Optimal. Leaf size=59 \[ -\frac {x (2 a+3 b)}{2 b^2}+\frac {(a+b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{\sqrt {a} b^2}-\frac {\sin (x) \cos (x)}{2 b} \]
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Rubi [A] time = 0.11, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3191, 414, 522, 203, 205} \[ -\frac {x (2 a+3 b)}{2 b^2}+\frac {(a+b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{\sqrt {a} b^2}-\frac {\sin (x) \cos (x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 414
Rule 522
Rule 3191
Rubi steps
\begin {align*} \int \frac {\cos ^4(x)}{a+b \sin ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2 \left (a+(a+b) x^2\right )} \, dx,x,\tan (x)\right )\\ &=-\frac {\cos (x) \sin (x)}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {a+2 b+(-a-b) x^2}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )} \, dx,x,\tan (x)\right )}{2 b}\\ &=-\frac {\cos (x) \sin (x)}{2 b}+\frac {(a+b)^2 \operatorname {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (x)\right )}{b^2}-\frac {(2 a+3 b) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right )}{2 b^2}\\ &=-\frac {(2 a+3 b) x}{2 b^2}+\frac {(a+b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{\sqrt {a} b^2}-\frac {\cos (x) \sin (x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 55, normalized size = 0.93 \[ \frac {\frac {4 (a+b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{\sqrt {a}}-2 (2 a x+3 b x+b \sin (x) \cos (x))}{4 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 239, normalized size = 4.05 \[ \left [-\frac {2 \, b \cos \relax (x) \sin \relax (x) - {\left (a + b\right )} \sqrt {-\frac {a + b}{a}} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \relax (x)^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \relax (x)^{2} - 4 \, {\left ({\left (2 \, a^{2} + a b\right )} \cos \relax (x)^{3} - {\left (a^{2} + a b\right )} \cos \relax (x)\right )} \sqrt {-\frac {a + b}{a}} \sin \relax (x) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \relax (x)^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \relax (x)^{2} + a^{2} + 2 \, a b + b^{2}}\right ) + 2 \, {\left (2 \, a + 3 \, b\right )} x}{4 \, b^{2}}, -\frac {b \cos \relax (x) \sin \relax (x) + {\left (a + b\right )} \sqrt {\frac {a + b}{a}} \arctan \left (\frac {{\left ({\left (2 \, a + b\right )} \cos \relax (x)^{2} - a - b\right )} \sqrt {\frac {a + b}{a}}}{2 \, {\left (a + b\right )} \cos \relax (x) \sin \relax (x)}\right ) + {\left (2 \, a + 3 \, b\right )} x}{2 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 92, normalized size = 1.56 \[ -\frac {{\left (2 \, a + 3 \, b\right )} x}{2 \, b^{2}} + \frac {{\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \relax (x) + b \tan \relax (x)}{\sqrt {a^{2} + a b}}\right )\right )} {\left (a^{2} + 2 \, a b + b^{2}\right )}}{\sqrt {a^{2} + a b} b^{2}} - \frac {\tan \relax (x)}{2 \, {\left (\tan \relax (x)^{2} + 1\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 111, normalized size = 1.88 \[ -\frac {\tan \relax (x )}{2 b \left (\tan ^{2}\relax (x )+1\right )}-\frac {3 \arctan \left (\tan \relax (x )\right )}{2 b}-\frac {\arctan \left (\tan \relax (x )\right ) a}{b^{2}}+\frac {\arctan \left (\frac {\left (a +b \right ) \tan \relax (x )}{\sqrt {a \left (a +b \right )}}\right ) a^{2}}{b^{2} \sqrt {a \left (a +b \right )}}+\frac {2 \arctan \left (\frac {\left (a +b \right ) \tan \relax (x )}{\sqrt {a \left (a +b \right )}}\right ) a}{b \sqrt {a \left (a +b \right )}}+\frac {\arctan \left (\frac {\left (a +b \right ) \tan \relax (x )}{\sqrt {a \left (a +b \right )}}\right )}{\sqrt {a \left (a +b \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 64, normalized size = 1.08 \[ -\frac {{\left (2 \, a + 3 \, b\right )} x}{2 \, b^{2}} - \frac {\tan \relax (x)}{2 \, {\left (b \tan \relax (x)^{2} + b\right )}} + \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \relax (x)}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.67, size = 119, normalized size = 2.02 \[ -\frac {3\,\mathrm {atan}\left (\frac {\sin \relax (x)}{\cos \relax (x)}\right )}{2\,b}-\frac {a\,\mathrm {atan}\left (\frac {\sin \relax (x)}{\cos \relax (x)}\right )}{b^2}-\frac {\cos \relax (x)\,\sin \relax (x)}{2\,b}-\frac {\mathrm {atanh}\left (\frac {\sin \relax (x)\,\sqrt {-a^4-3\,a^3\,b-3\,a^2\,b^2-a\,b^3}}{\cos \relax (x)\,a^2+b\,\cos \relax (x)\,a}\right )\,\sqrt {-a^4-3\,a^3\,b-3\,a^2\,b^2-a\,b^3}}{a\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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