Optimal. Leaf size=103 \[ -\frac {a^2 \coth (c+d x)}{d}+\frac {b (16 a+11 b) \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac {1}{16} b x (16 a+5 b)+\frac {b^2 \sinh (c+d x) \cosh ^5(c+d x)}{6 d}-\frac {13 b^2 \sinh (c+d x) \cosh ^3(c+d x)}{24 d} \]
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Rubi [A] time = 0.19, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {a^2 \coth (c+d x)}{d}+\frac {b (16 a+11 b) \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac {1}{16} b x (16 a+5 b)+\frac {b^2 \sinh (c+d x) \cosh ^5(c+d x)}{6 d}-\frac {13 b^2 \sinh (c+d x) \cosh ^3(c+d x)}{24 d} \]
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 )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a-2 a x^2+(a+b) x^4\right )^2}{x^2 \left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^2 \cosh ^5(c+d x) \sinh (c+d x)}{6 d}-\frac {\operatorname {Subst}\left (\int \frac {-6 a^2+\left (18 a^2+b^2\right ) x^2-6 (3 a-b) (a+b) x^4+6 (a+b)^2 x^6}{x^2 \left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{6 d}\\ &=-\frac {13 b^2 \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b^2 \cosh ^5(c+d x) \sinh (c+d x)}{6 d}+\frac {\operatorname {Subst}\left (\int \frac {24 a^2-3 \left (16 a^2-3 b^2\right ) x^2+24 (a+b)^2 x^4}{x^2 \left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{24 d}\\ &=\frac {b (16 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b^2 \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b^2 \cosh ^5(c+d x) \sinh (c+d x)}{6 d}-\frac {\operatorname {Subst}\left (\int \frac {-48 a^2+3 \left (16 a^2+16 a b+5 b^2\right ) x^2}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{48 d}\\ &=-\frac {a^2 \coth (c+d x)}{d}+\frac {b (16 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b^2 \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b^2 \cosh ^5(c+d x) \sinh (c+d x)}{6 d}-\frac {(b (16 a+5 b)) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{16 d}\\ &=-\frac {1}{16} b (16 a+5 b) x-\frac {a^2 \coth (c+d x)}{d}+\frac {b (16 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b^2 \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b^2 \cosh ^5(c+d x) \sinh (c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 77, normalized size = 0.75 \[ \frac {b ((96 a+45 b) \sinh (2 (c+d x))-192 a c-192 a d x-9 b \sinh (4 (c+d x))+b \sinh (6 (c+d x))-60 b c-60 b d x)-192 a^2 \coth (c+d x)}{192 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.99, size = 217, normalized size = 2.11 \[ \frac {b^{2} \cosh \left (d x + c\right )^{7} + 7 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - 10 \, b^{2} \cosh \left (d x + c\right )^{5} + 5 \, {\left (7 \, b^{2} \cosh \left (d x + c\right )^{3} - 10 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 6 \, {\left (16 \, a b + 9 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (21 \, b^{2} \cosh \left (d x + c\right )^{5} - 100 \, b^{2} \cosh \left (d x + c\right )^{3} + 18 \, {\left (16 \, a b + 9 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 3 \, {\left (128 \, a^{2} + 32 \, a b + 15 \, b^{2}\right )} \cosh \left (d x + c\right ) - 24 \, {\left ({\left (16 \, a b + 5 \, b^{2}\right )} d x - 16 \, a^{2}\right )} \sinh \left (d x + c\right )}{384 \, d \sinh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 179, normalized size = 1.74 \[ \frac {b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 9 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 96 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 45 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 24 \, {\left (16 \, a b + 5 \, b^{2}\right )} {\left (d x + c\right )} + {\left (352 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 110 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 96 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 45 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - \frac {768 \, a^{2}}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 91, normalized size = 0.88 \[ \frac {-a^{2} \coth \left (d x +c \right )+2 a b \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+b^{2} \left (\left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sinh ^{3}\left (d x +c \right )\right )}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 146, normalized size = 1.42 \[ -\frac {1}{4} \, a b {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{384} \, b^{2} {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} + \frac {2 \, a^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 148, normalized size = 1.44 \[ \frac {3\,b^2\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{128\,d}-\frac {2\,a^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (15\,b^2+32\,a\,b\right )}{128\,d}-x\,\left (\frac {5\,b^2}{16}+a\,b\right )-\frac {3\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}}{128\,d}-\frac {b^2\,{\mathrm {e}}^{-6\,c-6\,d\,x}}{384\,d}+\frac {b^2\,{\mathrm {e}}^{6\,c+6\,d\,x}}{384\,d}+\frac {b\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (32\,a+15\,b\right )}{128\,d} \]
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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