Optimal. Leaf size=48 \[ -\frac {a \tanh ^3(c+d x)}{3 d}-\frac {a \tanh (c+d x)}{d}+a x+\frac {b \tanh ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.06, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4141, 1802, 206} \[ -\frac {a \tanh ^3(c+d x)}{3 d}-\frac {a \tanh (c+d x)}{d}+a x+\frac {b \tanh ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 1802
Rule 4141
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right ) \tanh ^4(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (a+b \left (1-x^2\right )\right )}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a-a x^2+b x^4+\frac {a}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a \tanh (c+d x)}{d}-\frac {a \tanh ^3(c+d x)}{3 d}+\frac {b \tanh ^5(c+d x)}{5 d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a x-\frac {a \tanh (c+d x)}{d}-\frac {a \tanh ^3(c+d x)}{3 d}+\frac {b \tanh ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 57, normalized size = 1.19 \[ \frac {a \tanh ^{-1}(\tanh (c+d x))}{d}-\frac {a \tanh ^3(c+d x)}{3 d}-\frac {a \tanh (c+d x)}{d}+\frac {b \tanh ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 327, normalized size = 6.81 \[ \frac {{\left (15 \, a d x + 20 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (15 \, a d x + 20 \, a - 3 \, b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - {\left (20 \, a - 3 \, b\right )} \sinh \left (d x + c\right )^{5} + 5 \, {\left (15 \, a d x + 20 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{3} - 5 \, {\left (2 \, {\left (20 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 8 \, a + 3 \, b\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (2 \, {\left (15 \, a d x + 20 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (15 \, a d x + 20 \, a - 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, {\left (15 \, a d x + 20 \, a - 3 \, b\right )} \cosh \left (d x + c\right ) - 5 \, {\left ({\left (20 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (8 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 4 \, a - 6 \, b\right )} \sinh \left (d x + c\right )}{15 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 105, normalized size = 2.19 \[ \frac {15 \, a d x + \frac {2 \, {\left (30 \, a e^{\left (8 \, d x + 8 \, c\right )} - 15 \, b e^{\left (8 \, d x + 8 \, c\right )} + 90 \, a e^{\left (6 \, d x + 6 \, c\right )} + 110 \, a e^{\left (4 \, d x + 4 \, c\right )} - 30 \, b e^{\left (4 \, d x + 4 \, c\right )} + 70 \, a e^{\left (2 \, d x + 2 \, c\right )} + 20 \, a - 3 \, b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.33, size = 98, normalized size = 2.04 \[ \frac {a \left (d x +c -\tanh \left (d x +c \right )-\frac {\left (\tanh ^{3}\left (d x +c \right )\right )}{3}\right )+b \left (-\frac {\sinh ^{3}\left (d x +c \right )}{2 \cosh \left (d x +c \right )^{5}}-\frac {3 \sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{5}}+\frac {3 \left (\frac {8}{15}+\frac {\mathrm {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 92, normalized size = 1.92 \[ \frac {b \tanh \left (d x + c\right )^{5}}{5 \, d} + \frac {1}{3} \, a {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.54, size = 433, normalized size = 9.02 \[ a\,x+\frac {\frac {2\,\left (2\,a-3\,b\right )}{15\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (2\,a-b\right )}{5\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}+\frac {\frac {2\,\left (2\,a-b\right )}{5\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )}{5\,d}+\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a+b\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (2\,a-3\,b\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (2\,a-b\right )}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1}+\frac {\frac {2\,\left (a+b\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a-b\right )}{5\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+\frac {\frac {2\,\left (a+b\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a+b\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a-3\,b\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (2\,a-b\right )}{5\,d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}+\frac {2\,\left (2\,a-b\right )}{5\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \tanh ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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