Optimal. Leaf size=338 \[ \frac {(e x)^{2 n}}{2 a e n}+\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}-\frac {b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n} \]
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Rubi [A]
time = 0.46, antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4294, 4290,
4276, 3404, 2296, 2221, 2317, 2438} \begin {gather*} \frac {b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}-\frac {b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}+\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d e n \sqrt {b^2-a^2}}-\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d e n \sqrt {b^2-a^2}}+\frac {(e x)^{2 n}}{2 a e n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3404
Rule 4276
Rule 4290
Rule 4294
Rubi steps
\begin {align*} \int \frac {(e x)^{-1+2 n}}{a+b \csc \left (c+d x^n\right )} \, dx &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \int \frac {x^{-1+2 n}}{a+b \csc \left (c+d x^n\right )} \, dx}{e}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{a+b \csc (c+d x)} \, dx,x,x^n\right )}{e n}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \left (\frac {x}{a}-\frac {b x}{a (b+a \sin (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac {(e x)^{2 n}}{2 a e n}-\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{b+a \sin (c+d x)} \, dx,x,x^n\right )}{a e n}\\ &=\frac {(e x)^{2 n}}{2 a e n}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,x^n\right )}{a e n}\\ &=\frac {(e x)^{2 n}}{2 a e n}+\frac {\left (2 i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{2 b-2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^n\right )}{\sqrt {-a^2+b^2} e n}-\frac {\left (2 i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{2 b+2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^n\right )}{\sqrt {-a^2+b^2} e n}\\ &=\frac {(e x)^{2 n}}{2 a e n}+\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} d e n}\\ &=\frac {(e x)^{2 n}}{2 a e n}+\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i a x}{2 b-2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i a x}{2 b+2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a \sqrt {-a^2+b^2} d^2 e n}\\ &=\frac {(e x)^{2 n}}{2 a e n}+\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}-\frac {b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(1003\) vs. \(2(338)=676\).
time = 5.55, size = 1003, normalized size = 2.97 \begin {gather*} \frac {(e x)^{2 n} \csc \left (c+d x^n\right ) \left (1-\frac {2 b x^{-2 n} \left (\frac {\pi \text {ArcTan}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+\frac {2 \left (c-\text {ArcCos}\left (-\frac {b}{a}\right )\right ) \tanh ^{-1}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )+\left (-2 c+\pi -2 d x^n\right ) \tanh ^{-1}\left (\frac {(a+b) \tan \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )-\left (\text {ArcCos}\left (-\frac {b}{a}\right )-2 i \tanh ^{-1}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (\frac {(a+b) \left (a-b-i \sqrt {a^2-b^2}\right ) \left (1+i \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )\right )}\right )+\left (\text {ArcCos}\left (-\frac {b}{a}\right )+2 i \left (-\tanh ^{-1}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )+\tanh ^{-1}\left (\frac {(a+b) \tan \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )\right )\right ) \log \left (\frac {\sqrt [4]{-1} \sqrt {a^2-b^2} e^{-\frac {1}{2} i \left (c+d x^n\right )}}{\sqrt {2} \sqrt {a} \sqrt {b+a \sin \left (c+d x^n\right )}}\right )+\left (\text {ArcCos}\left (-\frac {b}{a}\right )+2 i \tanh ^{-1}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )-2 i \tanh ^{-1}\left (\frac {(a+b) \tan \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (-\frac {(-1)^{3/4} \sqrt {a^2-b^2} e^{\frac {1}{2} i \left (c+d x^n\right )}}{\sqrt {2} \sqrt {a} \sqrt {b+a \sin \left (c+d x^n\right )}}\right )-\left (\text {ArcCos}\left (-\frac {b}{a}\right )+2 i \tanh ^{-1}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (1+\frac {i \left (i b+\sqrt {a^2-b^2}\right ) \left (a+b+\sqrt {a^2-b^2} \tan \left (\frac {1}{4} \left (2 c-\pi +2 d x^n\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )\right )}\right )+i \left (\text {PolyLog}\left (2,\frac {\left (b-i \sqrt {a^2-b^2}\right ) \left (a+b+\sqrt {a^2-b^2} \tan \left (\frac {1}{4} \left (2 c-\pi +2 d x^n\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )\right )}\right )-\text {PolyLog}\left (2,\frac {\left (b+i \sqrt {a^2-b^2}\right ) \left (a+b+\sqrt {a^2-b^2} \tan \left (\frac {1}{4} \left (2 c-\pi +2 d x^n\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )\right )}\right )\right )}{\sqrt {a^2-b^2}}\right )}{d^2}\right ) \left (b+a \sin \left (c+d x^n\right )\right )}{2 a e n \left (a+b \csc \left (c+d x^n\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.29, size = 1340, normalized size = 3.96
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1340\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1218 vs. \(2 (302) = 604\).
