Optimal. Leaf size=309 \[ -\frac {i (c+d x)^3}{3 a^2 f}-\frac {2 d^2 (c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {2 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 i d^2 (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{a^2 f^3}+\frac {4 d^3 \text {Li}_3\left (i e^{i (e+f x)}\right )}{a^2 f^4} \]
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Rubi [A]
time = 0.25, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3399, 4271,
4269, 3556, 3798, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {4 i d^2 (c+d x) \text {PolyLog}\left (2,i e^{i (e+f x)}\right )}{a^2 f^3}+\frac {4 d^3 \text {PolyLog}\left (3,i e^{i (e+f x)}\right )}{a^2 f^4}-\frac {2 d^2 (c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a^2 f^3}+\frac {2 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a^2 f^2}-\frac {d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{2 a^2 f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 a^2 f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{6 a^2 f}-\frac {i (c+d x)^3}{3 a^2 f}+\frac {4 d^3 \log \left (\sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )\right )}{a^2 f^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3399
Rule 3556
Rule 3798
Rule 4269
Rule 4271
Rule 6724
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{(a+a \sin (e+f x))^2} \, dx &=\frac {\int (c+d x)^3 \csc ^4\left (\frac {1}{2} \left (e+\frac {\pi }{2}\right )+\frac {f x}{2}\right ) \, dx}{4 a^2}\\ &=-\frac {d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\int (c+d x)^3 \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{6 a^2}+\frac {d^2 \int (c+d x) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{a^2 f^2}\\ &=-\frac {2 d^2 (c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (2 d^3\right ) \int \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{a^2 f^3}+\frac {d \int (c+d x)^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{a^2 f}\\ &=-\frac {i (c+d x)^3}{3 a^2 f}-\frac {2 d^2 (c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {4 d^3 \log \left (\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}+\frac {(2 d) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1-i e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a^2 f}\\ &=-\frac {i (c+d x)^3}{3 a^2 f}-\frac {2 d^2 (c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {2 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {\left (4 d^2\right ) \int (c+d x) \log \left (1-i e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^2}\\ &=-\frac {i (c+d x)^3}{3 a^2 f}-\frac {2 d^2 (c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {2 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 i d^2 (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{a^2 f^3}+\frac {\left (4 i d^3\right ) \int \text {Li}_2\left (i e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^3}\\ &=-\frac {i (c+d x)^3}{3 a^2 f}-\frac {2 d^2 (c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {2 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 i d^2 (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{a^2 f^3}+\frac {\left (4 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a^2 f^4}\\ &=-\frac {i (c+d x)^3}{3 a^2 f}-\frac {2 d^2 (c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {2 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 i d^2 (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{a^2 f^3}+\frac {4 d^3 \text {Li}_3\left (i e^{i (e+f x)}\right )}{a^2 f^4}\\ \end {align*}
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Mathematica [A]
time = 1.34, size = 257, normalized size = 0.83 \begin {gather*} \frac {-2 i f (c+d x)^3+12 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )+\frac {24 d^3 \log \left (\cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )}{f^2}+\frac {24 d^2 \left (-i f (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right )+d \text {Li}_3\left (i e^{i (e+f x)}\right )\right )}{f^2}-3 d (c+d x)^2 \sec ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\frac {12 d^2 (c+d x) \tan \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{f}+2 f (c+d x)^3 \tan \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+f (c+d x)^3 \sec ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right ) \tan \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{6 a^2 f^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 806 vs. \(2 (254 ) = 508\).
time = 0.94, size = 807, normalized size = 2.61
method | result | size |
risch | \(-\frac {2 i \left (3 i c^{2} d \,f^{2} x +3 d^{3} f^{2} x^{3} {\mathrm e}^{i \left (f x +e \right )}+6 i f c \,d^{2} x \,{\mathrm e}^{i \left (f x +e \right )}+3 i f \,c^{2} d \,{\mathrm e}^{i \left (f x +e \right )}+9 c \,d^{2} f^{2} x^{2} {\mathrm e}^{i \left (f x +e \right )}+3 f \,d^{3} x^{2} {\mathrm e}^{2 i \left (f x +e \right )}-6 i d^{3} x \,{\mathrm e}^{2 i \left (f x +e \right )}+i c^{3} f^{2}+6 i c \,d^{2}+9 c^{2} d \,f^{2} x \,{\mathrm e}^{i \left (f x +e \right )}+6 f c \,d^{2} x \,{\mathrm e}^{2 i \left (f x +e \right )}-6 i c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+6 i d^{3} x +3 i c \,d^{2} f^{2} x^{2}+3 c^{3} f^{2} {\mathrm e}^{i \left (f x +e \right )}+3 f \,c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}+i d^{3} f^{2} x^{3}+12 d^{3} x \,{\mathrm e}^{i \left (f x +e \right )}+3 i f \,d^{3} x^{2} {\mathrm e}^{i \left (f x +e \right )}+12 c \,d^{2} {\mathrm e}^{i \left (f x +e \right )}\right )}{3 \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} f^{3} a^{2}}+\frac {4 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) d^{3}}{a^{2} f^{4}}-\frac {4 \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right ) d^{3}}{a^{2} f^{4}}-\frac {2 i d^{3} x^{3}}{3 a^{2} f}+\frac {2 \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) d^{3} x^{2}}{a^{2} f^{2}}+\frac {2 i d^{3} e^{2} x}{a^{2} f^{3}}+\frac {4 i d^{3} e^{3}}{3 a^{2} f^{4}}-\frac {2 i c \,d^{2} x^{2}}{a^{2} f}-\frac {4 i \polylog \left (2, i {\mathrm e}^{i \left (f x +e \right )}\right ) d^{3} x}{a^{2} f^{3}}-\frac {2 i c \,d^{2} e^{2}}{a^{2} f^{3}}-\frac {2 \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) d^{3} e^{2}}{a^{2} f^{4}}+\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) d^{3} e^{2}}{a^{2} f^{4}}-\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right ) d^{3} e^{2}}{a^{2} f^{4}}-\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right ) c^{2} d}{a^{2} f^{2}}+\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c^{2} d}{a^{2} f^{2}}-\frac {4 i c \,d^{2} e x}{a^{2} f^{2}}-\frac {4 i c \,d^{2} \polylog \left (2, i {\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{3}}+\frac {4 \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) c \,d^{2} x}{a^{2} f^{2}}+\frac {4 \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) c \,d^{2} e}{a^{2} f^{3}}-\frac {4 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c \,d^{2} e}{a^{2} f^{3}}+\frac {4 \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right ) c \,d^{2} e}{a^{2} f^{3}}+\frac {4 d^{3} \polylog \left (3, i {\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}\) | \(807\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 3779 vs. \(2 (257) = 514\).
time = 0.94, size = 3779, normalized size = 12.23 \begin {gather*} \text {Too large to display} \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. 1764 vs. \(2 (257) = 514\).
time = 0.46, size = 1764, normalized size = 5.71 \begin {gather*} \text {Too large to display} \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*} \frac {\int \frac {c^{3}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{3} x^{3}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c d^{2} x^{2}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c^{2} d x}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx}{a^{2}} \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(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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