Optimal. Leaf size=140 \[ \frac {1}{128} \left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) x-\frac {1}{384} b \left (608 a^3+808 a^2 b+480 a b^2+105 b^3\right ) \cos (x) \sin (x)-\frac {1}{192} b^2 \left (104 a^2+104 a b+35 b^2\right ) \cos (x) \sin ^3(x)-\frac {7}{48} b (2 a+b) \cos (x) \sin (x) \left (a+b \sin ^2(x)\right )^2-\frac {1}{8} b \cos (x) \sin (x) \left (a+b \sin ^2(x)\right )^3 \]
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Rubi [A]
time = 0.11, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3259, 3249,
3248} \begin {gather*} -\frac {1}{192} b^2 \left (104 a^2+104 a b+35 b^2\right ) \sin ^3(x) \cos (x)-\frac {1}{384} b \left (608 a^3+808 a^2 b+480 a b^2+105 b^3\right ) \sin (x) \cos (x)+\frac {1}{128} x \left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right )-\frac {1}{8} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^3-\frac {7}{48} b (2 a+b) \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 3248
Rule 3249
Rule 3259
Rubi steps
\begin {align*} \int \left (a+b \sin ^2(x)\right )^4 \, dx &=-\frac {1}{8} b \cos (x) \sin (x) \left (a+b \sin ^2(x)\right )^3+\frac {1}{8} \int \left (a+b \sin ^2(x)\right )^2 \left (a (8 a+b)+7 b (2 a+b) \sin ^2(x)\right ) \, dx\\ &=-\frac {7}{48} b (2 a+b) \cos (x) \sin (x) \left (a+b \sin ^2(x)\right )^2-\frac {1}{8} b \cos (x) \sin (x) \left (a+b \sin ^2(x)\right )^3+\frac {1}{48} \int \left (a+b \sin ^2(x)\right ) \left (a \left (48 a^2+20 a b+7 b^2\right )+b \left (104 a^2+104 a b+35 b^2\right ) \sin ^2(x)\right ) \, dx\\ &=\frac {1}{128} \left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) x-\frac {1}{384} b \left (608 a^3+808 a^2 b+480 a b^2+105 b^3\right ) \cos (x) \sin (x)-\frac {1}{192} b^2 \left (104 a^2+104 a b+35 b^2\right ) \cos (x) \sin ^3(x)-\frac {7}{48} b (2 a+b) \cos (x) \sin (x) \left (a+b \sin ^2(x)\right )^2-\frac {1}{8} b \cos (x) \sin (x) \left (a+b \sin ^2(x)\right )^3\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 113, normalized size = 0.81 \begin {gather*} \frac {24 \left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) x-96 b (2 a+b) \left (16 a^2+16 a b+7 b^2\right ) \sin (2 x)+24 b^2 \left (24 a^2+24 a b+7 b^2\right ) \sin (4 x)-32 b^3 (2 a+b) \sin (6 x)+3 b^4 \sin (8 x)}{3072} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 110, normalized size = 0.79
method | result | size |
default | \(b^{4} \left (-\frac {\left (\sin ^{7}\left (x \right )+\frac {7 \left (\sin ^{5}\left (x \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (x \right )\right )}{24}+\frac {35 \sin \left (x \right )}{16}\right ) \cos \left (x \right )}{8}+\frac {35 x}{128}\right )+4 a \,b^{3} \left (-\frac {\left (\sin ^{5}\left (x \right )+\frac {5 \left (\sin ^{3}\left (x \right )\right )}{4}+\frac {15 \sin \left (x \right )}{8}\right ) \cos \left (x \right )}{6}+\frac {5 x}{16}\right )+6 a^{2} b^{2} \left (-\frac {\left (\sin ^{3}\left (x \right )+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\right )+4 a^{3} b \left (-\frac {\sin \left (x \right ) \cos \left (x \right )}{2}+\frac {x}{2}\right )+a^{4} x\) | \(110\) |
risch | \(a^{4} x +2 x \,a^{3} b +\frac {9 x \,a^{2} b^{2}}{4}+\frac {5 x a \,b^{3}}{4}+\frac {35 x \,b^{4}}{128}+\frac {b^{4} \sin \left (8 x \right )}{1024}-\frac {\sin \left (6 x \right ) a \,b^{3}}{48}-\frac {\sin \left (6 x \right ) b^{4}}{96}+\frac {3 \sin \left (4 x \right ) a^{2} b^{2}}{16}+\frac {3 \sin \left (4 x \right ) a \,b^{3}}{16}+\frac {7 \sin \left (4 x \right ) b^{4}}{128}-\sin \left (2 x \right ) a^{3} b -\frac {3 \sin \left (2 x \right ) a^{2} b^{2}}{2}-\frac {15 \sin \left (2 x \right ) a \,b^{3}}{16}-\frac {7 \sin \left (2 x \right ) b^{4}}{32}\) | \(136\) |
