Optimal. Leaf size=243 \[ -\frac {i (c+d x)^2}{3 a^2 f}-\frac {2 d^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^3}-\frac {(c+d x)^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {(c+d x)^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {4 d (c+d x) \log \left (1-i e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac {4 i d^2 \text {Li}_2\left (i e^{i (e+f x)}\right )}{3 a^2 f^3} \]
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Rubi [A]
time = 0.18, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3399, 4271,
3852, 8, 4269, 3798, 2221, 2317, 2438} \begin {gather*} -\frac {4 i d^2 \text {PolyLog}\left (2,i e^{i (e+f x)}\right )}{3 a^2 f^3}+\frac {4 d (c+d x) \log \left (1-i e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac {d (c+d x) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 a^2 f^2}-\frac {(c+d x)^2 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 a^2 f}-\frac {(c+d x)^2 \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)^2}{3 a^2 f}-\frac {2 d^2 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 a^2 f^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 3399
Rule 3798
Rule 3852
Rule 4269
Rule 4271
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{(a+a \sin (e+f x))^2} \, dx &=\frac {\int (c+d x)^2 \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) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {(c+d x)^2 \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)^2 \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{6 a^2}+\frac {d^2 \int \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{3 a^2 f^2}\\ &=-\frac {(c+d x)^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {(c+d x)^2 \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^2\right ) \text {Subst}\left (\int 1 \, dx,x,\cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right )}{3 a^2 f^3}+\frac {(2 d) \int (c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{3 a^2 f}\\ &=-\frac {i (c+d x)^2}{3 a^2 f}-\frac {2 d^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^3}-\frac {(c+d x)^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {(c+d x)^2 \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) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)}{1-i e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{3 a^2 f}\\ &=-\frac {i (c+d x)^2}{3 a^2 f}-\frac {2 d^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^3}-\frac {(c+d x)^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {(c+d x)^2 \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 (c+d x) \log \left (1-i e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac {\left (4 d^2\right ) \int \log \left (1-i e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{3 a^2 f^2}\\ &=-\frac {i (c+d x)^2}{3 a^2 f}-\frac {2 d^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^3}-\frac {(c+d x)^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {(c+d x)^2 \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 (c+d x) \log \left (1-i e^{i (e+f x)}\right )}{3 a^2 f^2}+\frac {\left (4 i d^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{3 a^2 f^3}\\ &=-\frac {i (c+d x)^2}{3 a^2 f}-\frac {2 d^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^3}-\frac {(c+d x)^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {(c+d x)^2 \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 (c+d x) \log \left (1-i e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac {4 i d^2 \text {Li}_2\left (i e^{i (e+f x)}\right )}{3 a^2 f^3}\\ \end {align*}
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Mathematica [A]
time = 1.53, size = 175, normalized size = 0.72 \begin {gather*} \frac {-2 i f (c+d x) \left (f (c+d x)+4 i d \log \left (1-i e^{i (e+f x)}\right )\right )-8 i d^2 \text {Li}_2\left (i e^{i (e+f x)}\right )+2 \left (c^2 f^2+2 c d f^2 x+d^2 \left (2+f^2 x^2\right )\right ) \tan \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+f (c+d x) \sec ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right ) \left (-2 d+f (c+d x) \tan \left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )}{6 a^2 f^3} \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. 420 vs. \(2 (191 ) = 382\).
time = 0.88, size = 421, normalized size = 1.73
method | result | size |
risch | \(-\frac {2 i \left (i d^{2} x^{2} f^{2}+3 d^{2} f^{2} x^{2} {\mathrm e}^{i \left (f x +e \right )}+2 i c d \,f^{2} x +2 i f \,d^{2} x \,{\mathrm e}^{i \left (f x +e \right )}+6 c d \,f^{2} x \,{\mathrm e}^{i \left (f x +e \right )}+2 f \,d^{2} x \,{\mathrm e}^{2 i \left (f x +e \right )}+i c^{2} f^{2}+2 i f c d \,{\mathrm e}^{i \left (f x +e \right )}-2 i d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+3 c^{2} f^{2} {\mathrm e}^{i \left (f x +e \right )}+2 f c d \,{\mathrm e}^{2 i \left (f x +e \right )}+2 i d^{2}+4 d^{2} {\mathrm e}^{i \left (f x +e \right )}\right )}{3 \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} f^{3} a^{2}}-\frac {4 \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right ) c d}{3 a^{2} f^{2}}+\frac {4 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c d}{3 a^{2} f^{2}}-\frac {2 i d^{2} x^{2}}{3 a^{2} f}-\frac {4 i d^{2} e x}{3 a^{2} f^{2}}-\frac {2 i d^{2} e^{2}}{3 a^{2} f^{3}}+\frac {4 d^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{3 a^{2} f^{2}}+\frac {4 d^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) e}{3 a^{2} f^{3}}-\frac {4 i d^{2} \polylog \left (2, i {\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{3}}+\frac {4 d^{2} e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{3}}-\frac {4 d^{2} e \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{3 a^{2} f^{3}}\) | \(421\) |
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. 871 vs. \(2 (193) = 386\).
