Optimal. Leaf size=181 \[ -\frac {a^3 \coth (c+d x)}{d}+\frac {b \left (384 a^2+528 a b+193 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{256 d}-\frac {3}{256} b x \left (128 a^2+80 a b+21 b^2\right )+\frac {b^2 (80 a+171 b) \sinh (c+d x) \cosh ^5(c+d x)}{160 d}-\frac {b^2 (208 a+149 b) \sinh (c+d x) \cosh ^3(c+d x)}{128 d}+\frac {b^3 \sinh (c+d x) \cosh ^9(c+d x)}{10 d}-\frac {41 b^3 \sinh (c+d x) \cosh ^7(c+d x)}{80 d} \]
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Rubi [A] time = 0.44, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3217, 1259, 1805, 453, 206} \[ \frac {b \left (384 a^2+528 a b+193 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{256 d}-\frac {3}{256} b x \left (128 a^2+80 a b+21 b^2\right )-\frac {a^3 \coth (c+d x)}{d}+\frac {b^2 (80 a+171 b) \sinh (c+d x) \cosh ^5(c+d x)}{160 d}-\frac {b^2 (208 a+149 b) \sinh (c+d x) \cosh ^3(c+d x)}{128 d}+\frac {b^3 \sinh (c+d x) \cosh ^9(c+d x)}{10 d}-\frac {41 b^3 \sinh (c+d x) \cosh ^7(c+d x)}{80 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 453
Rule 1259
Rule 1805
Rule 3217
Rubi steps
\begin {align*} \int \text {csch}^2(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a-2 a x^2+(a+b) x^4\right )^3}{x^2 \left (1-x^2\right )^6} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^3 \cosh ^9(c+d x) \sinh (c+d x)}{10 d}-\frac {\operatorname {Subst}\left (\int \frac {-10 a^3+\left (50 a^3+b^3\right ) x^2-10 \left (10 a^3+3 a^2 b-b^3\right ) x^4+10 \left (10 a^3+9 a^2 b+b^3\right ) x^6-10 (5 a-b) (a+b)^2 x^8+10 (a+b)^3 x^{10}}{x^2 \left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{10 d}\\ &=-\frac {41 b^3 \cosh ^7(c+d x) \sinh (c+d x)}{80 d}+\frac {b^3 \cosh ^9(c+d x) \sinh (c+d x)}{10 d}+\frac {\operatorname {Subst}\left (\int \frac {80 a^3-\left (320 a^3-33 b^3\right ) x^2+240 \left (2 a^3+a^2 b+b^3\right ) x^4-160 (2 a-b) (a+b)^2 x^6+80 (a+b)^3 x^8}{x^2 \left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{80 d}\\ &=\frac {b^2 (80 a+171 b) \cosh ^5(c+d x) \sinh (c+d x)}{160 d}-\frac {41 b^3 \cosh ^7(c+d x) \sinh (c+d x)}{80 d}+\frac {b^3 \cosh ^9(c+d x) \sinh (c+d x)}{10 d}-\frac {\operatorname {Subst}\left (\int \frac {-480 a^3+15 \left (96 a^3+16 a b^2+21 b^3\right ) x^2-1440 (a-b) (a+b)^2 x^4+480 (a+b)^3 x^6}{x^2 \left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{480 d}\\ &=-\frac {b^2 (208 a+149 b) \cosh ^3(c+d x) \sinh (c+d x)}{128 d}+\frac {b^2 (80 a+171 b) \cosh ^5(c+d x) \sinh (c+d x)}{160 d}-\frac {41 b^3 \cosh ^7(c+d x) \sinh (c+d x)}{80 d}+\frac {b^3 \cosh ^9(c+d x) \sinh (c+d x)}{10 d}+\frac {\operatorname {Subst}\left (\int \frac {1920 a^3-15 \left (256 a^3-144 a b^2-65 b^3\right ) x^2+1920 (a+b)^3 x^4}{x^2 \left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{1920 d}\\ &=\frac {b \left (384 a^2+528 a b+193 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{256 d}-\frac {b^2 (208 a+149 b) \cosh ^3(c+d x) \sinh (c+d x)}{128 d}+\frac {b^2 (80 a+171 b) \cosh ^5(c+d x) \sinh (c+d x)}{160 d}-\frac {41 b^3 \cosh ^7(c+d x) \sinh (c+d x)}{80 d}+\frac {b^3 \cosh ^9(c+d x) \sinh (c+d x)}{10 d}-\frac {\operatorname {Subst}\left (\int \frac {-3840 a^3+15 \left (256 a^3+384 a^2 b+240 a b^2+63 b^3\right ) x^2}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{3840 d}\\ &=-\frac {a^3 \coth (c+d x)}{d}+\frac {b \left (384 a^2+528 a b+193 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{256 d}-\frac {b^2 (208 a+149 b) \cosh ^3(c+d x) \sinh (c+d x)}{128 d}+\frac {b^2 (80 a+171 b) \cosh ^5(c+d x) \sinh (c+d x)}{160 d}-\frac {41 b^3 \cosh ^7(c+d x) \sinh (c+d x)}{80 d}+\frac {b^3 \cosh ^9(c+d x) \sinh (c+d x)}{10 d}-\frac {\left (3 b \left (128 a^2+80 a b+21 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{256 d}\\ &=-\frac {3}{256} b \left (128 a^2+80 a b+21 b^2\right ) x-\frac {a^3 \coth (c+d x)}{d}+\frac {b \left (384 a^2+528 a b+193 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{256 d}-\frac {b^2 (208 a+149 b) \cosh ^3(c+d x) \sinh (c+d x)}{128 d}+\frac {b^2 (80 a+171 b) \cosh ^5(c+d x) \sinh (c+d x)}{160 d}-\frac {41 b^3 \cosh ^7(c+d x) \sinh (c+d x)}{80 d}+\frac {b^3 \cosh ^9(c+d x) \sinh (c+d x)}{10 d}\\ \end {align*}
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Mathematica [A] time = 0.78, size = 134, normalized size = 0.74 \[ \frac {-10240 a^3 \coth (c+d x)-120 b \left (128 a^2+80 a b+21 b^2\right ) (c+d x)+60 b \left (128 a^2+120 a b+35 b^2\right ) \sinh (2 (c+d x))-120 b^2 (12 a+5 b) \sinh (4 (c+d x))+10 b^2 (16 a+15 b) \sinh (6 (c+d x))-25 b^3 \sinh (8 (c+d x))+2 b^3 \sinh (10 (c+d x))}{10240 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 474, normalized size = 2.62 \[ \frac {2 \, b^{3} \cosh \left (d x + c\right )^{11} + 22 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{10} - 27 \, b^{3} \cosh \left (d x + c\right )^{9} + 3 \, {\left (110 \, b^{3} \cosh \left (d x + c\right )^{3} - 81 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{8} + 5 \, {\left (32 \, a b^{2} + 35 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 7 \, {\left (132 \, b^{3} \cosh \left (d x + c\right )^{5} - 324 \, b^{3} \cosh \left (d x + c\right )^{3} + 5 \, {\left (32 \, a b^{2} + 35 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} - 50 \, {\left (32 \, a b^{2} + 15 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + {\left (660 \, b^{3} \cosh \left (d x + c\right )^{7} - 3402 \, b^{3} \cosh \left (d x + c\right )^{5} + 175 \, {\left (32 \, a b^{2} + 35 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 250 \, {\left (32 \, a b^{2} + 15 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 60 \, {\left (128 \, a^{2} b + 144 \, a b^{2} + 45 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (110 \, b^{3} \cosh \left (d x + c\right )^{9} - 972 \, b^{3} \cosh \left (d x + c\right )^{7} + 105 \, {\left (32 \, a b^{2} + 35 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 500 \, {\left (32 \, a b^{2} + 15 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 180 \, {\left (128 \, a^{2} b + 144 \, a b^{2} + 45 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 20 \, {\left (1024 \, a^{3} + 384 \, a^{2} b + 360 \, a b^{2} + 105 \, b^{3}\right )} \cosh \left (d x + c\right ) + 80 \, {\left (256 \, a^{3} - 3 \, {\left (128 \, a^{2} b + 80 \, a b^{2} + 21 \, b^{3}\right )} d x\right )} \sinh \left (d x + c\right )}{20480 \, d \sinh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 355, normalized size = 1.96 \[ \frac {2 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 25 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 160 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 150 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 1440 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 600 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 7680 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 7200 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2100 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 240 \, {\left (128 \, a^{2} b + 80 \, a b^{2} + 21 \, b^{3}\right )} {\left (d x + c\right )} - \frac {40960 \, a^{3}}{e^{\left (2 \, d x + 2 \, c\right )} - 1} + {\left (35072 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 21920 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 5754 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 7680 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 7200 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 2100 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 1440 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 600 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 160 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 150 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 25 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b^{3}\right )} e^{\left (-10 \, d x - 10 \, c\right )}}{20480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 163, normalized size = 0.90 \[ \frac {-a^{3} \coth \left (d x +c \right )+3 a^{2} b \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+3 a \,b^{2} \left (\left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sinh ^{3}\left (d x +c \right )\right )}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )+b^{3} \left (\left (\frac {\left (\sinh ^{9}\left (d x +c \right )\right )}{10}-\frac {9 \left (\sinh ^{7}\left (d x +c \right )\right )}{80}+\frac {21 \left (\sinh ^{5}\left (d x +c \right )\right )}{160}-\frac {21 \left (\sinh ^{3}\left (d x +c \right )\right )}{128}+\frac {63 \sinh \left (d x +c \right )}{256}\right ) \cosh \left (d x +c \right )-\frac {63 d x}{256}-\frac {63 c}{256}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 284, normalized size = 1.57 \[ -\frac {3}{8} \, a^{2} b {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{20480} \, b^{3} {\left (\frac {{\left (25 \, e^{\left (-2 \, d x - 2 \, c\right )} - 150 \, e^{\left (-4 \, d x - 4 \, c\right )} + 600 \, e^{\left (-6 \, d x - 6 \, c\right )} - 2100 \, e^{\left (-8 \, d x - 8 \, c\right )} - 2\right )} e^{\left (10 \, d x + 10 \, c\right )}}{d} + \frac {5040 \, {\left (d x + c\right )}}{d} + \frac {2100 \, e^{\left (-2 \, d x - 2 \, c\right )} - 600 \, e^{\left (-4 \, d x - 4 \, c\right )} + 150 \, e^{\left (-6 \, d x - 6 \, c\right )} - 25 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d}\right )} - \frac {1}{128} \, a b^{2} {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} + \frac {2 \, a^{3}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 265, normalized size = 1.46 \[ \frac {5\,b^3\,{\mathrm {e}}^{-8\,c-8\,d\,x}}{4096\,d}-\frac {2\,a^3}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {5\,b^3\,{\mathrm {e}}^{8\,c+8\,d\,x}}{4096\,d}-\frac {b^3\,{\mathrm {e}}^{-10\,c-10\,d\,x}}{10240\,d}+\frac {b^3\,{\mathrm {e}}^{10\,c+10\,d\,x}}{10240\,d}-\frac {3\,b\,x\,\left (128\,a^2+80\,a\,b+21\,b^2\right )}{256}-\frac {3\,b\,{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (128\,a^2+120\,a\,b+35\,b^2\right )}{1024\,d}+\frac {3\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (128\,a^2+120\,a\,b+35\,b^2\right )}{1024\,d}+\frac {3\,b^2\,{\mathrm {e}}^{-4\,c-4\,d\,x}\,\left (12\,a+5\,b\right )}{512\,d}-\frac {3\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (12\,a+5\,b\right )}{512\,d}-\frac {b^2\,{\mathrm {e}}^{-6\,c-6\,d\,x}\,\left (16\,a+15\,b\right )}{2048\,d}+\frac {b^2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (16\,a+15\,b\right )}{2048\,d} \]
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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