Integrand size = 15, antiderivative size = 60 \[ \int \frac {\sin ^4(x)}{a+b \cos ^2(x)} \, dx=-\frac {(2 a+3 b) x}{2 b^2}-\frac {(a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} b^2}+\frac {\cos (x) \sin (x)}{2 b} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3270, 425, 536, 209, 211} \[ \int \frac {\sin ^4(x)}{a+b \cos ^2(x)} \, dx=-\frac {(a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} b^2}-\frac {x (2 a+3 b)}{2 b^2}+\frac {\sin (x) \cos (x)}{2 b} \]
[In]
[Out]
Rule 209
Rule 211
Rule 425
Rule 536
Rule 3270
Rubi steps \begin{align*} \text {integral}& = -\text {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 {\text {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 \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )}{b^2}+\frac {(2 a+3 b) \text {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} \arctan \left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} b^2}+\frac {\cos (x) \sin (x)}{2 b} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87 \[ \int \frac {\sin ^4(x)}{a+b \cos ^2(x)} \, dx=\frac {-4 a x-6 b x+\frac {4 (a+b)^{3/2} \arctan \left (\frac {\sqrt {a} \tan (x)}{\sqrt {a+b}}\right )}{\sqrt {a}}+b \sin (2 x)}{4 b^2} \]
[In]
[Out]
Time = 0.65 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {\left (a +b \right )^{2} \arctan \left (\frac {a \tan \left (x \right )}{\sqrt {\left (a +b \right ) a}}\right )}{b^{2} \sqrt {\left (a +b \right ) a}}-\frac {-\frac {b \tan \left (x \right )}{2 \left (\tan ^{2}\left (x \right )+1\right )}+\frac {\left (2 a +3 b \right ) \arctan \left (\tan \left (x \right )\right )}{2}}{b^{2}}\) | \(61\) |
risch | \(-\frac {x a}{b^{2}}-\frac {3 x}{2 b}-\frac {i {\mathrm e}^{2 i x}}{8 b}+\frac {i {\mathrm e}^{-2 i x}}{8 b}+\frac {\sqrt {-\left (a +b \right ) a}\, \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {-\left (a +b \right ) a}-2 a -b}{b}\right )}{2 b^{2}}+\frac {\sqrt {-\left (a +b \right ) a}\, \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {-\left (a +b \right ) a}-2 a -b}{b}\right )}{2 a b}-\frac {\sqrt {-\left (a +b \right ) a}\, \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {-\left (a +b \right ) a}+2 a +b}{b}\right )}{2 b^{2}}-\frac {\sqrt {-\left (a +b \right ) a}\, \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {-\left (a +b \right ) a}+2 a +b}{b}\right )}{2 a b}\) | \(209\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.52 \[ \int \frac {\sin ^4(x)}{a+b \cos ^2(x)} \, dx=\left [\frac {2 \, b \cos \left (x\right ) \sin \left (x\right ) + {\left (a + b\right )} \sqrt {-\frac {a + b}{a}} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (x\right )^{4} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} \cos \left (x\right )^{2} - 4 \, {\left ({\left (2 \, a^{2} + a b\right )} \cos \left (x\right )^{3} - a^{2} \cos \left (x\right )\right )} \sqrt {-\frac {a + b}{a}} \sin \left (x\right ) + a^{2}}{b^{2} \cos \left (x\right )^{4} + 2 \, a b \cos \left (x\right )^{2} + a^{2}}\right ) - 2 \, {\left (2 \, a + 3 \, b\right )} x}{4 \, b^{2}}, \frac {b \cos \left (x\right ) \sin \left (x\right ) - {\left (a + b\right )} \sqrt {\frac {a + b}{a}} \arctan \left (\frac {{\left ({\left (2 \, a + b\right )} \cos \left (x\right )^{2} - a\right )} \sqrt {\frac {a + b}{a}}}{2 \, {\left (a + b\right )} \cos \left (x\right ) \sin \left (x\right )}\right ) - {\left (2 \, a + 3 \, b\right )} x}{2 \, b^{2}}\right ] \]
[In]
[Out]
Timed out. \[ \int \frac {\sin ^4(x)}{a+b \cos ^2(x)} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.03 \[ \int \frac {\sin ^4(x)}{a+b \cos ^2(x)} \, dx=-\frac {{\left (2 \, a + 3 \, b\right )} x}{2 \, b^{2}} + \frac {\tan \left (x\right )}{2 \, {\left (b \tan \left (x\right )^{2} + b\right )}} + \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {a \tan \left (x\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} b^{2}} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.33 \[ \int \frac {\sin ^4(x)}{a+b \cos ^2(x)} \, dx=-\frac {{\left (2 \, a + 3 \, b\right )} x}{2 \, b^{2}} + \frac {{\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (x\right )}{\sqrt {a^{2} + a b}}\right )\right )} {\left (a^{2} + 2 \, a b + b^{2}\right )}}{\sqrt {a^{2} + a b} b^{2}} + \frac {\tan \left (x\right )}{2 \, {\left (\tan \left (x\right )^{2} + 1\right )} b} \]
[In]
[Out]
Time = 2.63 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.10 \[ \int \frac {\sin ^4(x)}{a+b \cos ^2(x)} \, dx=\frac {\cos \left (x\right )\,\sin \left (x\right )}{2\,b}-\frac {a\,\mathrm {atan}\left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )}{b^2}-\frac {3\,\mathrm {atan}\left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )}{2\,b}-\frac {\mathrm {atanh}\left (\frac {\sin \left (x\right )\,\sqrt {-a^4-3\,a^3\,b-3\,a^2\,b^2-a\,b^3}}{\cos \left (x\right )\,a^2+2\,\cos \left (x\right )\,a\,b+\cos \left (x\right )\,b^2}\right )\,\sqrt {-a^4-3\,a^3\,b-3\,a^2\,b^2-a\,b^3}}{a\,b^2} \]
[In]
[Out]