Optimal. Leaf size=60 \[ -\frac {x (2 a+3 b)}{2 b^2}-\frac {(a+b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} b^2}+\frac {\sin (x) \cos (x)}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 60, 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} \[ -\frac {x (2 a+3 b)}{2 b^2}-\frac {(a+b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} b^2}+\frac {\sin (x) \cos (x)}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 205
Rule 414
Rule 522
Rule 3191
Rubi steps
\begin {align*} \int \frac {\sin ^4(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2 \left (a+(a+b) x^2\right )} \, dx,x,\cot (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,\cot (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,\cot (x)\right )}{b^2}+\frac {(2 a+3 b) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (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} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} b^2}+\frac {\cos (x) \sin (x)}{2 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 52, normalized size = 0.87 \[ \frac {\frac {4 (a+b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a+b}}\right )}{\sqrt {a}}-4 a x-6 b x+b \sin (2 x)}{4 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.62, size = 211, normalized size = 3.52 \[ \left [\frac {2 \, b \cos \relax (x) \sin \relax (x) + {\left (a + b\right )} \sqrt {-\frac {a + b}{a}} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \relax (x)^{4} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} \cos \relax (x)^{2} - 4 \, {\left ({\left (2 \, a^{2} + a b\right )} \cos \relax (x)^{3} - a^{2} \cos \relax (x)\right )} \sqrt {-\frac {a + b}{a}} \sin \relax (x) + a^{2}}{b^{2} \cos \relax (x)^{4} + 2 \, a b \cos \relax (x)^{2} + a^{2}}\right ) - 2 \, {\left (2 \, a + 3 \, b\right )} x}{4 \, b^{2}}, \frac {b \cos \relax (x) \sin \relax (x) - {\left (a + b\right )} \sqrt {\frac {a + b}{a}} \arctan \left (\frac {{\left ({\left (2 \, a + b\right )} \cos \relax (x)^{2} - a\right )} \sqrt {\frac {a + b}{a}}}{2 \, {\left (a + b\right )} \cos \relax (x) \sin \relax (x)}\right ) - {\left (2 \, a + 3 \, b\right )} x}{2 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 80, normalized size = 1.33 \[ -\frac {{\left (2 \, a + 3 \, b\right )} x}{2 \, b^{2}} + \frac {{\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \relax (x)}{\sqrt {a^{2} + a b}}\right )\right )} {\left (a^{2} + 2 \, a b + b^{2}\right )}}{\sqrt {a^{2} + a b} b^{2}} + \frac {\tan \relax (x)}{2 \, {\left (\tan \relax (x)^{2} + 1\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.08, size = 105, normalized size = 1.75 \[ \frac {\arctan \left (\frac {a \tan \relax (x )}{\sqrt {\left (a +b \right ) a}}\right ) a^{2}}{b^{2} \sqrt {\left (a +b \right ) a}}+\frac {2 \arctan \left (\frac {a \tan \relax (x )}{\sqrt {\left (a +b \right ) a}}\right ) a}{b \sqrt {\left (a +b \right ) a}}+\frac {\arctan \left (\frac {a \tan \relax (x )}{\sqrt {\left (a +b \right ) a}}\right )}{\sqrt {\left (a +b \right ) a}}+\frac {\tan \relax (x )}{2 b \left (\tan ^{2}\relax (x )+1\right )}-\frac {3 \arctan \left (\tan \relax (x )\right )}{2 b}-\frac {\arctan \left (\tan \relax (x )\right ) a}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.77, size = 62, normalized size = 1.03 \[ -\frac {{\left (2 \, a + 3 \, b\right )} x}{2 \, b^{2}} + \frac {\tan \relax (x)}{2 \, {\left (b \tan \relax (x)^{2} + b\right )}} + \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {a \tan \relax (x)}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.45, size = 126, normalized size = 2.10 \[ \frac {\cos \relax (x)\,\sin \relax (x)}{2\,b}-\frac {a\,\mathrm {atan}\left (\frac {\sin \relax (x)}{\cos \relax (x)}\right )}{b^2}-\frac {3\,\mathrm {atan}\left (\frac {\sin \relax (x)}{\cos \relax (x)}\right )}{2\,b}-\frac {\mathrm {atanh}\left (\frac {\sin \relax (x)\,\sqrt {-a^4-3\,a^3\,b-3\,a^2\,b^2-a\,b^3}}{\cos \relax (x)\,a^2+2\,\cos \relax (x)\,a\,b+\cos \relax (x)\,b^2}\right )\,\sqrt {-a^4-3\,a^3\,b-3\,a^2\,b^2-a\,b^3}}{a\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________