norman | \(\frac {\left (-36 a^{3} b -\frac {153}{2} a^{2} b^{2}-\frac {383}{6} a \,b^{3}-\frac {2681}{192} b^{4}\right ) \left (\tan ^{5}\left (\frac {x}{2}\right )\right )+\left (-20 a^{3} b -\frac {93}{2} a^{2} b^{2}-\frac {283}{6} a \,b^{3}-\frac {5053}{192} b^{4}\right ) \left (\tan ^{7}\left (\frac {x}{2}\right )\right )+\left (-20 a^{3} b -\frac {69}{2} a^{2} b^{2}-\frac {115}{6} a \,b^{3}-\frac {805}{192} b^{4}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+\left (-4 a^{3} b -\frac {9}{2} a^{2} b^{2}-\frac {5}{2} a \,b^{3}-\frac {35}{64} b^{4}\right ) \tan \left (\frac {x}{2}\right )+\left (4 a^{3} b +\frac {9}{2} a^{2} b^{2}+\frac {5}{2} a \,b^{3}+\frac {35}{64} b^{4}\right ) \left (\tan ^{15}\left (\frac {x}{2}\right )\right )+\left (20 a^{3} b +\frac {69}{2} a^{2} b^{2}+\frac {115}{6} a \,b^{3}+\frac {805}{192} b^{4}\right ) \left (\tan ^{13}\left (\frac {x}{2}\right )\right )+\left (20 a^{3} b +\frac {93}{2} a^{2} b^{2}+\frac {283}{6} a \,b^{3}+\frac {5053}{192} b^{4}\right ) \left (\tan ^{9}\left (\frac {x}{2}\right )\right )+\left (36 a^{3} b +\frac {153}{2} a^{2} b^{2}+\frac {383}{6} a \,b^{3}+\frac {2681}{192} b^{4}\right ) \left (\tan ^{11}\left (\frac {x}{2}\right )\right )+\left (a^{4}+2 a^{3} b +\frac {9}{4} a^{2} b^{2}+\frac {5}{4} a \,b^{3}+\frac {35}{128} b^{4}\right ) x +\left (a^{4}+2 a^{3} b +\frac {9}{4} a^{2} b^{2}+\frac {5}{4} a \,b^{3}+\frac {35}{128} b^{4}\right ) x \left (\tan ^{16}\left (\frac {x}{2}\right )\right )+\left (8 a^{4}+16 a^{3} b +18 a^{2} b^{2}+10 a \,b^{3}+\frac {35}{16} b^{4}\right ) x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\left (8 a^{4}+16 a^{3} b +18 a^{2} b^{2}+10 a \,b^{3}+\frac {35}{16} b^{4}\right ) x \left (\tan ^{14}\left (\frac {x}{2}\right )\right )+\left (28 a^{4}+56 a^{3} b +63 a^{2} b^{2}+35 a \,b^{3}+\frac {245}{32} b^{4}\right ) x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\left (28 a^{4}+56 a^{3} b +63 a^{2} b^{2}+35 a \,b^{3}+\frac {245}{32} b^{4}\right ) x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )+\left (56 a^{4}+112 a^{3} b +126 a^{2} b^{2}+70 a \,b^{3}+\frac {245}{16} b^{4}\right ) x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+\left (56 a^{4}+112 a^{3} b +126 a^{2} b^{2}+70 a \,b^{3}+\frac {245}{16} b^{4}\right ) x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )+\left (70 a^{4}+140 a^{3} b +\frac {315}{2} a^{2} b^{2}+\frac {175}{2} a \,b^{3}+\frac {1225}{64} b^{4}\right ) x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{8}}\) | \(616\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 108, normalized size = 0.77 \begin {gather*} \frac {1}{48} \, {\left (4 \, \sin \left (2 \, x\right )^{3} + 60 \, x + 9 \, \sin \left (4 \, x\right ) - 48 \, \sin \left (2 \, x\right )\right )} a b^{3} + \frac {1}{3072} \, {\left (128 \, \sin \left (2 \, x\right )^{3} + 840 \, x + 3 \, \sin \left (8 \, x\right ) + 168 \, \sin \left (4 \, x\right ) - 768 \, \sin \left (2 \, x\right )\right )} b^{4} + \frac {3}{16} \, a^{2} b^{2} {\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} + a^{3} b {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 123, normalized size = 0.88 \begin {gather*} \frac {1}{128} \, {\left (128 \, a^{4} + 256 \, a^{3} b + 288 \, a^{2} b^{2} + 160 \, a b^{3} + 35 \, b^{4}\right )} x + \frac {1}{384} \, {\left (48 \, b^{4} \cos \left (x\right )^{7} - 8 \, {\left (32 \, a b^{3} + 25 \, b^{4}\right )} \cos \left (x\right )^{5} + 2 \, {\left (288 \, a^{2} b^{2} + 416 \, a b^{3} + 163 \, b^{4}\right )} \cos \left (x\right )^{3} - 3 \, {\left (256 \, a^{3} b + 480 \, a^{2} b^{2} + 352 \, a b^{3} + 93 \, b^{4}\right )} \cos \left (x\right )\right )} \sin \left (x\right ) \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. 410 vs.
