Optimal. Leaf size=69 \[ -\frac {8 e^{3 a} x \left (c x^n\right )^{3 b} \, _2F_1\left (3,\frac {3 b+\frac {1}{n}}{2 b};\frac {1}{2} \left (5+\frac {1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{3 b n+1} \]
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Rubi [A] time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5546, 5548, 263, 364} \[ -\frac {8 e^{3 a} x \left (c x^n\right )^{3 b} \, _2F_1\left (3,\frac {3 b+\frac {1}{n}}{2 b};\frac {1}{2} \left (5+\frac {1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{3 b n+1} \]
Antiderivative was successfully verified.
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Rule 263
Rule 364
Rule 5546
Rule 5548
Rubi steps
\begin {align*} \int \text {csch}^3\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {1}{n}} \text {csch}^3(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (8 e^{-3 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1-3 b+\frac {1}{n}}}{\left (1-e^{-2 a} x^{-2 b}\right )^3} \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (8 e^{-3 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+3 b+\frac {1}{n}}}{\left (-e^{-2 a}+x^{2 b}\right )^3} \, dx,x,c x^n\right )}{n}\\ &=-\frac {8 e^{3 a} x \left (c x^n\right )^{3 b} \, _2F_1\left (3,\frac {3 b+\frac {1}{n}}{2 b};\frac {1}{2} \left (5+\frac {1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+3 b n}\\ \end {align*}
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Mathematica [A] time = 5.10, size = 101, normalized size = 1.46 \[ \frac {8 e^a x (b n-1) \left (c x^n\right )^b \, _2F_1\left (1,\frac {1}{2} \left (1+\frac {1}{b n}\right );\frac {1}{2} \left (3+\frac {1}{b n}\right );e^{2 \left (a+b \log \left (c x^n\right )\right )}\right )-4 x \left (b n \coth \left (a+b \log \left (c x^n\right )\right )+1\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{8 b^2 n^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.11, size = 0, normalized size = 0.00 \[ \int \mathrm {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -8 \, {\left (b^{2} n^{2} - 1\right )} \int \frac {1}{16 \, {\left (b^{2} c^{b} n^{2} e^{\left (b \log \left (x^{n}\right ) + a\right )} + b^{2} n^{2}\right )}}\,{d x} - 8 \, {\left (b^{2} n^{2} - 1\right )} \int \frac {1}{16 \, {\left (b^{2} c^{b} n^{2} e^{\left (b \log \left (x^{n}\right ) + a\right )} - b^{2} n^{2}\right )}}\,{d x} - \frac {{\left (b c^{3 \, b} n + c^{3 \, b}\right )} x e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} + {\left (b c^{b} n - c^{b}\right )} x e^{\left (b \log \left (x^{n}\right ) + a\right )}}{b^{2} c^{4 \, b} n^{2} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 2 \, b^{2} c^{2 \, b} n^{2} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b^{2} n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]
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 \[ \int \operatorname {csch}^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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