Optimal. Leaf size=42 \[ -x \csc \left (a+b \log \left (c x^n\right )\right )-b n x \cot \left (a+b \log \left (c x^n\right )\right ) \csc \left (a+b \log \left (c x^n\right )\right ) \]
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Rubi [C] time = 0.13, antiderivative size = 172, normalized size of antiderivative = 4.10, number of steps used = 7, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {4504, 4506, 364} \[ 2 e^{i a} x (b n+i) \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {i}{b n}\right );\frac {1}{2} \left (3-\frac {i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )-\frac {16 e^{3 i a} b^2 n^2 x \left (c x^n\right )^{3 i b} \, _2F_1\left (3,\frac {1}{2} \left (3-\frac {i}{b n}\right );\frac {1}{2} \left (5-\frac {i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-3 b n+i} \]
Warning: Unable to verify antiderivative.
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Rule 364
Rule 4504
Rule 4506
Rubi steps
\begin {align*} \int \left (-\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\left (2 b^2 n^2\right ) \int \csc ^3\left (a+b \log \left (c x^n\right )\right ) \, dx+\left (-1-b^2 n^2\right ) \int \csc \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\left (2 b^2 n x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {1}{n}} \csc ^3(a+b \log (x)) \, dx,x,c x^n\right )+\frac {\left (\left (-1-b^2 n^2\right ) x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {1}{n}} \csc (a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\left (16 i b^2 e^{3 i a} n x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+3 i b+\frac {1}{n}}}{\left (1-e^{2 i a} x^{2 i b}\right )^3} \, dx,x,c x^n\right )-\frac {\left (2 i e^{i a} \left (-1-b^2 n^2\right ) x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+i b+\frac {1}{n}}}{1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n}\\ &=2 e^{i a} (i+b n) x \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {i}{b n}\right );\frac {1}{2} \left (3-\frac {i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )-\frac {16 b^2 e^{3 i a} n^2 x \left (c x^n\right )^{3 i b} \, _2F_1\left (3,\frac {1}{2} \left (3-\frac {i}{b n}\right );\frac {1}{2} \left (5-\frac {i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{i-3 b n}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 30, normalized size = 0.71 \[ -x \left (b n \cot \left (a+b \log \left (c x^n\right )\right )+1\right ) \csc \left (a+b \log \left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 3.56, size = 50, normalized size = 1.19 \[ \frac {b n x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + x \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{\cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 1} \]
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 \[ \int 2 \, b^{2} n^{2} \csc \left (b \log \left (c x^{n}\right ) + a\right )^{3} - {\left (b^{2} n^{2} + 1\right )} \csc \left (b \log \left (c x^{n}\right ) + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.65, size = 523, normalized size = 12.45 \[ \frac {2 c^{i b} \left (x^{n}\right )^{i b} x \left (n b \,c^{2 i b} \left (x^{n}\right )^{2 i b} {\mathrm e}^{\frac {3 b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{-\frac {3 b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {3 b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )}{2}} {\mathrm e}^{\frac {3 b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )}{2}} {\mathrm e}^{3 i a}+b n \,{\mathrm e}^{\frac {b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{-\frac {b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )}{2}} {\mathrm e}^{\frac {b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )}{2}} {\mathrm e}^{i a}-i c^{2 i b} \left (x^{n}\right )^{2 i b} {\mathrm e}^{\frac {3 b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{-\frac {3 b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {3 b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )}{2}} {\mathrm e}^{\frac {3 b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )}{2}} {\mathrm e}^{3 i a}+i {\mathrm e}^{\frac {b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{-\frac {b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )}{2}} {\mathrm e}^{\frac {b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )}{2}} {\mathrm e}^{i a}\right )}{\left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{-b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{b \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )} {\mathrm e}^{2 i a}-1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.66, size = 1701, normalized size = 40.50 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.26, size = 85, normalized size = 2.02 \[ \frac {2\,x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,\left (b\,n+1{}\mathrm {i}\right )+2\,x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\,\left (b\,n-\mathrm {i}\right )}{{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )}^2} \]
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 \[ \int \left (2 b^{2} n^{2} \csc ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} - b^{2} n^{2} - 1\right ) \csc {\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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