Optimal. Leaf size=140 \[ \frac{1}{128} x \left (288 a^2 b^2+256 a^3 b+128 a^4+160 a b^3+35 b^4\right )-\frac{1}{192} b^2 \left (104 a^2+104 a b+35 b^2\right ) \sin ^3(x) \cos (x)-\frac{1}{384} b \left (808 a^2 b+608 a^3+480 a b^2+105 b^3\right ) \sin (x) \cos (x)-\frac{1}{8} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^3-\frac{7}{48} b (2 a+b) \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^2 \]
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Rubi [A] time = 0.166894, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3180, 3170, 3169} \[ \frac{1}{128} x \left (288 a^2 b^2+256 a^3 b+128 a^4+160 a b^3+35 b^4\right )-\frac{1}{192} b^2 \left (104 a^2+104 a b+35 b^2\right ) \sin ^3(x) \cos (x)-\frac{1}{384} b \left (808 a^2 b+608 a^3+480 a b^2+105 b^3\right ) \sin (x) \cos (x)-\frac{1}{8} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^3-\frac{7}{48} b (2 a+b) \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^2 \]
Antiderivative was successfully verified.
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Rule 3180
Rule 3170
Rule 3169
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*}
Mathematica [A] time = 0.14971, size = 113, normalized size = 0.81 \[ \frac{24 x \left (288 a^2 b^2+256 a^3 b+128 a^4+160 a b^3+35 b^4\right )+24 b^2 \left (24 a^2+24 a b+7 b^2\right ) \sin (4 x)-96 b (2 a+b) \left (16 a^2+16 a b+7 b^2\right ) \sin (2 x)-32 b^3 (2 a+b) \sin (6 x)+3 b^4 \sin (8 x)}{3072} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 110, normalized size = 0.8 \begin{align*}{b}^{4} \left ( -{\frac{\cos \left ( x \right ) }{8} \left ( \left ( \sin \left ( x \right ) \right ) ^{7}+{\frac{7\, \left ( \sin \left ( x \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \sin \left ( x \right ) \right ) ^{3}}{24}}+{\frac{35\,\sin \left ( x \right ) }{16}} \right ) }+{\frac{35\,x}{128}} \right ) +4\,a{b}^{3} \left ( -1/6\, \left ( \left ( \sin \left ( x \right ) \right ) ^{5}+5/4\, \left ( \sin \left ( x \right ) \right ) ^{3}+{\frac{15\,\sin \left ( x \right ) }{8}} \right ) \cos \left ( x \right ) +{\frac{5\,x}{16}} \right ) +6\,{a}^{2}{b}^{2} \left ( -1/4\, \left ( \left ( \sin \left ( x \right ) \right ) ^{3}+3/2\,\sin \left ( x \right ) \right ) \cos \left ( x \right ) +3/8\,x \right ) +4\,{a}^{3}b \left ( -1/2\,\sin \left ( x \right ) \cos \left ( x \right ) +x/2 \right ) +{a}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955661, size = 146, normalized size = 1.04 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74816, size = 323, normalized size = 2.31 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 18.4127, size = 410, normalized size = 2.93 \begin{align*} 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10373, size = 159, normalized size = 1.14 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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