time = 0.60, size = 871, normalized size = 3.58 \begin {gather*} -\frac {2 \, {\left (i \, c^{2} f^{2} + 2 i \, d^{2} - 2 \, {\left (c d f \cos \left (3 \, f x + 3 \, e\right ) + 3 i \, c d f \cos \left (2 \, f x + 2 \, e\right ) - 3 \, c d f \cos \left (f x + e\right ) + i \, c d f \sin \left (3 \, f x + 3 \, e\right ) - 3 \, c d f \sin \left (2 \, f x + 2 \, e\right ) - 3 i \, c d f \sin \left (f x + e\right ) - i \, c d f\right )} \arctan \left (\sin \left (f x + e\right ) + 1, \cos \left (f x + e\right )\right ) + 2 \, {\left (d^{2} f x \cos \left (3 \, f x + 3 \, e\right ) + 3 i \, d^{2} f x \cos \left (2 \, f x + 2 \, e\right ) - 3 \, d^{2} f x \cos \left (f x + e\right ) + i \, d^{2} f x \sin \left (3 \, f x + 3 \, e\right ) - 3 \, d^{2} f x \sin \left (2 \, f x + 2 \, e\right ) - 3 i \, d^{2} f x \sin \left (f x + e\right ) - i \, d^{2} f x\right )} \arctan \left (\cos \left (f x + e\right ), \sin \left (f x + e\right ) + 1\right ) + {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x\right )} \cos \left (3 \, f x + 3 \, e\right ) + {\left (3 i \, d^{2} f^{2} x^{2} + 2 \, c d f - 2 i \, d^{2} + 2 \, {\left (3 i \, c d f^{2} + d^{2} f\right )} x\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (3 \, c^{2} f^{2} + 2 i \, d^{2} f x + 2 i \, c d f + 4 \, d^{2}\right )} \cos \left (f x + e\right ) + 2 \, {\left (d^{2} \cos \left (3 \, f x + 3 \, e\right ) + 3 i \, d^{2} \cos \left (2 \, f x + 2 \, e\right ) - 3 \, d^{2} \cos \left (f x + e\right ) + i \, d^{2} \sin \left (3 \, f x + 3 \, e\right ) - 3 \, d^{2} \sin \left (2 \, f x + 2 \, e\right ) - 3 i \, d^{2} \sin \left (f x + e\right ) - i \, d^{2}\right )} {\rm Li}_2\left (i \, e^{\left (i \, f x + i \, e\right )}\right ) + {\left (d^{2} f x + c d f + {\left (i \, d^{2} f x + i \, c d f\right )} \cos \left (3 \, f x + 3 \, e\right ) - 3 \, {\left (d^{2} f x + c d f\right )} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, {\left (-i \, d^{2} f x - i \, c d f\right )} \cos \left (f x + e\right ) - {\left (d^{2} f x + c d f\right )} \sin \left (3 \, f x + 3 \, e\right ) + 3 \, {\left (-i \, d^{2} f x - i \, c d f\right )} \sin \left (2 \, f x + 2 \, e\right ) + 3 \, {\left (d^{2} f x + c d f\right )} \sin \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) + 1\right ) + {\left (i \, d^{2} f^{2} x^{2} + 2 i \, c d f^{2} x\right )} \sin \left (3 \, f x + 3 \, e\right ) - {\left (3 \, d^{2} f^{2} x^{2} - 2 i \, c d f - 2 \, d^{2} + 2 \, {\left (3 \, c d f^{2} - i \, d^{2} f\right )} x\right )} \sin \left (2 \, f x + 2 \, e\right ) + {\left (3 i \, c^{2} f^{2} - 2 \, d^{2} f x - 2 \, c d f + 4 i \, d^{2}\right )} \sin \left (f x + e\right )\right )}}{-3 i \, a^{2} f^{3} \cos \left (3 \, f x + 3 \, e\right ) + 9 \, a^{2} f^{3} \cos \left (2 \, f x + 2 \, e\right ) + 9 i \, a^{2} f^{3} \cos \left (f x + e\right ) + 3 \, a^{2} f^{3} \sin \left (3 \, f x + 3 \, e\right ) + 9 i \, a^{2} f^{3} \sin \left (2 \, f x + 2 \, e\right ) - 9 \, a^{2} f^{3} \sin \left (f x + e\right ) - 3 \, a^{2} f^{3}} \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. 940 vs. \(2 (193) = 386\).
