Optimal. Leaf size=156 \[ \frac {1}{8} a^2 \left (12 c^2+16 c d+7 d^2\right ) x-\frac {a^2 \left (12 c^2+16 c d+7 d^2\right ) \cos (e+f x)}{6 f}-\frac {a^2 \left (12 c^2+16 c d+7 d^2\right ) \cos (e+f x) \sin (e+f x)}{24 f}-\frac {(8 c-d) d \cos (e+f x) (a+a \sin (e+f x))^2}{12 f}-\frac {d^2 \cos (e+f x) (a+a \sin (e+f x))^3}{4 a f} \]
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Rubi [A]
time = 0.13, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2840, 2830,
2723} \begin {gather*} -\frac {a^2 \left (12 c^2+16 c d+7 d^2\right ) \cos (e+f x)}{6 f}-\frac {a^2 \left (12 c^2+16 c d+7 d^2\right ) \sin (e+f x) \cos (e+f x)}{24 f}+\frac {1}{8} a^2 x \left (12 c^2+16 c d+7 d^2\right )-\frac {d (8 c-d) \cos (e+f x) (a \sin (e+f x)+a)^2}{12 f}-\frac {d^2 \cos (e+f x) (a \sin (e+f x)+a)^3}{4 a f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2723
Rule 2830
Rule 2840
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^2 \, dx &=-\frac {d^2 \cos (e+f x) (a+a \sin (e+f x))^3}{4 a f}+\frac {\int (a+a \sin (e+f x))^2 \left (a \left (4 c^2+3 d^2\right )+a (8 c-d) d \sin (e+f x)\right ) \, dx}{4 a}\\ &=-\frac {(8 c-d) d \cos (e+f x) (a+a \sin (e+f x))^2}{12 f}-\frac {d^2 \cos (e+f x) (a+a \sin (e+f x))^3}{4 a f}+\frac {1}{12} \left (12 c^2+16 c d+7 d^2\right ) \int (a+a \sin (e+f x))^2 \, dx\\ &=\frac {1}{8} a^2 \left (12 c^2+16 c d+7 d^2\right ) x-\frac {a^2 \left (12 c^2+16 c d+7 d^2\right ) \cos (e+f x)}{6 f}-\frac {a^2 \left (12 c^2+16 c d+7 d^2\right ) \cos (e+f x) \sin (e+f x)}{24 f}-\frac {(8 c-d) d \cos (e+f x) (a+a \sin (e+f x))^2}{12 f}-\frac {d^2 \cos (e+f x) (a+a \sin (e+f x))^3}{4 a f}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 148, normalized size = 0.95 \begin {gather*} -\frac {a^2 \cos (e+f x) \left (6 \left (12 c^2+16 c d+7 d^2\right ) \sin ^{-1}\left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (16 \left (3 c^2+5 c d+2 d^2\right )+3 \left (4 c^2+16 c d+7 d^2\right ) \sin (e+f x)+16 d (c+d) \sin ^2(e+f x)+6 d^2 \sin ^3(e+f x)\right )\right )}{24 f \sqrt {\cos ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 219, normalized size = 1.40
method | result | size |
risch | \(\frac {3 a^{2} c^{2} x}{2}+2 a^{2} c d x +\frac {7 a^{2} d^{2} x}{8}-\frac {2 a^{2} \cos \left (f x +e \right ) c^{2}}{f}-\frac {7 a^{2} \cos \left (f x +e \right ) c d}{2 f}-\frac {3 a^{2} \cos \left (f x +e \right ) d^{2}}{2 f}+\frac {d^{2} a^{2} \sin \left (4 f x +4 e \right )}{32 f}+\frac {d \,a^{2} \cos \left (3 f x +3 e \right ) c}{6 f}+\frac {d^{2} a^{2} \cos \left (3 f x +3 e \right )}{6 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} c^{2}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} c d}{f}-\frac {\sin \left (2 f x +2 e \right ) d^{2} a^{2}}{2 f}\) | \(196\) |
derivativedivides | \(\frac {a^{2} c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 a^{2} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+d^{2} a^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-2 a^{2} c^{2} \cos \left (f x +e \right )+4 a^{2} c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 d^{2} a^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} c^{2} \left (f x +e \right )-2 a^{2} c d \cos \left (f x +e \right )+d^{2} a^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\) | \(219\) |
default | \(\frac {a^{2} c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 a^{2} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+d^{2} a^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-2 a^{2} c^{2} \cos \left (f x +e \right )+4 a^{2} c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 d^{2} a^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} c^{2} \left (f x +e \right )-2 a^{2} c d \cos \left (f x +e \right )+d^{2} a^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\) | \(219\) |
norman | \(\frac {\left (\frac {3}{2} a^{2} c^{2}+2 a^{2} c d +\frac {7}{8} d^{2} a^{2}\right ) x +\left (6 a^{2} c^{2}+8 a^{2} c d +\frac {7}{2} d^{2} a^{2}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (6 a^{2} c^{2}+8 a^{2} c d +\frac {7}{2} d^{2} a^{2}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (9 a^{2} c^{2}+12 a^{2} c d +\frac {21}{4} d^{2} a^{2}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {3}{2} a^{2} c^{2}+2 a^{2} c d +\frac {7}{8} d^{2} a^{2}\right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {12 a^{2} c^{2}+20 a^{2} c d +8 d^{2} a^{2}}{3 f}-\frac {4 \left (a^{2} c^{2}+a^{2} c d \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (6 a^{2} c^{2}+10 a^{2} c d +4 d^{2} a^{2}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {4 \left (9 a^{2} c^{2}+17 a^{2} c d +8 d^{2} a^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {a^{2} \left (4 c^{2}+16 c d +7 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a^{2} \left (4 c^{2}+16 c d +7 