Optimal. Leaf size=83 \[ -\frac{b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac{(a+b)^2 \tanh ^3(c+d x)}{3 d}-\frac{(a+b)^2 \tanh (c+d x)}{d}+x (a+b)^2-\frac{b^2 \tanh ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.0788679, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3670, 461, 206} \[ -\frac{b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac{(a+b)^2 \tanh ^3(c+d x)}{3 d}-\frac{(a+b)^2 \tanh (c+d x)}{d}+x (a+b)^2-\frac{b^2 \tanh ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 461
Rule 206
Rubi steps
\begin{align*} \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (a+b x^2\right )^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-(a+b)^2-(a+b)^2 x^2-b (2 a+b) x^4-b^2 x^6+\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{(a+b)^2 \tanh ^3(c+d x)}{3 d}-\frac{b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac{b^2 \tanh ^7(c+d x)}{7 d}+\frac{(a+b)^2 \operatorname{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{(a+b)^2 \tanh ^3(c+d x)}{3 d}-\frac{b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac{b^2 \tanh ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [B] time = 0.0767448, size = 190, normalized size = 2.29 \[ -\frac{a^2 \tanh ^3(c+d x)}{3 d}+\frac{a^2 \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{a^2 \tanh (c+d x)}{d}-\frac{2 a b \tanh ^5(c+d x)}{5 d}-\frac{2 a b \tanh ^3(c+d x)}{3 d}+\frac{2 a b \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{2 a b \tanh (c+d x)}{d}-\frac{b^2 \tanh ^7(c+d x)}{7 d}-\frac{b^2 \tanh ^5(c+d x)}{5 d}-\frac{b^2 \tanh ^3(c+d x)}{3 d}+\frac{b^2 \tanh ^{-1}(\tanh (c+d x))}{d}-\frac{b^2 \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 236, normalized size = 2.8 \begin{align*} -{\frac{{a}^{2}\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{2\,d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) ab}{d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ){b}^{2}}{2\,d}}-{\frac{{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{3}{a}^{2}}{3\,d}}-{\frac{{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-2\,{\frac{ab\tanh \left ( dx+c \right ) }{d}}-{\frac{2\, \left ( \tanh \left ( dx+c \right ) \right ) ^{5}ab}{5\,d}}-{\frac{2\,ab \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){a}^{2}}{2\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) ab}{d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){b}^{2}}{2\,d}}-{\frac{{a}^{2}\tanh \left ( dx+c \right ) }{d}}-{\frac{{b}^{2}\tanh \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16148, size = 498, normalized size = 6. \begin{align*} \frac{1}{105} \, b^{2}{\left (105 \, x + \frac{105 \, c}{d} - \frac{8 \,{\left (203 \, e^{\left (-2 \, d x - 2 \, c\right )} + 609 \, e^{\left (-4 \, d x - 4 \, c\right )} + 770 \, e^{\left (-6 \, d x - 6 \, c\right )} + 770 \, e^{\left (-8 \, d x - 8 \, c\right )} + 315 \, e^{\left (-10 \, d x - 10 \, c\right )} + 105 \, e^{\left (-12 \, d x - 12 \, c\right )} + 44\right )}}{d{\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}}\right )} + \frac{2}{15} \, a b{\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{1}{3} \, a^{2}{\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97528, size = 2140, normalized size = 25.78 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.51795, size = 165, normalized size = 1.99 \begin{align*} \begin{cases} a^{2} x - \frac{a^{2} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac{a^{2} \tanh{\left (c + d x \right )}}{d} + 2 a b x - \frac{2 a b \tanh ^{5}{\left (c + d x \right )}}{5 d} - \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 ^{7}{\left (c + d x \right )}}{7 d} - \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 ^{4}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31415, size = 405, normalized size = 4.88 \begin{align*} \frac{{\left (a^{2} + 2 \, a b + b^{2}\right )}{\left (d x + c\right )}}{d} + \frac{4 \,{\left (105 \, a^{2} e^{\left (12 \, d x + 12 \, c\right )} + 315 \, a b e^{\left (12 \, d x + 12 \, c\right )} + 210 \, b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 525 \, a^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1260 \, a b e^{\left (10 \, d x + 10 \, c\right )} + 630 \, b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1120 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 2555 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 1540 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 1330 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3080 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 1540 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 945 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2121 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 1218 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 385 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 812 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 406 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 70 \, a^{2} + 161 \, a b + 88 \, b^{2}\right )}}{105 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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