Optimal. Leaf size=63 \[ (a+b)^2 x-\frac {(a+b)^2 \tanh (c+d x)}{d}-\frac {b (2 a+b) \tanh ^3(c+d x)}{3 d}-\frac {b^2 \tanh ^5(c+d x)}{5 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 472, 212}
\begin {gather*} -\frac {b (2 a+b) \tanh ^3(c+d x)}{3 d}-\frac {(a+b)^2 \tanh (c+d x)}{d}+x (a+b)^2-\frac {b^2 \tanh ^5(c+d x)}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 472
Rule 3751
Rubi steps
\begin {align*} \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {x^2 \left (a+b x^2\right )^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-(a+b)^2-b (2 a+b) x^2-b^2 x^4+\frac {a^2+2 a b+b^2}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {(a+b)^2 \tanh (c+d x)}{d}-\frac {b (2 a+b) \tanh ^3(c+d x)}{3 d}-\frac {b^2 \tanh ^5(c+d x)}{5 d}+\frac {(a+b)^2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=(a+b)^2 x-\frac {(a+b)^2 \tanh (c+d x)}{d}-\frac {b (2 a+b) \tanh ^3(c+d x)}{3 d}-\frac {b^2 \tanh ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(137\) vs. \(2(63)=126\).
time = 0.04, size = 137, normalized size = 2.17 \begin {gather*} \frac {a^2 \tanh ^{-1}(\tanh (c+d x))}{d}+\frac {2 a b \tanh ^{-1}(\tanh (c+d x))}{d}+\frac {b^2 \tanh ^{-1}(\tanh (c+d x))}{d}-\frac {a^2 \tanh (c+d x)}{d}-\frac {2 a b \tanh (c+d x)}{d}-\frac {b^2 \tanh (c+d x)}{d}-\frac {2 a b \tanh ^3(c+d x)}{3 d}-\frac {b^2 \tanh ^3(c+d x)}{3 d}-\frac {b^2 \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. \(119\) vs.
\(2(59)=118\).
time = 0.36, size = 120, normalized size = 1.90
method | result | size |
derivativedivides | \(\frac {-\frac {2 a b \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-a^{2} \tanh \left (d x +c \right )-2 a b \tanh \left (d x +c \right )+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (1+\tanh \left (d x +c \right )\right )}{2}-\frac {b^{2} \left (\tanh ^{5}\left (d x +c \right )\right )}{5}-\frac {b^{2} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-b^{2} \tanh \left (d x +c \right )-\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}}{d}\) | \(120\) |
default | \(\frac {-\frac {2 a b \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-a^{2} \tanh \left (d x +c \right )-2 a b \tanh \left (d x +c \right )+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (1+\tanh \left (d x +c \right )\right )}{2}-\frac {b^{2} \left (\tanh ^{5}\left (d x +c \right )\right )}{5}-\frac {b^{2} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-b^{2} \tanh \left (d x +c \right )-\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}}{d}\) | \(120\) |
risch | \(a^{2} x +2 a b x +b^{2} x +\frac {2 a^{2} {\mathrm e}^{8 d x +8 c}+8 a b \,{\mathrm e}^{8 d x +8 c}+6 b^{2} {\mathrm e}^{8 d x +8 c}+8 a^{2} {\mathrm e}^{6 d x +6 c}+24 a b \,{\mathrm e}^{6 d x +6 c}+12 b^{2} {\mathrm e}^{6 d x +6 c}+12 a^{2} {\mathrm e}^{4 d x +4 c}+\frac {88 a b \,{\mathrm e}^{4 d x +4 c}}{3}+\frac {56 b^{2} {\mathrm e}^{4 d x +4 c}}{3}+8 a^{2} {\mathrm e}^{2 d x +2 c}+\frac {56 a b \,{\mathrm e}^{2 d x +2 c}}{3}+\frac {28 b^{2} {\mathrm e}^{2 d x +2 c}}{3}+2 a^{2}+\frac {16 a b}{3}+\frac {46 b^{2}}{15}}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{5}}\) | \(214\) |
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. 231 vs.
\(2 (59) = 118\).
time = 0.29, size = 231, normalized size = 3.67 \begin {gather*} \frac {1}{15} \, b^{2} {\left (15 \, x + \frac {15 \, c}{d} - \frac {2 \, {\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} + 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} + 45 \, e^{\left (-8 \, d x - 8 \, c\right )} + 23\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 )}}\right )} + \frac {2}{3} \, a b {\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 )} + a^{2} {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, 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. 483 vs.
\(2 (59) = 118\).
time = 0.39, size = 483, normalized size = 7.67 \begin {gather*} \frac {{\left (15 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 15 \, a^{2} + 40 \, a b + 23 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (15 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 15 \, a^{2} + 40 \, a b + 23 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - {\left (15 \, a^{2} + 40 \, a b + 23 \, b^{2}\right )} \sinh \left (d x + c\right )^{5} + 5 \, {\left (15 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 15 \, a^{2} + 40 \, a b + 23 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 5 \, {\left (2 \, {\left (15 \, a^{2} + 40 \, a b + 23 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 9 \, a^{2} + 16 \, a b + 5 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (2 \, {\left (15 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 15 \, a^{2} + 40 \, a b + 23 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (15 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 15 \, a^{2} + 40 \, a b + 23 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, {\left (15 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 15 \, a^{2} + 40 \, a b + 23 \, b^{2}\right )} \cosh \left (d x + c\right ) - 5 \, {\left ({\left (15 \, a^{2} + 40 \, a b + 23 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (9 \, a^{2} + 16 \, a b + 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 6 \, a^{2} + 8 \, a b + 10 \, b^{2}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs.
\(2 (53) = 106\).
time = 0.17, size = 117, normalized size = 1.86 \begin {gather*} \begin {cases} a^{2} x - \frac {a^{2} \tanh {\left (c + d x \right )}}{d} + 2 a b x - \frac {2 a b \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {2 a b \tanh {\left (c + d x \right )}}{d} + b^{2} x - \frac {b^{2} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {b^{2} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{2} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{2} \tanh ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \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. 218 vs.
\(2 (59) = 118\).
time = 0.48, size = 218, normalized size = 3.46 \begin {gather*} \frac {15 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (15 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 60 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 45 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 60 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 180 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 90 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 220 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 140 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 60 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 140 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 70 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 15 \, a^{2} + 40 \, a b + 23 \, b^{2}\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 = 1.30, size = 67, normalized size = 1.06 \begin {gather*} x\,\left (a^2+2\,a\,b+b^2\right )-\frac {\mathrm {tanh}\left (c+d\,x\right )\,{\left (a+b\right )}^2}{d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (b^2+2\,a\,b\right )}{3\,d}-\frac {b^2\,{\mathrm {tanh}\left (c+d\,x\right )}^5}{5\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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