Optimal. Leaf size=100 \[ \frac {4}{3} a b \left (a^2-2 b^2\right ) \cos (x)+\frac {1}{3} b^2 \left (2 a^2-3 b^2\right ) \sin (x) \cos (x)-\frac {1}{3} \sec (x) (a+b \sin (x))^2 \left (a b-\left (2 a^2-3 b^2\right ) \sin (x)\right )+\frac {1}{3} \sec ^3(x) (a \sin (x)+b) (a+b \sin (x))^3+b^4 x \]
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Rubi [A] time = 0.20, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4391, 2691, 2861, 2734} \[ \frac {4}{3} a b \left (a^2-2 b^2\right ) \cos (x)+\frac {1}{3} b^2 \left (2 a^2-3 b^2\right ) \sin (x) \cos (x)-\frac {1}{3} \sec (x) (a+b \sin (x))^2 \left (a b-\left (2 a^2-3 b^2\right ) \sin (x)\right )+\frac {1}{3} \sec ^3(x) (a \sin (x)+b) (a+b \sin (x))^3+b^4 x \]
Antiderivative was successfully verified.
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Rule 2691
Rule 2734
Rule 2861
Rule 4391
Rubi steps
\begin {align*} \int (a \sec (x)+b \tan (x))^4 \, dx &=\int \sec ^4(x) (a+b \sin (x))^4 \, dx\\ &=\frac {1}{3} \sec ^3(x) (b+a \sin (x)) (a+b \sin (x))^3-\frac {1}{3} \int \sec ^2(x) (a+b \sin (x))^2 \left (-2 a^2+3 b^2+a b \sin (x)\right ) \, dx\\ &=\frac {1}{3} \sec ^3(x) (b+a \sin (x)) (a+b \sin (x))^3-\frac {1}{3} \sec (x) (a+b \sin (x))^2 \left (a b-\left (2 a^2-3 b^2\right ) \sin (x)\right )+\frac {1}{3} \int (a+b \sin (x)) \left (2 a b^2-2 b \left (2 a^2-3 b^2\right ) \sin (x)\right ) \, dx\\ &=b^4 x+\frac {4}{3} a b \left (a^2-2 b^2\right ) \cos (x)+\frac {1}{3} b^2 \left (2 a^2-3 b^2\right ) \cos (x) \sin (x)+\frac {1}{3} \sec ^3(x) (b+a \sin (x)) (a+b \sin (x))^3-\frac {1}{3} \sec (x) (a+b \sin (x))^2 \left (a b-\left (2 a^2-3 b^2\right ) \sin (x)\right )\\ \end {align*}
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Mathematica [A] time = 0.20, size = 96, normalized size = 0.96 \[ \frac {1}{12} \sec ^3(x) \left (6 a^4 \sin (x)+2 a^4 \sin (3 x)+16 a^3 b+18 a^2 b^2 \sin (x)-6 a^2 b^2 \sin (3 x)-24 a b^3 \cos (2 x)-8 a b^3-4 b^4 \sin (3 x)+9 b^4 x \cos (x)+3 b^4 x \cos (3 x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 80, normalized size = 0.80 \[ \frac {3 \, b^{4} x \cos \relax (x)^{3} - 12 \, a b^{3} \cos \relax (x)^{2} + 4 \, a^{3} b + 4 \, a b^{3} + {\left (a^{4} + 6 \, a^{2} b^{2} + b^{4} + 2 \, {\left (a^{4} - 3 \, a^{2} b^{2} - 2 \, b^{4}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)}{3 \, \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 131, normalized size = 1.31 \[ b^{4} x - \frac {2 \, {\left (3 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{5} - 3 \, b^{4} \tan \left (\frac {1}{2} \, x\right )^{5} + 12 \, a^{3} b \tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{3} + 24 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + 10 \, b^{4} \tan \left (\frac {1}{2} \, x\right )^{3} + 24 \, a b^{3} \tan \left (\frac {1}{2} \, x\right )^{2} + 3 \, a^{4} \tan \left (\frac {1}{2} \, x\right ) - 3 \, b^{4} \tan \left (\frac {1}{2} \, x\right ) + 4 \, a^{3} b - 8 \, a b^{3}\right )}}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 96, normalized size = 0.96 \[ -a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\relax (x )\right )}{3}\right ) \tan \relax (x )+\frac {4 a^{3} b}{3 \cos \relax (x )^{3}}+\frac {2 a^{2} b^{2} \left (\sin ^{3}\relax (x )\right )}{\cos \relax (x )^{3}}+4 a \,b^{3} \left (\frac {\sin ^{4}\relax (x )}{3 \cos \relax (x )^{3}}-\frac {\sin ^{4}\relax (x )}{3 \cos \relax (x )}-\frac {\left (2+\sin ^{2}\relax (x )\right ) \cos \relax (x )}{3}\right )+b^{4} \left (\frac {\left (\tan ^{3}\relax (x )\right )}{3}-\tan \relax (x )+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 72, normalized size = 0.72 \[ 2 \, a^{2} b^{2} \tan \relax (x)^{3} + \frac {1}{3} \, {\left (\tan \relax (x)^{3} + 3 \, \tan \relax (x)\right )} a^{4} + \frac {1}{3} \, {\left (\tan \relax (x)^{3} + 3 \, x - 3 \, \tan \relax (x)\right )} b^{4} - \frac {4 \, {\left (3 \, \cos \relax (x)^{2} - 1\right )} a b^{3}}{3 \, \cos \relax (x)^{3}} + \frac {4 \, a^{3} b}{3 \, \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.53, size = 115, normalized size = 1.15 \[ b^4\,x-\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (2\,a^4-2\,b^4\right )-\frac {16\,a\,b^3}{3}+\frac {8\,a^3\,b}{3}+{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (-\frac {4\,a^4}{3}+16\,a^2\,b^2+\frac {20\,b^4}{3}\right )+{\mathrm {tan}\left (\frac {x}{2}\right )}^5\,\left (2\,a^4-2\,b^4\right )+16\,a\,b^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+8\,a^3\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2-1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.22, size = 97, normalized size = 0.97 \[ \frac {a^{4} \tan ^{3}{\relax (x )}}{3} + a^{4} \tan {\relax (x )} + \frac {4 a^{3} b \sec ^{3}{\relax (x )}}{3} + 2 a^{2} b^{2} \tan ^{3}{\relax (x )} + \frac {4 a b^{3} \sec ^{3}{\relax (x )}}{3} - 4 a b^{3} \sec {\relax (x )} + b^{4} x + \frac {b^{4} \sin ^{3}{\relax (x )}}{3 \cos ^{3}{\relax (x )}} - \frac {b^{4} \sin {\relax (x )}}{\cos {\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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