\(2 (146) = 292\).
time = 0.72, size = 410, normalized size = 2.93 \begin {gather*} a^{4} x + 2 a^{3} b x \sin ^{2}{\left (x \right )} + 2 a^{3} b x \cos ^{2}{\left (x \right )} - 2 a^{3} b \sin {\left (x \right )} \cos {\left (x \right )} + \frac {9 a^{2} b^{2} x \sin ^{4}{\left (x \right )}}{4} + \frac {9 a^{2} b^{2} x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{2} + \frac {9 a^{2} b^{2} x \cos ^{4}{\left (x \right )}}{4} - \frac {15 a^{2} b^{2} \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{4} - \frac {9 a^{2} b^{2} \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{4} + \frac {5 a b^{3} x \sin ^{6}{\left (x \right )}}{4} + \frac {15 a b^{3} x \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{4} + \frac {15 a b^{3} x \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{4} + \frac {5 a b^{3} x \cos ^{6}{\left (x \right )}}{4} - \frac {11 a b^{3} \sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{4} - \frac {10 a b^{3} \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{3} - \frac {5 a b^{3} \sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{4} + \frac {35 b^{4} x \sin ^{8}{\left (x \right )}}{128} + \frac {35 b^{4} x \sin ^{6}{\left (x \right )} \cos ^{2}{\left (x \right )}}{32} + \frac {105 b^{4} x \sin ^{4}{\left (x \right )} \cos ^{4}{\left (x \right )}}{64} + \frac {35 b^{4} x \sin ^{2}{\left (x \right )} \cos ^{6}{\left (x \right )}}{32} + \frac {35 b^{4} x \cos ^{8}{\left (x \right )}}{128} - \frac {93 b^{4} \sin ^{7}{\left (x \right )} \cos {\left (x \right )}}{128} - \frac {511 b^{4} \sin ^{5}{\left (x \right )} \cos ^{3}{\left (x \right )}}{384} - \frac {385 b^{4} \sin ^{3}{\left (x \right )} \cos ^{5}{\left (x \right )}}{384} - \frac {35 b^{4} \sin {\left (x \right )} \cos ^{7}{\left (x \right )}}{128} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 118, normalized size = 0.84 \begin {gather*} \frac {1}{1024} \, b^{4} \sin \left (8 \, x\right ) + \frac {1}{128} \, {\left (128 \, a^{4} + 256 \, a^{3} b + 288 \, a^{2} b^{2} + 160 \, a b^{3} + 35 \, b^{4}\right )} x - \frac {1}{96} \, {\left (2 \, a b^{3} + b^{4}\right )} \sin \left (6 \, x\right ) + \frac {1}{128} \, {\left (24 \, a^{2} b^{2} + 24 \, a b^{3} + 7 \, b^{4}\right )} \sin \left (4 \, x\right ) - \frac {1}{32} \, {\left (32 \, a^{3} b + 48 \, a^{2} b^{2} + 30 \, a b^{3} + 7 \, b^{4}\right )} \sin \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.63, size = 147, normalized size = 1.05 \begin {gather*} x\,a^4-2\,\sin \left (x\right )\,a^3\,b\,\cos \left (x\right )+2\,x\,a^3\,b+\frac {3\,\sin \left (x\right )\,a^2\,b^2\,{\cos \left (x\right )}^3}{2}-\frac {15\,\sin \left (x\right )\,a^2\,b^2\,\cos \left (x\right )}{4}+\frac {9\,x\,a^2\,b^2}{4}-\frac {2\,\sin \left (x\right )\,a\,b^3\,{\cos \left (x\right )}^5}{3}+\frac {13\,\sin \left (x\right )\,a\,b^3\,{\cos \left (x\right )}^3}{6}-\frac {11\,\sin \left (x\right )\,a\,b^3\,\cos \left (x\right )}{4}+\frac {5\,x\,a\,b^3}{4}+\frac {\sin \left (x\right )\,b^4\,{\cos \left (x\right )}^7}{8}-\frac {25\,\sin \left (x\right )\,b^4\,{\cos \left (x\right )}^5}{48}+\frac {163\,\sin \left (x\right )\,b^4\,{\cos \left (x\right )}^3}{192}-\frac {93\,\sin \left (x\right )\,b^4\,\cos \left (x\right )}{128}+\frac {35\,x\,b^4}{128} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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