Optimal. Leaf size=94 \[ -\frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac{b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac{(a+b)^3 \tanh (c+d x)}{d}+x (a+b)^3-\frac{b^3 \tanh ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.0927085, antiderivative size = 94, 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} \[ -\frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac{b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac{(a+b)^3 \tanh (c+d x)}{d}+x (a+b)^3-\frac{b^3 \tanh ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 461
Rule 206
Rubi steps
\begin{align*} \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (a+b x^2\right )^3}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-(a+b)^3-b \left (3 a^2+3 a b+b^2\right ) x^2-b^2 (3 a+b) x^4-b^3 x^6+\frac{a^3+3 a^2 b+3 a b^2+b^3}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{(a+b)^3 \tanh (c+d x)}{d}-\frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac{b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac{b^3 \tanh ^7(c+d x)}{7 d}+\frac{(a+b)^3 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=(a+b)^3 x-\frac{(a+b)^3 \tanh (c+d x)}{d}-\frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac{b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac{b^3 \tanh ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 1.61293, size = 108, normalized size = 1.15 \[ \frac{\tanh (c+d x) \left (-35 b \left (3 a^2+3 a b+b^2\right ) \tanh ^2(c+d x)-21 b^2 (3 a+b) \tanh ^4(c+d x)+\frac{105 (a+b)^3 \tanh ^{-1}\left (\sqrt{\tanh ^2(c+d x)}\right )}{\sqrt{\tanh ^2(c+d x)}}-105 (a+b)^3-15 b^3 \tanh ^6(c+d x)\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 299, normalized size = 3.2 \begin{align*} -{\frac{{a}^{3}\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{2\,d}}-{\frac{3\,\ln \left ( \tanh \left ( dx+c \right ) -1 \right ){a}^{2}b}{2\,d}}-{\frac{3\,\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) a{b}^{2}}{2\,d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ){b}^{3}}{2\,d}}-{\frac{{b}^{3} \left ( \tanh \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-{\frac{{b}^{3} \left ( \tanh \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{{b}^{3} \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{b}^{3}\tanh \left ( dx+c \right ) }{d}}-{\frac{{a}^{2}b \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{a{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{3\,a{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-3\,{\frac{{a}^{2}b\tanh \left ( dx+c \right ) }{d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){a}^{3}}{2\,d}}+{\frac{3\,\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){a}^{2}b}{2\,d}}+{\frac{3\,\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) a{b}^{2}}{2\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){b}^{3}}{2\,d}}-{\frac{{a}^{3}\tanh \left ( dx+c \right ) }{d}}-3\,{\frac{a{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 = 1.06923, size = 540, normalized size = 5.74 \begin{align*} \frac{1}{105} \, b^{3}{\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{1}{5} \, a 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 )} + a^{2} 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^{3}{\left (x + \frac{c}{d} - \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97666, size = 2693, normalized size = 28.65 \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 = 1.69827, size = 192, normalized size = 2.04 \begin{align*} \begin{cases} a^{3} x - \frac{a^{3} \tanh{\left (c + d x \right )}}{d} + 3 a^{2} b x - \frac{a^{2} b \tanh ^{3}{\left (c + d x \right )}}{d} - \frac{3 a^{2} b \tanh{\left (c + d x \right )}}{d} + 3 a b^{2} x - \frac{3 a b^{2} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac{a b^{2} \tanh ^{3}{\left (c + d x \right )}}{d} - \frac{3 a b^{2} \tanh{\left (c + d x \right )}}{d} + b^{3} x - \frac{b^{3} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac{b^{3} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac{b^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac{b^{3} \tanh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{3} \tanh ^{2}{\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 = 1.46132, size = 564, normalized size = 6. \begin{align*} \frac{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}{\left (d x + c\right )}}{d} + \frac{2 \,{\left (105 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 945 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 420 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 3150 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 3780 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1260 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 6720 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 7665 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 3080 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 2100 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 7980 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 9240 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3080 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 5670 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 6363 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2436 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 630 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2310 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2436 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 812 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3}\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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