time = 0.38, size = 940, normalized size = 3.87 \begin {gather*} \frac {d^{2} f^{2} x^{2} + c^{2} f^{2} + 2 \, c d f + {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2} + 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (c d f^{2} + d^{2} f\right )} x + 2 \, {\left (d^{2} f^{2} x^{2} + c^{2} f^{2} + c d f + d^{2} + {\left (2 \, c d f^{2} + d^{2} f\right )} x\right )} \cos \left (f x + e\right ) + 2 \, {\left (-i \, d^{2} \cos \left (f x + e\right )^{2} + i \, d^{2} \cos \left (f x + e\right ) + 2 i \, d^{2} + {\left (i \, d^{2} \cos \left (f x + e\right ) + 2 i \, d^{2}\right )} \sin \left (f x + e\right )\right )} {\rm Li}_2\left (i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) + 2 \, {\left (i \, d^{2} \cos \left (f x + e\right )^{2} - i \, d^{2} \cos \left (f x + e\right ) - 2 i \, d^{2} + {\left (-i \, d^{2} \cos \left (f x + e\right ) - 2 i \, d^{2}\right )} \sin \left (f x + e\right )\right )} {\rm Li}_2\left (-i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) - 2 \, {\left (2 \, c d f - {\left (c d f - d^{2} e\right )} \cos \left (f x + e\right )^{2} - 2 \, d^{2} e + {\left (c d f - d^{2} e\right )} \cos \left (f x + e\right ) + {\left (2 \, c d f - 2 \, d^{2} e + {\left (c d f - d^{2} e\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) - 2 \, {\left (2 \, d^{2} f x - {\left (d^{2} f x + d^{2} e\right )} \cos \left (f x + e\right )^{2} + 2 \, d^{2} e + {\left (d^{2} f x + d^{2} e\right )} \cos \left (f x + e\right ) + {\left (2 \, d^{2} f x + 2 \, d^{2} e + {\left (d^{2} f x + d^{2} e\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \log \left (i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (2 \, d^{2} f x - {\left (d^{2} f x + d^{2} e\right )} \cos \left (f x + e\right )^{2} + 2 \, d^{2} e + {\left (d^{2} f x + d^{2} e\right )} \cos \left (f x + e\right ) + {\left (2 \, d^{2} f x + 2 \, d^{2} e + {\left (d^{2} f x + d^{2} e\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \log \left (-i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (2 \, c d f - {\left (c d f - d^{2} e\right )} \cos \left (f x + e\right )^{2} - 2 \, d^{2} e + {\left (c d f - d^{2} e\right )} \cos \left (f x + e\right ) + {\left (2 \, c d f - 2 \, d^{2} e + {\left (c d f - d^{2} e\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \log \left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) - {\left (d^{2} f^{2} x^{2} + c^{2} f^{2} - 2 \, c d f + 2 \, {\left (c d f^{2} - d^{2} f\right )} x - {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2} + 2 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f^{3} \cos \left (f x + e\right )^{2} - a^{2} f^{3} \cos \left (f x + e\right ) - 2 \, a^{2} f^{3} - {\left (a^{2} f^{3} \cos \left (f x + e\right ) + 2 \, a^{2} f^{3}\right )} \sin \left (f x + e\right )\right )}} \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^{2}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} x^{2}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c 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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