Optimal. Leaf size=84 \[ -\frac{4 e^{2 i a} x \left (c x^n\right )^{2 i b} \text{Hypergeometric2F1}\left (2,\frac{1}{2} \left (2-\frac{i}{b n}\right ),\frac{1}{2} \left (4-\frac{i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+2 i b n} \]
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Rubi [A] time = 0.0599466, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4504, 4506, 364} \[ -\frac{4 e^{2 i a} x \left (c x^n\right )^{2 i b} \, _2F_1\left (2,\frac{1}{2} \left (2-\frac{i}{b n}\right );\frac{1}{2} \left (4-\frac{i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+2 i b n} \]
Antiderivative was successfully verified.
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Rule 4504
Rule 4506
Rule 364
Rubi steps
\begin{align*} \int \csc ^2\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}} \csc ^2(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=-\frac{\left (4 e^{2 i a} x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+2 i b+\frac{1}{n}}}{\left (1-e^{2 i a} x^{2 i b}\right )^2} \, dx,x,c x^n\right )}{n}\\ &=-\frac{4 e^{2 i a} x \left (c x^n\right )^{2 i b} \, _2F_1\left (2,\frac{1}{2} \left (2-\frac{i}{b n}\right );\frac{1}{2} \left (4-\frac{i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+2 i b n}\\ \end{align*}
Mathematica [A] time = 5.20936, size = 146, normalized size = 1.74 \[ \frac{x \left (-\frac{e^{2 i a} \left (c x^n\right )^{2 i b} \text{Hypergeometric2F1}\left (1,1-\frac{i}{2 b n},2-\frac{i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b n-i}-i \text{Hypergeometric2F1}\left (1,-\frac{i}{2 b n},1-\frac{i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )-\cot \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.417, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\csc \left (b \log \left (c x^{n}\right ) + a\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{2}{\left (a + b \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc \left (b \log \left (c x^{n}\right ) + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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