d^{2}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a^{2} \left (4 c^{2}+16 c d +15 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {a^{2} \left (4 c^{2}+16 c d +15 d^{2}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4}}\) | \(465\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 227, normalized size = 1.46 \begin {gather*} \frac {24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} + 96 \, {\left (f x + e\right )} a^{2} c^{2} + 64 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c d + 96 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d + 64 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} d^{2} + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{2} + 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{2} - 192 \, a^{2} c^{2} \cos \left (f x + e\right ) - 192 \, a^{2} c d \cos \left (f x + e\right )}{96 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 150, normalized size = 0.96 \begin {gather*} \frac {16 \, {\left (a^{2} c d + a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (12 \, a^{2} c^{2} + 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} f x - 48 \, {\left (a^{2} c^{2} + 2 \, a^{2} c d + a^{2} d^{2}\right )} \cos \left (f x + e\right ) + 3 \, {\left (2 \, a^{2} d^{2} \cos \left (f x + e\right )^{3} - {\left (4 \, a^{2} c^{2} + 16 \, a^{2} c d + 9 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 459 vs.
\(2 (143) = 286\).
time = 0.26, size = 459, normalized size = 2.94 \begin {gather*} \begin {cases} \frac {a^{2} c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{2} x - \frac {a^{2} c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a^{2} c^{2} \cos {\left (e + f x \right )}}{f} + 2 a^{2} c d x \sin ^{2}{\left (e + f x \right )} + 2 a^{2} c d x \cos ^{2}{\left (e + f x \right )} - \frac {2 a^{2} c d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 a^{2} c d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 a^{2} c d \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 a^{2} c d \cos {\left (e + f x \right )}}{f} + \frac {3 a^{2} d^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 a^{2} d^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {a^{2} d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 a^{2} d^{2} x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {a^{2} d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {5 a^{2} d^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {2 a^{2} d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 a^{2} d^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {a^{2} d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {4 a^{2} d^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (c + d \sin {\left (e \right )}\right )^{2} \left (a \sin {\left (e \right )} + a\right )^{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 208, normalized size = 1.33 \begin {gather*} -\frac {2 \, a^{2} c d \cos \left (f x + e\right )}{f} + \frac {a^{2} d^{2} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {a^{2} d^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{8} \, {\left (4 \, a^{2} c^{2} + 16 \, a^{2} c d + 3 \, a^{2} d^{2}\right )} x + \frac {1}{2} \, {\left (2 \, a^{2} c^{2} + a^{2} d^{2}\right )} x + \frac {{\left (a^{2} c d + a^{2} d^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{6 \, f} - \frac {{\left (4 \, a^{2} c^{2} + 3 \, a^{2} c d + 3 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )}{2 \, f} - \frac {{\left (a^{2} c^{2} + 4 \, a^{2} c d + a^{2} d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.34, size = 440, normalized size = 2.82 \begin {gather*} \frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (12\,c^2+16\,c\,d+7\,d^2\right )}{4\,\left (3\,a^2\,c^2+4\,a^2\,c\,d+\frac {7\,a^2\,d^2}{4}\right )}\right )\,\left (12\,c^2+16\,c\,d+7\,d^2\right )}{4\,f}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a^2\,c^2+4\,a^2\,c\,d+\frac {7\,a^2\,d^2}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (a^2\,c^2+4\,a^2\,c\,d+\frac {7\,a^2\,d^2}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (a^2\,c^2+4\,a^2\,c\,d+\frac {15\,a^2\,d^2}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (a^2\,c^2+4\,a^2\,c\,d+\frac {15\,a^2\,d^2}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (12\,a^2\,c^2+20\,a^2\,c\,d+8\,a^2\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (12\,a^2\,c^2+\frac {68\,a^2\,c\,d}{3}+\frac {32\,a^2\,d^2}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (4\,a^2\,c^2+4\,d\,a^2\,c\right )+4\,a^2\,c^2+\frac {8\,a^2\,d^2}{3}+\frac {20\,a^2\,c\,d}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {a^2\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )\,\left (12\,c^2+16\,c\,d+7\,d^2\right )}{4\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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