Optimal. Leaf size=48 \[ a x-\frac {a \tanh (c+d x)}{d}-\frac {a \tanh ^3(c+d x)}{3 d}+\frac {b \tanh ^5(c+d x)}{5 d} \]
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Rubi [A]
time = 0.04, 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 = {4226, 1816,
212} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 1816
Rule 4226
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right ) \tanh ^4(c+d x) \, dx &=\frac {\text {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 {\text {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 \text {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 \begin {gather*} \frac {a \tanh ^{-1}(\tanh (c+d x))}{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} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(101\) vs.
\(2(44)=88\).
time = 1.92, size = 102, normalized size = 2.12
method | result | size |
risch | \(a x +\frac {4 a \,{\mathrm e}^{8 d x +8 c}-2 b \,{\mathrm e}^{8 d x +8 c}+12 a \,{\mathrm e}^{6 d x +6 c}+\frac {44 a \,{\mathrm e}^{4 d x +4 c}}{3}-4 b \,{\mathrm e}^{4 d x +4 c}+\frac {28 a \,{\mathrm e}^{2 d x +2 c}}{3}+\frac {8 a}{3}-\frac {2 b}{5}}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{5}}\) | \(102\) |
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. 92 vs.
\(2 (44) = 88\).
time = 0.29, size = 92, normalized size = 1.92 \begin {gather*} \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 )} \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. 327 vs.
\(2 (44) = 88\).
time = 0.38, size = 327, normalized size = 6.81 \begin {gather*} \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 )}} \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 ) \tanh ^{4}{\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. 108 vs.
\(2 (44) = 88\).
time = 0.42, size = 108, normalized size = 2.25 \begin {gather*} \frac {15 \, {\left (d x + c\right )} a + \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} \end {gather*}
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 \begin {gather*} 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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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