Optimal. Leaf size=65 \[ \frac {16 \coth (x)}{35 \sqrt {-\text {csch}^2(x)}}+\frac {8 \coth (x)}{35 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {6 \coth (x)}{35 \left (-\text {csch}^2(x)\right )^{5/2}}+\frac {\coth (x)}{7 \left (-\text {csch}^2(x)\right )^{7/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4122, 192, 191} \[ \frac {16 \coth (x)}{35 \sqrt {-\text {csch}^2(x)}}+\frac {8 \coth (x)}{35 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {6 \coth (x)}{35 \left (-\text {csch}^2(x)\right )^{5/2}}+\frac {\coth (x)}{7 \left (-\text {csch}^2(x)\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\left (-\text {csch}^2(x)\right )^{7/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{9/2}} \, dx,x,\coth (x)\right )\\ &=\frac {\coth (x)}{7 \left (-\text {csch}^2(x)\right )^{7/2}}+\frac {6}{7} \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{7/2}} \, dx,x,\coth (x)\right )\\ &=\frac {\coth (x)}{7 \left (-\text {csch}^2(x)\right )^{7/2}}+\frac {6 \coth (x)}{35 \left (-\text {csch}^2(x)\right )^{5/2}}+\frac {24}{35} \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{5/2}} \, dx,x,\coth (x)\right )\\ &=\frac {\coth (x)}{7 \left (-\text {csch}^2(x)\right )^{7/2}}+\frac {6 \coth (x)}{35 \left (-\text {csch}^2(x)\right )^{5/2}}+\frac {8 \coth (x)}{35 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {16}{35} \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{3/2}} \, dx,x,\coth (x)\right )\\ &=\frac {\coth (x)}{7 \left (-\text {csch}^2(x)\right )^{7/2}}+\frac {6 \coth (x)}{35 \left (-\text {csch}^2(x)\right )^{5/2}}+\frac {8 \coth (x)}{35 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {16 \coth (x)}{35 \sqrt {-\text {csch}^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 39, normalized size = 0.60 \[ \frac {(1225 \cosh (x)-245 \cosh (3 x)+49 \cosh (5 x)-5 \cosh (7 x)) \text {csch}(x)}{2240 \sqrt {-\text {csch}^2(x)}} \]
Antiderivative was successfully verified.
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fricas [C] time = 1.13, size = 50, normalized size = 0.77 \[ \frac {1}{4480} \, {\left (5 i \, e^{\left (14 \, x\right )} - 49 i \, e^{\left (12 \, x\right )} + 245 i \, e^{\left (10 \, x\right )} - 1225 i \, e^{\left (8 \, x\right )} - 1225 i \, e^{\left (6 \, x\right )} + 245 i \, e^{\left (4 \, x\right )} - 49 i \, e^{\left (2 \, x\right )} + 5 i\right )} e^{\left (-7 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.17, size = 76, normalized size = 1.17 \[ \frac {i \, {\left (1225 \, e^{\left (6 \, x\right )} - 245 \, e^{\left (4 \, x\right )} + 49 \, e^{\left (2 \, x\right )} - 5\right )} e^{\left (-7 \, x\right )}}{4480 \, \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} - \frac {i \, {\left (5 \, e^{\left (7 \, x\right )} - 49 \, e^{\left (5 \, x\right )} + 245 \, e^{\left (3 \, x\right )} - 1225 \, e^{x}\right )}}{4480 \, \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 238, normalized size = 3.66 \[ -\frac {{\mathrm e}^{8 x}}{896 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {7 \,{\mathrm e}^{6 x}}{640 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {7 \,{\mathrm e}^{4 x}}{128 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {35 \,{\mathrm e}^{2 x}}{128 \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}+\frac {35}{128 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {7 \,{\mathrm e}^{-2 x}}{128 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {7 \,{\mathrm e}^{-4 x}}{640 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {{\mathrm e}^{-6 x}}{896 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.03, size = 47, normalized size = 0.72 \[ -\frac {1}{896} i \, e^{\left (7 \, x\right )} + \frac {7}{640} i \, e^{\left (5 \, x\right )} - \frac {7}{128} i \, e^{\left (3 \, x\right )} + \frac {35}{128} i \, e^{\left (-x\right )} - \frac {7}{128} i \, e^{\left (-3 \, x\right )} + \frac {7}{640} i \, e^{\left (-5 \, x\right )} - \frac {1}{896} i \, e^{\left (-7 \, x\right )} + \frac {35}{128} i \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (-\frac {1}{{\mathrm {sinh}\relax (x)}^2}\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \operatorname {csch}^{2}{\relax (x )}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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