Optimal. Leaf size=181 \[ \frac{b \left (384 a^2+528 a b+193 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{256 d}-\frac{3}{256} b x \left (128 a^2+80 a b+21 b^2\right )-\frac{a^3 \coth (c+d x)}{d}+\frac{b^2 (80 a+171 b) \sinh (c+d x) \cosh ^5(c+d x)}{160 d}-\frac{b^2 (208 a+149 b) \sinh (c+d x) \cosh ^3(c+d x)}{128 d}+\frac{b^3 \sinh (c+d x) \cosh ^9(c+d x)}{10 d}-\frac{41 b^3 \sinh (c+d x) \cosh ^7(c+d x)}{80 d} \]
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Rubi [A] time = 0.438191, antiderivative size = 181, normalized size of antiderivative = 1., 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 3217
Rule 1259
Rule 1805
Rule 453
Rule 206
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*}
Mathematica [A] time = 0.770606, size = 134, normalized size = 0.74 \[ \frac{-120 b \left (128 a^2+80 a b+21 b^2\right ) (c+d x)+60 b \left (128 a^2+120 a b+35 b^2\right ) \sinh (2 (c+d x))-10240 a^3 \coth (c+d x)-120 b^2 (12 a+5 b) \sinh (4 (c+d x))+10 b^2 (16 a+15 b) \sinh (6 (c+d x))-25 b^3 \sinh (8 (c+d x))+2 b^3 \sinh (10 (c+d x))}{10240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 163, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -{a}^{3}{\rm coth} \left (dx+c\right )+3\,{a}^{2}b \left ( 1/2\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) -1/2\,dx-c/2 \right ) +3\,a{b}^{2} \left ( \left ( 1/6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}-{\frac{5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{5\,\sinh \left ( dx+c \right ) }{16}} \right ) \cosh \left ( dx+c \right ) -{\frac{5\,dx}{16}}-{\frac{5\,c}{16}} \right ) +{b}^{3} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{9}}{10}}-{\frac{9\, \left ( \sinh \left ( dx+c \right ) \right ) ^{7}}{80}}+{\frac{21\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{160}}-{\frac{21\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{128}}+{\frac{63\,\sinh \left ( dx+c \right ) }{256}} \right ) \cosh \left ( dx+c \right ) -{\frac{63\,dx}{256}}-{\frac{63\,c}{256}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09203, size = 383, normalized size = 2.12 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.44548, size = 1247, normalized size = 6.89 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.70399, size = 531, normalized size = 2.93 \begin{align*} -\frac{3 \,{\left (128 \, a^{2} b + 80 \, a b^{2} + 21 \, b^{3}\right )}{\left (d x + c\right )}}{256 \, d} - \frac{2 \, a^{3}}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}} + \frac{{\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} + \frac{2 \, b^{3} d^{4} e^{\left (10 \, d x + 10 \, c\right )} - 25 \, b^{3} d^{4} e^{\left (8 \, d x + 8 \, c\right )} + 160 \, a b^{2} d^{4} e^{\left (6 \, d x + 6 \, c\right )} + 150 \, b^{3} d^{4} e^{\left (6 \, d x + 6 \, c\right )} - 1440 \, a b^{2} d^{4} e^{\left (4 \, d x + 4 \, c\right )} - 600 \, b^{3} d^{4} e^{\left (4 \, d x + 4 \, c\right )} + 7680 \, a^{2} b d^{4} e^{\left (2 \, d x + 2 \, c\right )} + 7200 \, a b^{2} d^{4} e^{\left (2 \, d x + 2 \, c\right )} + 2100 \, b^{3} d^{4} e^{\left (2 \, d x + 2 \, c\right )}}{20480 \, d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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