Optimal. Leaf size=81 \[ \frac {\left (a^3+b^3\right ) \log (\sinh (c+d x))}{d}-\frac {(a+b)^3 \text {csch}^4(c+d x)}{4 d}-\frac {(2 a-b) (a+b)^2 \text {csch}^2(c+d x)}{2 d}-\frac {b^3 \log (\cosh (c+d x))}{d} \]
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Rubi [A] time = 0.12, antiderivative size = 81, 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 = {4138, 446, 88} \[ \frac {\left (a^3+b^3\right ) \log (\sinh (c+d x))}{d}-\frac {(a+b)^3 \text {csch}^4(c+d x)}{4 d}-\frac {(2 a-b) (a+b)^2 \text {csch}^2(c+d x)}{2 d}-\frac {b^3 \log (\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rule 4138
Rubi steps
\begin {align*} \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (b+a x^2\right )^3}{x \left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(b+a x)^3}{(1-x)^3 x} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-\frac {(a+b)^3}{(-1+x)^3}-\frac {(2 a-b) (a+b)^2}{(-1+x)^2}+\frac {-a^3-b^3}{-1+x}+\frac {b^3}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {(2 a-b) (a+b)^2 \text {csch}^2(c+d x)}{2 d}-\frac {(a+b)^3 \text {csch}^4(c+d x)}{4 d}-\frac {b^3 \log (\cosh (c+d x))}{d}+\frac {\left (a^3+b^3\right ) \log (\sinh (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.91, size = 101, normalized size = 1.25 \[ -\frac {2 \left (a \cosh ^2(c+d x)+b\right )^3 \left (-4 \left (a^3+b^3\right ) \log (\sinh (c+d x))+(a+b)^3 \text {csch}^4(c+d x)+2 (2 a-b) (a+b)^2 \text {csch}^2(c+d x)+4 b^3 \log (\cosh (c+d x))\right )}{d (a \cosh (2 (c+d x))+a+2 b)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 1830, normalized size = 22.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 248, normalized size = 3.06 \[ -\frac {12 \, a^{3} d x + 12 \, b^{3} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 12 \, {\left (a^{3} e^{\left (2 \, c\right )} + b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, c\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {25 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 25 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 52 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 72 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 124 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 102 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 144 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 246 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 52 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 72 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 124 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 25 \, a^{3} + 25 \, b^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 153, normalized size = 1.89 \[ \frac {a^{3} \ln \left (\sinh \left (d x +c \right )\right )}{d}-\frac {a^{3} \left (\coth ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a^{3} \left (\coth ^{4}\left (d x +c \right )\right )}{4 d}-\frac {3 a^{2} b \left (\cosh ^{2}\left (d x +c \right )\right )}{2 d \sinh \left (d x +c \right )^{4}}+\frac {3 a^{2} b}{4 d \sinh \left (d x +c \right )^{4}}-\frac {3 a \,b^{2}}{4 d \sinh \left (d x +c \right )^{4}}-\frac {b^{3}}{4 d \sinh \left (d x +c \right )^{4}}+\frac {b^{3}}{2 d \sinh \left (d x +c \right )^{2}}+\frac {b^{3} \ln \left (\tanh \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 422, normalized size = 5.21 \[ a^{3} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {4 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + b^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac {2 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 4 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + 6 \, a^{2} b {\left (\frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}} + \frac {e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - \frac {12 \, a b^{2}}{d {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.64, size = 384, normalized size = 4.74 \[ -a^3\,x-\frac {2\,\left (4\,a^3+9\,a^2\,b+6\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )\,\left (b^3\,d-d\,\left (a^3+b^3\right )\right )}{2\,d^2}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (a^6\,\sqrt {-d^2}+4\,b^6\,\sqrt {-d^2}+4\,a^3\,b^3\,\sqrt {-d^2}\right )}{a^3\,d\,\sqrt {a^6+4\,a^3\,b^3+4\,b^6}+2\,b^3\,d\,\sqrt {a^6+4\,a^3\,b^3+4\,b^6}}\right )\,\sqrt {a^6+4\,a^3\,b^3+4\,b^6}}{\sqrt {-d^2}}-\frac {2\,\left (2\,a^3+3\,a^2\,b-b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {8\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {4\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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