d \sinh \left (d x + c\right )^{7} + 3 \, d \cosh \left (d x + c\right )^{5} + {\left (21 \, d \cosh \left (d x + c\right )^{2} + 5 \, d\right )} \sinh \left (d x + c\right )^{5} + 5 \, {\left (7 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + d \cosh \left (d x + c\right )^{3} + {\left (35 \, d \cosh \left (d x + c\right )^{4} + 50 \, d \cosh \left (d x + c\right )^{2} + 9 \, d\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (7 \, d \cosh \left (d x + c\right )^{5} + 10 \, d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 5 \, d \cosh \left (d x + c\right ) + {\left (7 \, d \cosh \left (d x + c\right )^{6} + 25 \, d \cosh \left (d x + c\right )^{4} + 27 \, d \cosh \left (d x + c\right )^{2} + 5 \, d\right )} \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 122, normalized size = 2.60 \[ -\frac {2 \, {\left (\frac {5 \, a^{2}}{e^{\left (2 \, d x + 2 \, c\right )} - 1} + \frac {10 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 5 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 30 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 30 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 10 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 10 \, a b e^{\left (2 \, d x + 2 \, c\right )} + b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}\right )}}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 98, normalized size = 2.09 \[ \frac {-a^{2} \coth \left (d x +c \right )-\frac {a b}{\cosh \left (d x +c \right )^{2}}+b^{2} \left (-\frac {\sinh ^{3}\left (d x +c \right )}{2 \cosh \left (d x +c \right )^{5}}-\frac {3 \sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{5}}+\frac {3 \left (\frac {8}{15}+\frac {\mathrm {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 256, normalized size = 5.45 \[ \frac {2}{5} \, b^{2} {\left (\frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {5 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + \frac {2 \, a^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} - \frac {4 \, a b}{d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 483, normalized size = 10.28 \[ -\frac {\frac {2\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (b^2+2\,a\,b\right )}{5\,d}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^2+a\,b\right )}{5\,d}-\frac {2\,\left (2\,a\,b-b^2\right )}{5\,d}+\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a\,b-b^2\right )}{5\,d}+\frac {12\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1}-\frac {\frac {2\,b^2}{5\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (b^2+2\,a\,b\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b-b^2\right )}{5\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {\frac {2\,\left (a\,b-b^2\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^2+2\,a\,b\right )}{5\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (b^2+2\,a\,b\right )}{5\,d}-\frac {2\,\left (b^2+a\,b\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a\,b-b^2\right )}{5\,d}+\frac {6\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{5\,d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {2\,a^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,\left (b^2+2\,a\,b\right )}{5\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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 \[ \int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{2} \operatorname {csch}^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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