time = 5.78, size = 1218, normalized size = 3.60 \begin {gather*} -\frac {a b c \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} e^{\left (2 \, n - 1\right )} \log \left (2 \, a \cos \left (d x^{n} + c\right ) + 2 i \, a \sin \left (d x^{n} + c\right ) + 2 \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} + 2 i \, b\right ) + a b c \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} e^{\left (2 \, n - 1\right )} \log \left (2 \, a \cos \left (d x^{n} + c\right ) - 2 i \, a \sin \left (d x^{n} + c\right ) + 2 \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - 2 i \, b\right ) - a b c \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} e^{\left (2 \, n - 1\right )} \log \left (-2 \, a \cos \left (d x^{n} + c\right ) + 2 i \, a \sin \left (d x^{n} + c\right ) + 2 \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} + 2 i \, b\right ) - a b c \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} e^{\left (2 \, n - 1\right )} \log \left (-2 \, a \cos \left (d x^{n} + c\right ) - 2 i \, a \sin \left (d x^{n} + c\right ) + 2 \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - 2 i \, b\right ) - {\left (a^{2} - b^{2}\right )} d^{2} x^{2 \, n} e^{\left (2 \, n - 1\right )} - i \, a b \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} {\rm Li}_2\left (\frac {{\left (a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} + i \, b\right )} \cos \left (d x^{n} + c\right ) + {\left (i \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - b\right )} \sin \left (d x^{n} + c\right ) - a}{a} + 1\right ) e^{\left (2 \, n - 1\right )} - i \, a b \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} {\rm Li}_2\left (-\frac {{\left (a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} + i \, b\right )} \cos \left (d x^{n} + c\right ) - {\left (i \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - b\right )} \sin \left (d x^{n} + c\right ) + a}{a} + 1\right ) e^{\left (2 \, n - 1\right )} + i \, a b \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} {\rm Li}_2\left (\frac {{\left (a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - i \, b\right )} \cos \left (d x^{n} + c\right ) + {\left (-i \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - b\right )} \sin \left (d x^{n} + c\right ) - a}{a} + 1\right ) e^{\left (2 \, n - 1\right )} + i \, a b \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} {\rm Li}_2\left (-\frac {{\left (a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - i \, b\right )} \cos \left (d x^{n} + c\right ) - {\left (-i \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - b\right )} \sin \left (d x^{n} + c\right ) + a}{a} + 1\right ) e^{\left (2 \, n - 1\right )} + {\left (a b d x^{n} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} e^{\left (2 \, n - 1\right )} + a b c \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} e^{\left (2 \, n - 1\right )}\right )} \log \left (-\frac {{\left (a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} + i \, b\right )} \cos \left (d x^{n} + c\right ) + {\left (i \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - b\right )} \sin \left (d x^{n} + c\right ) - a}{a}\right ) - {\left (a b d x^{n} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} e^{\left (2 \, n - 1\right )} + a b c \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} e^{\left (2 \, n - 1\right )}\right )} \log \left (\frac {{\left (a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} + i \, b\right )} \cos \left (d x^{n} + c\right ) - {\left (i \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - b\right )} \sin \left (d x^{n} + c\right ) + a}{a}\right ) + {\left (a b d x^{n} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} e^{\left (2 \, n - 1\right )} + a b c \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} e^{\left (2 \, n - 1\right )}\right )} \log \left (-\frac {{\left (a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - i \, b\right )} \cos \left (d x^{n} + c\right ) + {\left (-i \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - b\right )} \sin \left (d x^{n} + c\right ) - a}{a}\right ) - {\left (a b d x^{n} \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} e^{\left (2 \, n - 1\right )} + a b c \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} e^{\left (2 \, n - 1\right )}\right )} \log \left (\frac {{\left (a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - i \, b\right )} \cos \left (d x^{n} + c\right ) - {\left (-i \, a \sqrt {\frac {a^{2} - b^{2}}{a^{2}}} - b\right )} \sin \left (d x^{n} + c\right ) + a}{a}\right )}{2 \, {\left (a^{3} - a b^{2}\right )} d^{2} n} \end {gather*}
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 \begin {gather*} \int \frac {\left (e x\right )^{2 n - 1}}{a + b \csc {\left (c + d x^{n} \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,x\right )}^{2\,n-1}}{a+\frac {b}{\sin \left (c+d\,x^n\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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