Integrand size = 13, antiderivative size = 110 \[ \int \frac {\sin ^4(x)}{a+b \sin (x)} \, dx=-\frac {a \left (2 a^2+b^2\right ) x}{2 b^4}+\frac {2 a^4 \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \sqrt {a^2-b^2}}-\frac {\left (3 a^2+2 b^2\right ) \cos (x)}{3 b^3}+\frac {a \cos (x) \sin (x)}{2 b^2}-\frac {\cos (x) \sin ^2(x)}{3 b} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2872, 3128, 3102, 2814, 2739, 632, 210} \[ \int \frac {\sin ^4(x)}{a+b \sin (x)} \, dx=-\frac {a x \left (2 a^2+b^2\right )}{2 b^4}-\frac {\left (3 a^2+2 b^2\right ) \cos (x)}{3 b^3}+\frac {2 a^4 \arctan \left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{b^4 \sqrt {a^2-b^2}}+\frac {a \sin (x) \cos (x)}{2 b^2}-\frac {\sin ^2(x) \cos (x)}{3 b} \]
[In]
[Out]
Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2872
Rule 3102
Rule 3128
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (x) \sin ^2(x)}{3 b}+\frac {\int \frac {\sin (x) \left (2 a+2 b \sin (x)-3 a \sin ^2(x)\right )}{a+b \sin (x)} \, dx}{3 b} \\ & = \frac {a \cos (x) \sin (x)}{2 b^2}-\frac {\cos (x) \sin ^2(x)}{3 b}+\frac {\int \frac {-3 a^2+a b \sin (x)+2 \left (3 a^2+2 b^2\right ) \sin ^2(x)}{a+b \sin (x)} \, dx}{6 b^2} \\ & = -\frac {\left (3 a^2+2 b^2\right ) \cos (x)}{3 b^3}+\frac {a \cos (x) \sin (x)}{2 b^2}-\frac {\cos (x) \sin ^2(x)}{3 b}+\frac {\int \frac {-3 a^2 b-3 a \left (2 a^2+b^2\right ) \sin (x)}{a+b \sin (x)} \, dx}{6 b^3} \\ & = -\frac {a \left (2 a^2+b^2\right ) x}{2 b^4}-\frac {\left (3 a^2+2 b^2\right ) \cos (x)}{3 b^3}+\frac {a \cos (x) \sin (x)}{2 b^2}-\frac {\cos (x) \sin ^2(x)}{3 b}+\frac {a^4 \int \frac {1}{a+b \sin (x)} \, dx}{b^4} \\ & = -\frac {a \left (2 a^2+b^2\right ) x}{2 b^4}-\frac {\left (3 a^2+2 b^2\right ) \cos (x)}{3 b^3}+\frac {a \cos (x) \sin (x)}{2 b^2}-\frac {\cos (x) \sin ^2(x)}{3 b}+\frac {\left (2 a^4\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^4} \\ & = -\frac {a \left (2 a^2+b^2\right ) x}{2 b^4}-\frac {\left (3 a^2+2 b^2\right ) \cos (x)}{3 b^3}+\frac {a \cos (x) \sin (x)}{2 b^2}-\frac {\cos (x) \sin ^2(x)}{3 b}-\frac {\left (4 a^4\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {x}{2}\right )\right )}{b^4} \\ & = -\frac {a \left (2 a^2+b^2\right ) x}{2 b^4}+\frac {2 a^4 \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \sqrt {a^2-b^2}}-\frac {\left (3 a^2+2 b^2\right ) \cos (x)}{3 b^3}+\frac {a \cos (x) \sin (x)}{2 b^2}-\frac {\cos (x) \sin ^2(x)}{3 b} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.89 \[ \int \frac {\sin ^4(x)}{a+b \sin (x)} \, dx=\frac {-6 a \left (2 a^2+b^2\right ) x+\frac {24 a^4 \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-3 b \left (4 a^2+3 b^2\right ) \cos (x)+b^3 \cos (3 x)+3 a b^2 \sin (2 x)}{12 b^4} \]
[In]
[Out]
Time = 0.60 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.30
method | result | size |
default | \(\frac {2 a^{4} \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{4} \sqrt {a^{2}-b^{2}}}-\frac {2 \left (\frac {\frac {a \,b^{2} \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{2}+a^{2} b \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\left (2 a^{2} b +2 b^{3}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\frac {a \,b^{2} \tan \left (\frac {x}{2}\right )}{2}+a^{2} b +\frac {2 b^{3}}{3}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3}}+\frac {a \left (2 a^{2}+b^{2}\right ) \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{2}\right )}{b^{4}}\) | \(143\) |
risch | \(-\frac {a^{3} x}{b^{4}}-\frac {a x}{2 b^{2}}-\frac {{\mathrm e}^{i x} a^{2}}{2 b^{3}}-\frac {3 \,{\mathrm e}^{i x}}{8 b}-\frac {{\mathrm e}^{-i x} a^{2}}{2 b^{3}}-\frac {3 \,{\mathrm e}^{-i x}}{8 b}-\frac {a^{4} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, b^{4}}+\frac {a^{4} \ln \left ({\mathrm e}^{i x}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, b^{4}}+\frac {\cos \left (3 x \right )}{12 b}+\frac {a \sin \left (2 x \right )}{4 b^{2}}\) | \(212\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.03 \[ \int \frac {\sin ^4(x)}{a+b \sin (x)} \, dx=\left [-\frac {3 \, \sqrt {-a^{2} + b^{2}} a^{4} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (x\right ) \sin \left (x\right ) + b \cos \left (x\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left (a^{2} b^{3} - b^{5}\right )} \cos \left (x\right )^{3} - 3 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (x\right ) \sin \left (x\right ) + 3 \, {\left (2 \, a^{5} - a^{3} b^{2} - a b^{4}\right )} x + 6 \, {\left (a^{4} b - b^{5}\right )} \cos \left (x\right )}{6 \, {\left (a^{2} b^{4} - b^{6}\right )}}, -\frac {6 \, \sqrt {a^{2} - b^{2}} a^{4} \arctan \left (-\frac {a \sin \left (x\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (x\right )}\right ) - 2 \, {\left (a^{2} b^{3} - b^{5}\right )} \cos \left (x\right )^{3} - 3 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (x\right ) \sin \left (x\right ) + 3 \, {\left (2 \, a^{5} - a^{3} b^{2} - a b^{4}\right )} x + 6 \, {\left (a^{4} b - b^{5}\right )} \cos \left (x\right )}{6 \, {\left (a^{2} b^{4} - b^{6}\right )}}\right ] \]
[In]
[Out]
Timed out. \[ \int \frac {\sin ^4(x)}{a+b \sin (x)} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {\sin ^4(x)}{a+b \sin (x)} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.35 \[ \int \frac {\sin ^4(x)}{a+b \sin (x)} \, dx=\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} a^{4}}{\sqrt {a^{2} - b^{2}} b^{4}} - \frac {{\left (2 \, a^{3} + a b^{2}\right )} x}{2 \, b^{4}} - \frac {3 \, a b \tan \left (\frac {1}{2} \, x\right )^{5} + 6 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 12 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 12 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 3 \, a b \tan \left (\frac {1}{2} \, x\right ) + 6 \, a^{2} + 4 \, b^{2}}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3} b^{3}} \]
[In]
[Out]
Time = 6.86 (sec) , antiderivative size = 1075, normalized size of antiderivative = 9.77 \[ \int \frac {\sin ^4(x)}{a+b \sin (x)} \, dx=\text {Too large to display} \]
[In]
[Out]