Optimal. Leaf size=362 \[ -\frac {i x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d}+\frac {i x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d}-\frac {x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^2}+\frac {x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^2}-\frac {i \text {Li}_3\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^3}+\frac {i \text {Li}_3\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^3} \]
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Rubi [A]
time = 0.56, antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3460, 3404,
2296, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {i \text {PolyLog}\left (3,\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{d^3 \sqrt {a^2-b^2}}+\frac {i \text {PolyLog}\left (3,\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{d^3 \sqrt {a^2-b^2}}-\frac {x^2 \text {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{d^2 \sqrt {a^2-b^2}}+\frac {x^2 \text {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{d^2 \sqrt {a^2-b^2}}-\frac {i x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 d \sqrt {a^2-b^2}}+\frac {i x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{2 d \sqrt {a^2-b^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3404
Rule 3460
Rule 6724
Rubi steps
\begin {align*} \int \frac {x^5}{a+b \sin \left (c+d x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{a+b \sin (c+d x)} \, dx,x,x^2\right )\\ &=\text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx,x,x^2\right )\\ &=-\frac {(i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx,x,x^2\right )}{\sqrt {a^2-b^2}}+\frac {(i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx,x,x^2\right )}{\sqrt {a^2-b^2}}\\ &=-\frac {i x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d}+\frac {i x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d}+\frac {i \text {Subst}\left (\int x \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx,x,x^2\right )}{\sqrt {a^2-b^2} d}-\frac {i \text {Subst}\left (\int x \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx,x,x^2\right )}{\sqrt {a^2-b^2} d}\\ &=-\frac {i x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d}+\frac {i x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d}-\frac {x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^2}+\frac {x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^2}+\frac {\text {Subst}\left (\int \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx,x,x^2\right )}{\sqrt {a^2-b^2} d^2}-\frac {\text {Subst}\left (\int \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx,x,x^2\right )}{\sqrt {a^2-b^2} d^2}\\ &=-\frac {i x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d}+\frac {i x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d}-\frac {x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^2}+\frac {x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^2}-\frac {i \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{\sqrt {a^2-b^2} d^3}+\frac {i \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{\sqrt {a^2-b^2} d^3}\\ &=-\frac {i x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d}+\frac {i x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{2 \sqrt {a^2-b^2} d}-\frac {x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^2}+\frac {x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^2}-\frac {i \text {Li}_3\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^3}+\frac {i \text {Li}_3\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d^3}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 289, normalized size = 0.80 \begin {gather*} \frac {-2 d x^2 \text {Li}_2\left (-\frac {i b e^{i \left (c+d x^2\right )}}{-a+\sqrt {a^2-b^2}}\right )-i \left (d^2 x^4 \log \left (1+\frac {i b e^{i \left (c+d x^2\right )}}{-a+\sqrt {a^2-b^2}}\right )-d^2 x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )+2 i d x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )+2 \text {Li}_3\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )-2 \text {Li}_3\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )\right )}{2 \sqrt {a^2-b^2} d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{5}}{a +b \sin \left (d \,x^{2}+c \right )}\, dx\]
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 \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. 1435 vs. \(2 (300) = 600\).
time = 0.51, size = 1435, normalized size = 3.96 \begin {gather*} \frac {2 i \, b d x^{2} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {i \, a \cos \left (d x^{2} + c\right ) - a \sin \left (d x^{2} + c\right ) + {\left (b \cos \left (d x^{2} + c\right ) + i \, b \sin \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) - 2 i \, b d x^{2} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {i \, a \cos \left (d x^{2} + c\right ) - a \sin \left (d x^{2} + c\right ) - {\left (b \cos \left (d x^{2} + c\right ) + i \, b \sin \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) - 2 i \, b d x^{2} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {-i \, a \cos \left (d x^{2} + c\right ) - a \sin \left (d x^{2} + c\right ) + {\left (b \cos \left (d x^{2} + c\right ) - i \, b \sin \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) + 2 i \, b d x^{2} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {-i \, a \cos \left (d x^{2} + c\right ) - a \sin \left (d x^{2} + c\right ) - {\left (b \cos \left (d x^{2} + c\right ) - i \, b \sin \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) + b c^{2} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (2 \, b \cos \left (d x^{2} + c\right ) + 2 i \, b \sin \left (d x^{2} + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} + 2 i \, a\right ) + b c^{2} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (2 \, b \cos \left (d x^{2} + c\right ) - 2 i \, b \sin \left (d x^{2} + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - 2 i \, a\right ) - b c^{2} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-2 \, b \cos \left (d x^{2} + c\right ) + 2 i \, b \sin \left (d x^{2} + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} + 2 i \, a\right ) - b c^{2} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-2 \, b \cos \left (d x^{2} + c\right ) - 2 i \, b \sin \left (d x^{2} + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - 2 i \, a\right ) - {\left (b d^{2} x^{4} - b c^{2}\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-\frac {i \, a \cos \left (d x^{2} + c\right ) - a \sin \left (d x^{2} + c\right ) + {\left (b \cos \left (d x^{2} + c\right ) + i \, b \sin \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) + {\left (b d^{2} x^{4} - b c^{2}\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-\frac {i \, a \cos \left (d x^{2} + c\right ) - a \sin \left (d x^{2} + c\right ) - {\left (b \cos \left (d x^{2} + c\right ) + i \, b \sin \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) - {\left (b d^{2} x^{4} - b c^{2}\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-\frac {-i \, a \cos \left (d x^{2} + c\right ) - a \sin \left (d x^{2} + c\right ) + {\left (b \cos \left (d x^{2} + c\right ) - i \, b \sin \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) + {\left (b d^{2} x^{4} - b c^{2}\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-\frac {-i \, a \cos \left (d x^{2} + c\right ) - a \sin \left (d x^{2} + c\right ) - {\left (b \cos \left (d x^{2} + c\right ) - i \, b \sin \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm polylog}\left (3, -\frac {i \, a \cos \left (d x^{2} + c\right ) + a \sin \left (d x^{2} + c\right ) + {\left (b \cos \left (d x^{2} + c\right ) - i \, b \sin \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}}}{b}\right ) - 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm polylog}\left (3, -\frac {i \, a \cos \left (d x^{2} + c\right ) + a \sin \left (d x^{2} + c\right ) - {\left (b \cos \left (d x^{2} + c\right ) - i \, b \sin \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}}}{b}\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm polylog}\left (3, -\frac {-i \, a \cos \left (d x^{2} + c\right ) + a \sin \left (d x^{2} + c\right ) + {\left (b \cos \left (d x^{2} + c\right ) + i \, b \sin \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}}}{b}\right ) - 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm polylog}\left (3, -\frac {-i \, a \cos \left (d x^{2} + c\right ) + a \sin \left (d x^{2} + c\right ) - {\left (b \cos \left (d x^{2} + c\right ) + i \, b \sin \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}}}{b}\right )}{4 \, {\left (a^{2} - b^{2}\right )} d^{3}} \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 {x^{5}}{a + b \sin {\left (c + d x^{2} \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 {x^5}{a+b\,\sin \left (d\,x^2+c\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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