Optimal. Leaf size=69 \[ -\frac {\left (a^3+b^3\right ) \coth (c+d x)}{d}+a^3 x-\frac {(a+b)^3 \coth ^5(c+d x)}{5 d}-\frac {(a-2 b) (a+b)^2 \coth ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.11, antiderivative size = 69, 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 = {4141, 1802, 207} \[ -\frac {\left (a^3+b^3\right ) \coth (c+d x)}{d}+a^3 x-\frac {(a+b)^3 \coth ^5(c+d x)}{5 d}-\frac {(a-2 b) (a+b)^2 \coth ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 1802
Rule 4141
Rubi steps
\begin {align*} \int \coth ^6(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \left (1-x^2\right )\right )^3}{x^6 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {(a+b)^3}{x^6}+\frac {(a-2 b) (a+b)^2}{x^4}+\frac {a^3+b^3}{x^2}-\frac {a^3}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {\left (a^3+b^3\right ) \coth (c+d x)}{d}-\frac {(a-2 b) (a+b)^2 \coth ^3(c+d x)}{3 d}-\frac {(a+b)^3 \coth ^5(c+d x)}{5 d}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^3 x-\frac {\left (a^3+b^3\right ) \coth (c+d x)}{d}-\frac {(a-2 b) (a+b)^2 \coth ^3(c+d x)}{3 d}-\frac {(a+b)^3 \coth ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [B] time = 1.09, size = 303, normalized size = 4.39 \[ \frac {\text {csch}(c) \text {csch}^5(c+d x) \left (180 a^3 \sinh (2 c+d x)-140 a^3 \sinh (2 c+3 d x)-90 a^3 \sinh (4 c+3 d x)+46 a^3 \sinh (4 c+5 d x)+150 a^3 d x \cosh (2 c+d x)+75 a^3 d x \cosh (2 c+3 d x)-75 a^3 d x \cosh (4 c+3 d x)-15 a^3 d x \cosh (4 c+5 d x)+15 a^3 d x \cosh (6 c+5 d x)+280 a^3 \sinh (d x)-150 a^3 d x \cosh (d x)-90 a^2 b \sinh (4 c+3 d x)+18 a^2 b \sinh (4 c+5 d x)+180 a^2 b \sinh (d x)-180 a b^2 \sinh (2 c+d x)+60 a b^2 \sinh (2 c+3 d x)-12 a b^2 \sinh (4 c+5 d x)+60 a b^2 \sinh (d x)-80 b^3 \sinh (2 c+3 d x)+16 b^3 \sinh (4 c+5 d x)+160 b^3 \sinh (d x)\right )}{480 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 521, normalized size = 7.55 \[ -\frac {{\left (23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - {\left (15 \, a^{3} d x + 23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \sinh \left (d x + c\right )^{5} - 5 \, {\left (5 \, a^{3} - 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 5 \, {\left (15 \, a^{3} d x + 23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3} - 2 \, {\left (15 \, a^{3} d x + 23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (2 \, {\left (23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (5 \, a^{3} - 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, {\left (5 \, a^{3} + 9 \, a^{2} b + 12 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right ) - 5 \, {\left (30 \, a^{3} d x + {\left (15 \, a^{3} d x + 23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 46 \, a^{3} + 18 \, a^{2} b - 12 \, a b^{2} + 16 \, b^{3} - 3 \, {\left (15 \, a^{3} d x + 23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{15 \, {\left (d \sinh \left (d x + c\right )^{5} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (d \cosh \left (d x + c\right )^{4} - 3 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 210, normalized size = 3.04 \[ \frac {15 \, a^{3} d x - \frac {2 \, {\left (45 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 45 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 90 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 140 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 90 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 30 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 80 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 70 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 30 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 40 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 23 \, a^{3} + 9 \, a^{2} b - 6 \, a b^{2} + 8 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.42, size = 199, normalized size = 2.88 \[ \frac {a^{3} \left (d x +c -\coth \left (d x +c \right )-\frac {\left (\coth ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\coth ^{5}\left (d x +c \right )\right )}{5}\right )+3 a^{2} b \left (-\frac {\cosh ^{3}\left (d x +c \right )}{2 \sinh \left (d x +c \right )^{5}}+\frac {3 \cosh \left (d x +c \right )}{8 \sinh \left (d x +c \right )^{5}}+\frac {3 \left (-\frac {8}{15}-\frac {\mathrm {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )}{8}\right )+3 a \,b^{2} \left (-\frac {\cosh \left (d x +c \right )}{4 \sinh \left (d x +c \right )^{5}}-\frac {\left (-\frac {8}{15}-\frac {\mathrm {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )}{4}\right )+b^{3} \left (-\frac {8}{15}-\frac {\mathrm {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 826, normalized size = 11.97 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.60, size = 547, normalized size = 7.93 \[ a^3\,x-\frac {\frac {6\,\left (a^3+b\,a^2\right )}{5\,d}+\frac {24\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2\,b+a\,b^2\right )}{5\,d}+\frac {24\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a^2\,b+a\,b^2\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a^3+b\,a^2\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (5\,a^3+9\,a^2\,b+12\,a\,b^2+8\,b^3\right )}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1}-\frac {\frac {6\,\left (a^2\,b+a\,b^2\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3+b\,a^2\right )}{5\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {6\,\left (a^2\,b+a\,b^2\right )}{5\,d}+\frac {18\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^2\,b+a\,b^2\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a^3+b\,a^2\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (5\,a^3+9\,a^2\,b+12\,a\,b^2+8\,b^3\right )}{5\,d}}{6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {\frac {2\,\left (5\,a^3+9\,a^2\,b+12\,a\,b^2+8\,b^3\right )}{15\,d}+\frac {12\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2\,b+a\,b^2\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^3+b\,a^2\right )}{5\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}-\frac {6\,\left (a^3+b\,a^2\right )}{5\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
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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