Optimal. Leaf size=156 \[ \frac {2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 d e n \left (a^2-b^2\right )^{3/2}}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^n\right )\right )}+\frac {(e x)^n}{a^2 e n} \]
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Rubi [A] time = 0.29, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4209, 4205, 3785, 3919, 3831, 2660, 618, 206} \[ \frac {2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 d e n \left (a^2-b^2\right )^{3/2}}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^n\right )\right )}+\frac {(e x)^n}{a^2 e n} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 2660
Rule 3785
Rule 3831
Rule 3919
Rule 4205
Rule 4209
Rubi steps
\begin {align*} \int \frac {(e x)^{-1+n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx &=\frac {\left (x^{-n} (e x)^n\right ) \int \frac {x^{-1+n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx}{e}\\ &=\frac {\left (x^{-n} (e x)^n\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b \csc (c+d x))^2} \, dx,x,x^n\right )}{e n}\\ &=-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \csc \left (c+d x^n\right )\right )}-\frac {\left (x^{-n} (e x)^n\right ) \operatorname {Subst}\left (\int \frac {-a^2+b^2+a b \csc (c+d x)}{a+b \csc (c+d x)} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) e n}\\ &=\frac {(e x)^n}{a^2 e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \csc \left (c+d x^n\right )\right )}+\frac {\left (\left (-a^2 b+b \left (-a^2+b^2\right )\right ) x^{-n} (e x)^n\right ) \operatorname {Subst}\left (\int \frac {\csc (c+d x)}{a+b \csc (c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) e n}\\ &=\frac {(e x)^n}{a^2 e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \csc \left (c+d x^n\right )\right )}+\frac {\left (\left (-a^2 b+b \left (-a^2+b^2\right )\right ) x^{-n} (e x)^n\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a \sin (c+d x)}{b}} \, dx,x,x^n\right )}{a^2 b \left (a^2-b^2\right ) e n}\\ &=\frac {(e x)^n}{a^2 e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \csc \left (c+d x^n\right )\right )}+\frac {\left (2 \left (-a^2 b+b \left (-a^2+b^2\right )\right ) x^{-n} (e x)^n\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}+x^2} \, dx,x,\tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{a^2 b \left (a^2-b^2\right ) d e n}\\ &=\frac {(e x)^n}{a^2 e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \csc \left (c+d x^n\right )\right )}-\frac {\left (4 \left (-a^2 b+b \left (-a^2+b^2\right )\right ) x^{-n} (e x)^n\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (1-\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}+2 \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{a^2 b \left (a^2-b^2\right ) d e n}\\ &=\frac {(e x)^n}{a^2 e n}+\frac {2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \tanh ^{-1}\left (\frac {b \left (\frac {a}{b}+\tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2} d e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \csc \left (c+d x^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.95, size = 176, normalized size = 1.13 \[ \frac {x^{-n} (e x)^n \left (\sqrt {b^2-a^2} \left (\left (a^2-b^2\right ) \left (c+d x^n\right ) \left (a+b \csc \left (c+d x^n\right )\right )-a b^2 \cot \left (c+d x^n\right )\right )+2 b \left (b^2-2 a^2\right ) \tan ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {b^2-a^2}}\right ) \left (a+b \csc \left (c+d x^n\right )\right )\right )}{a^2 d e n (a-b) (a+b) \sqrt {b^2-a^2} \left (a+b \csc \left (c+d x^n\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 630, normalized size = 4.04 \[ \left [\frac {2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d e^{n - 1} x^{n} \sin \left (d x^{n} + c\right ) + 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d e^{n - 1} x^{n} - 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} e^{n - 1} \cos \left (d x^{n} + c\right ) + {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {a^{2} - b^{2}} e^{n - 1} \sin \left (d x^{n} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {a^{2} - b^{2}} e^{n - 1}\right )} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x^{n} + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} a \cos \left (d x^{n} + c\right ) + a^{2} + b^{2} + 2 \, {\left (\sqrt {a^{2} - b^{2}} b \cos \left (d x^{n} + c\right ) + a b\right )} \sin \left (d x^{n} + c\right )}{a^{2} \cos \left (d x^{n} + c\right )^{2} - 2 \, a b \sin \left (d x^{n} + c\right ) - a^{2} - b^{2}}\right )}{2 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d n \sin \left (d x^{n} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d n\right )}}, \frac {{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d e^{n - 1} x^{n} \sin \left (d x^{n} + c\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d e^{n - 1} x^{n} - {\left (a^{3} b^{2} - a b^{4}\right )} e^{n - 1} \cos \left (d x^{n} + c\right ) + {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {-a^{2} + b^{2}} e^{n - 1} \sin \left (d x^{n} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {-a^{2} + b^{2}} e^{n - 1}\right )} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} b \sin \left (d x^{n} + c\right ) + \sqrt {-a^{2} + b^{2}} a}{{\left (a^{2} - b^{2}\right )} \cos \left (d x^{n} + c\right )}\right )}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d n \sin \left (d x^{n} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d n}\right ] \]
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 \[ \int \frac {\left (e x\right )^{n - 1}}{{\left (b \csc \left (d x^{n} + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.95, size = 712, normalized size = 4.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (e\,x\right )}^{n-1}}{{\left (a+\frac {b}{\sin \left (c+d\,x^n\right )}\right )}^2} \,d x \]
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 \[ \int \frac {\left (e x\right )^{n - 1}}{\left (a + b \csc {\left (c + d x^{n} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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