Optimal. Leaf size=72 \[ -\frac {a^3 \coth ^2(c+d x)}{2 d}+\frac {a^2 (a+3 b) \log (\tanh (c+d x))}{d}+\frac {(a+b)^3 \log (\cosh (c+d x))}{d}-\frac {b^3 \tanh ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 72, 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 = {3670, 446, 88} \[ \frac {a^2 (a+3 b) \log (\tanh (c+d x))}{d}-\frac {a^3 \coth ^2(c+d x)}{2 d}+\frac {(a+b)^3 \log (\cosh (c+d x))}{d}-\frac {b^3 \tanh ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rule 3670
Rubi steps
\begin {align*} \int \coth ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^3}{x^3 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^3}{(1-x) x^2} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-b^3-\frac {(a+b)^3}{-1+x}+\frac {a^3}{x^2}+\frac {a^2 (a+3 b)}{x}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=-\frac {a^3 \coth ^2(c+d x)}{2 d}+\frac {(a+b)^3 \log (\cosh (c+d x))}{d}+\frac {a^2 (a+3 b) \log (\tanh (c+d x))}{d}-\frac {b^3 \tanh ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 63, normalized size = 0.88 \[ -\frac {a^3 \coth ^2(c+d x)-2 a^2 (a+3 b) \log (\tanh (c+d x))-2 (a+b)^3 \log (\cosh (c+d x))+b^3 \tanh ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 1686, normalized size = 23.42 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 274, normalized size = 3.81 \[ \frac {2 \, {\left (3 \, a b^{2} + b^{3}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right ) + 2 \, {\left (a^{3} + 3 \, a^{2} b\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right ) - \frac {a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 3 \, a^{2} b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 3 \, a b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 8 \, a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} - 8 \, b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 12 \, a^{3} - 12 \, a^{2} b - 12 \, a b^{2} + 12 \, b^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} - 4}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 94, normalized size = 1.31 \[ \frac {a^{3} \ln \left (\sinh \left (d x +c \right )\right )}{d}-\frac {a^{3} \left (\coth ^{2}\left (d x +c \right )\right )}{2 d}+\frac {3 a^{2} b \ln \left (\sinh \left (d x +c \right )\right )}{d}+\frac {3 a \,b^{2} \ln \left (\cosh \left (d x +c \right )\right )}{d}+\frac {b^{3} \ln \left (\cosh \left (d x +c \right )\right )}{d}-\frac {b^{3} \left (\tanh ^{2}\left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 203, normalized size = 2.82 \[ a^{3} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} + b^{3} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {3 \, a b^{2} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}{d} + \frac {3 \, a^{2} b \log \left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.48, size = 327, normalized size = 4.54 \[ \frac {\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )\,\left (d\,a^3+3\,d\,a^2\,b+3\,d\,a\,b^2+d\,b^3\right )}{2\,d^2}-\frac {\frac {4\,\left (a^3+b^3\right )}{d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3-b^3\right )}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1}-\frac {\frac {4\,\left (a^3+b^3\right )}{d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3-b^3\right )}{d}}{{\mathrm {e}}^{8\,c+8\,d\,x}-2\,{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (a^3\,\sqrt {-d^2}-b^3\,\sqrt {-d^2}-3\,a\,b^2\,\sqrt {-d^2}+3\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^6+6\,a^5\,b+3\,a^4\,b^2-20\,a^3\,b^3+3\,a^2\,b^4+6\,a\,b^5+b^6}}\right )\,\sqrt {a^6+6\,a^5\,b+3\,a^4\,b^2-20\,a^3\,b^3+3\,a^2\,b^4+6\,a\,b^5+b^6}}{\sqrt {-d^2}}-x\,{\left (a+b\right )}^3 \]
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 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \coth ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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