Integrand size = 18, antiderivative size = 374 \[ \int \frac {x^3}{\sqrt {a+a \cos (c+d x)}} \, dx=-\frac {4 i x^3 \arctan \left (e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cos (c+d x)}}+\frac {12 i x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{\frac {1}{2} i (c+d x)}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}-\frac {12 i x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{\frac {1}{2} i (c+d x)}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}-\frac {48 x \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{\frac {1}{2} i (c+d x)}\right )}{d^3 \sqrt {a+a \cos (c+d x)}}+\frac {48 x \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{\frac {1}{2} i (c+d x)}\right )}{d^3 \sqrt {a+a \cos (c+d x)}}-\frac {96 i \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,-i e^{\frac {1}{2} i (c+d x)}\right )}{d^4 \sqrt {a+a \cos (c+d x)}}+\frac {96 i \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,i e^{\frac {1}{2} i (c+d x)}\right )}{d^4 \sqrt {a+a \cos (c+d x)}} \]
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Time = 0.26 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3400, 4266, 2611, 6744, 2320, 6724} \[ \int \frac {x^3}{\sqrt {a+a \cos (c+d x)}} \, dx=-\frac {4 i x^3 \arctan \left (e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a \cos (c+d x)+a}}-\frac {96 i \operatorname {PolyLog}\left (4,-i e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^4 \sqrt {a \cos (c+d x)+a}}+\frac {96 i \operatorname {PolyLog}\left (4,i e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^4 \sqrt {a \cos (c+d x)+a}}-\frac {48 x \operatorname {PolyLog}\left (3,-i e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^3 \sqrt {a \cos (c+d x)+a}}+\frac {48 x \operatorname {PolyLog}\left (3,i e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^3 \sqrt {a \cos (c+d x)+a}}+\frac {12 i x^2 \operatorname {PolyLog}\left (2,-i e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cos (c+d x)+a}}-\frac {12 i x^2 \operatorname {PolyLog}\left (2,i e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cos (c+d x)+a}} \]
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Rule 2320
Rule 2611
Rule 3400
Rule 4266
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {\sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right ) \int x^3 \csc \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{\sqrt {a+a \cos (c+d x)}} \\ & = -\frac {4 i x^3 \arctan \left (e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cos (c+d x)}}-\frac {\left (6 \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int x^2 \log \left (1-i e^{i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{d \sqrt {a+a \cos (c+d x)}}+\frac {\left (6 \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int x^2 \log \left (1+i e^{i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{d \sqrt {a+a \cos (c+d x)}} \\ & = -\frac {4 i x^3 \arctan \left (e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cos (c+d x)}}+\frac {12 i x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{\frac {1}{2} i (c+d x)}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}-\frac {12 i x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{\frac {1}{2} i (c+d x)}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}-\frac {\left (24 i \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int x \operatorname {PolyLog}\left (2,-i e^{i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{d^2 \sqrt {a+a \cos (c+d x)}}+\frac {\left (24 i \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int x \operatorname {PolyLog}\left (2,i e^{i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{d^2 \sqrt {a+a \cos (c+d x)}} \\ & = -\frac {4 i x^3 \arctan \left (e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cos (c+d x)}}+\frac {12 i x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{\frac {1}{2} i (c+d x)}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}-\frac {12 i x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{\frac {1}{2} i (c+d x)}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}-\frac {48 x \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{\frac {1}{2} i (c+d x)}\right )}{d^3 \sqrt {a+a \cos (c+d x)}}+\frac {48 x \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{\frac {1}{2} i (c+d x)}\right )}{d^3 \sqrt {a+a \cos (c+d x)}}+\frac {\left (48 \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int \operatorname {PolyLog}\left (3,-i e^{i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{d^3 \sqrt {a+a \cos (c+d x)}}-\frac {\left (48 \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \int \operatorname {PolyLog}\left (3,i e^{i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{d^3 \sqrt {a+a \cos (c+d x)}} \\ & = -\frac {4 i x^3 \arctan \left (e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cos (c+d x)}}+\frac {12 i x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{\frac {1}{2} i (c+d x)}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}-\frac {12 i x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{\frac {1}{2} i (c+d x)}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}-\frac {48 x \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{\frac {1}{2} i (c+d x)}\right )}{d^3 \sqrt {a+a \cos (c+d x)}}+\frac {48 x \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{\frac {1}{2} i (c+d x)}\right )}{d^3 \sqrt {a+a \cos (c+d x)}}-\frac {\left (96 i \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{d^4 \sqrt {a+a \cos (c+d x)}}+\frac {\left (96 i \sin \left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {d x}{2}\right )\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{d^4 \sqrt {a+a \cos (c+d x)}} \\ & = -\frac {4 i x^3 \arctan \left (e^{\frac {1}{2} i (c+d x)}\right ) \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cos (c+d x)}}+\frac {12 i x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{\frac {1}{2} i (c+d x)}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}-\frac {12 i x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{\frac {1}{2} i (c+d x)}\right )}{d^2 \sqrt {a+a \cos (c+d x)}}-\frac {48 x \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{\frac {1}{2} i (c+d x)}\right )}{d^3 \sqrt {a+a \cos (c+d x)}}+\frac {48 x \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{\frac {1}{2} i (c+d x)}\right )}{d^3 \sqrt {a+a \cos (c+d x)}}-\frac {96 i \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,-i e^{\frac {1}{2} i (c+d x)}\right )}{d^4 \sqrt {a+a \cos (c+d x)}}+\frac {96 i \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,i e^{\frac {1}{2} i (c+d x)}\right )}{d^4 \sqrt {a+a \cos (c+d x)}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.53 \[ \int \frac {x^3}{\sqrt {a+a \cos (c+d x)}} \, dx=-\frac {4 i \cos \left (\frac {1}{2} (c+d x)\right ) \left (d^3 x^3 \arctan \left (e^{\frac {1}{2} i (c+d x)}\right )-3 d^2 x^2 \operatorname {PolyLog}\left (2,-i e^{\frac {1}{2} i (c+d x)}\right )+3 d^2 x^2 \operatorname {PolyLog}\left (2,i e^{\frac {1}{2} i (c+d x)}\right )-12 i d x \operatorname {PolyLog}\left (3,-i e^{\frac {1}{2} i (c+d x)}\right )+12 i d x \operatorname {PolyLog}\left (3,i e^{\frac {1}{2} i (c+d x)}\right )+24 \operatorname {PolyLog}\left (4,-i e^{\frac {1}{2} i (c+d x)}\right )-24 \operatorname {PolyLog}\left (4,i e^{\frac {1}{2} i (c+d x)}\right )\right )}{d^4 \sqrt {a (1+\cos (c+d x))}} \]
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\[\int \frac {x^{3}}{\sqrt {a +\cos \left (d x +c \right ) a}}d x\]
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\[ \int \frac {x^3}{\sqrt {a+a \cos (c+d x)}} \, dx=\int { \frac {x^{3}}{\sqrt {a \cos \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {x^3}{\sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {x^{3}}{\sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )}}\, dx \]
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\[ \int \frac {x^3}{\sqrt {a+a \cos (c+d x)}} \, dx=\int { \frac {x^{3}}{\sqrt {a \cos \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {x^3}{\sqrt {a+a \cos (c+d x)}} \, dx=\int { \frac {x^{3}}{\sqrt {a \cos \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {x^3}{\sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {x^3}{\sqrt {a+a\,\cos \left (c+d\,x\right )}} \,d x \]
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