Optimal. Leaf size=70 \[ -\frac {15 \text {csch}(a+b x)}{8 b}-\frac {15 \tan ^{-1}(\sinh (a+b x))}{8 b}+\frac {\text {csch}(a+b x) \text {sech}^4(a+b x)}{4 b}+\frac {5 \text {csch}(a+b x) \text {sech}^2(a+b x)}{8 b} \]
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Rubi [A] time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2621, 288, 321, 207} \[ -\frac {15 \text {csch}(a+b x)}{8 b}-\frac {15 \tan ^{-1}(\sinh (a+b x))}{8 b}+\frac {\text {csch}(a+b x) \text {sech}^4(a+b x)}{4 b}+\frac {5 \text {csch}(a+b x) \text {sech}^2(a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 207
Rule 288
Rule 321
Rule 2621
Rubi steps
\begin {align*} \int \text {csch}^2(a+b x) \text {sech}^5(a+b x) \, dx &=-\frac {i \operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^3} \, dx,x,-i \text {csch}(a+b x)\right )}{b}\\ &=\frac {\text {csch}(a+b x) \text {sech}^4(a+b x)}{4 b}-\frac {(5 i) \operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,-i \text {csch}(a+b x)\right )}{4 b}\\ &=\frac {5 \text {csch}(a+b x) \text {sech}^2(a+b x)}{8 b}+\frac {\text {csch}(a+b x) \text {sech}^4(a+b x)}{4 b}-\frac {(15 i) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,-i \text {csch}(a+b x)\right )}{8 b}\\ &=-\frac {15 \text {csch}(a+b x)}{8 b}+\frac {5 \text {csch}(a+b x) \text {sech}^2(a+b x)}{8 b}+\frac {\text {csch}(a+b x) \text {sech}^4(a+b x)}{4 b}-\frac {(15 i) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,-i \text {csch}(a+b x)\right )}{8 b}\\ &=-\frac {15 \tan ^{-1}(\sinh (a+b x))}{8 b}-\frac {15 \text {csch}(a+b x)}{8 b}+\frac {5 \text {csch}(a+b x) \text {sech}^2(a+b x)}{8 b}+\frac {\text {csch}(a+b x) \text {sech}^4(a+b x)}{4 b}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 29, normalized size = 0.41 \[ -\frac {\text {csch}(a+b x) \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};-\sinh ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 1183, normalized size = 16.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 124, normalized size = 1.77 \[ -\frac {15 \, \pi + \frac {4 \, {\left (7 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3} + 36 \, e^{\left (b x + a\right )} - 36 \, e^{\left (-b x - a\right )}\right )}}{{\left ({\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4\right )}^{2}} + \frac {32}{e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}} + 30 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 71, normalized size = 1.01 \[ -\frac {1}{b \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}-\frac {5 \mathrm {sech}\left (b x +a \right )^{3} \tanh \left (b x +a \right )}{4 b}-\frac {15 \,\mathrm {sech}\left (b x +a \right ) \tanh \left (b x +a \right )}{8 b}-\frac {15 \arctan \left ({\mathrm e}^{b x +a}\right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 136, normalized size = 1.94 \[ \frac {15 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{4 \, b} - \frac {15 \, e^{\left (-b x - a\right )} + 40 \, e^{\left (-3 \, b x - 3 \, a\right )} + 18 \, e^{\left (-5 \, b x - 5 \, a\right )} + 40 \, e^{\left (-7 \, b x - 7 \, a\right )} + 15 \, e^{\left (-9 \, b x - 9 \, a\right )}}{4 \, b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 2 \, e^{\left (-4 \, b x - 4 \, a\right )} - 2 \, e^{\left (-6 \, b x - 6 \, a\right )} - 3 \, e^{\left (-8 \, b x - 8 \, a\right )} - e^{\left (-10 \, b x - 10 \, a\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 210, normalized size = 3.00 \[ \frac {3\,{\mathrm {e}}^{a+b\,x}}{2\,b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}-\frac {15\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{4\,\sqrt {b^2}}+\frac {6\,{\mathrm {e}}^{a+b\,x}}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}+3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}+1\right )}-\frac {4\,{\mathrm {e}}^{a+b\,x}}{b\,\left (4\,{\mathrm {e}}^{2\,a+2\,b\,x}+6\,{\mathrm {e}}^{4\,a+4\,b\,x}+4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {7\,{\mathrm {e}}^{a+b\,x}}{4\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
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 \[ \int \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}^{5}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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