Integrand size = 10, antiderivative size = 487 \[ \int \frac {1}{a+b \cos ^4(x)} \, dx=\frac {\left (\sqrt {a}+\sqrt {a+b}\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}-\sqrt {2} (a+b)^{3/4} \cot (x)}{\sqrt [4]{a} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}-\frac {\left (\sqrt {a}+\sqrt {a+b}\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}+\sqrt {2} (a+b)^{3/4} \cot (x)}{\sqrt [4]{a} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}-\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}-\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \cot (x)+(a+b)^{3/4} \cot ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}+\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}+\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \cot (x)+(a+b)^{3/4} \cot ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}} \]
-1/8*ln((a+b)^(3/4)*cot(x)^2+(a+b)^(1/4)*a^(1/2)-a^(1/4)*cot(x)*2^(1/2)*(a +b-a^(1/2)*(a+b)^(1/2))^(1/2))*(a^(1/2)-(a+b)^(1/2))/a^(3/4)/(a+b)^(1/4)*2 ^(1/2)/(a+b-a^(1/2)*(a+b)^(1/2))^(1/2)+1/8*ln((a+b)^(3/4)*cot(x)^2+(a+b)^( 1/4)*a^(1/2)+a^(1/4)*cot(x)*2^(1/2)*(a+b-a^(1/2)*(a+b)^(1/2))^(1/2))*(a^(1 /2)-(a+b)^(1/2))/a^(3/4)/(a+b)^(1/4)*2^(1/2)/(a+b-a^(1/2)*(a+b)^(1/2))^(1/ 2)+1/4*arctan((-(a+b)^(3/4)*cot(x)*2^(1/2)+a^(1/4)*(a+b-a^(1/2)*(a+b)^(1/2 ))^(1/2))/a^(1/4)/(a+b+a^(1/2)*(a+b)^(1/2))^(1/2))*(a^(1/2)+(a+b)^(1/2))/a ^(3/4)/(a+b)^(1/4)*2^(1/2)/(a+b+a^(1/2)*(a+b)^(1/2))^(1/2)-1/4*arctan(((a+ b)^(3/4)*cot(x)*2^(1/2)+a^(1/4)*(a+b-a^(1/2)*(a+b)^(1/2))^(1/2))/a^(1/4)/( a+b+a^(1/2)*(a+b)^(1/2))^(1/2))*(a^(1/2)+(a+b)^(1/2))/a^(3/4)/(a+b)^(1/4)* 2^(1/2)/(a+b+a^(1/2)*(a+b)^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 2.85 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.25 \[ \int \frac {1}{a+b \cos ^4(x)} \, dx=\frac {\arctan \left (\frac {\sqrt {a} \tan (x)}{\sqrt {a+i \sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {a+i \sqrt {a} \sqrt {b}}}-\frac {\text {arctanh}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {-a+i \sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {-a+i \sqrt {a} \sqrt {b}}} \]
ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + I*Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a]*Sqrt[a + I *Sqrt[a]*Sqrt[b]]) - ArcTanh[(Sqrt[a]*Tan[x])/Sqrt[-a + I*Sqrt[a]*Sqrt[b]] ]/(2*Sqrt[a]*Sqrt[-a + I*Sqrt[a]*Sqrt[b]])
Time = 1.26 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {3042, 3688, 1483, 1142, 25, 27, 1083, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{a+b \cos ^4(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a+b \sin \left (x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 3688 |
| \(\displaystyle -\int \frac {\cot ^2(x)+1}{(a+b) \cot ^4(x)+2 a \cot ^2(x)+a}d\cot (x)\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle -\frac {\sqrt [4]{a+b} \int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}-\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \cot (x)}{\cot ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}-\frac {\sqrt [4]{a+b} \int \frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \cot (x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}}{\cot ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {\sqrt [4]{a+b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{\cot ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{\sqrt {2} (a+b)^{5/4}}-\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int -\frac {\sqrt {2} \left (\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}-\sqrt {2} \cot (x)\right )}{\cot ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}-\frac {\sqrt [4]{a+b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{\cot ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{\sqrt {2} (a+b)^{5/4}}+\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\sqrt {2} \left (\sqrt {2} \cot (x)+\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}\right )}{\cot ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a+b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{\cot ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{\sqrt {2} (a+b)^{5/4}}+\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\sqrt {2} \left (\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}-\sqrt {2} \cot (x)\right )}{\cot ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}-\frac {\sqrt [4]{a+b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{\cot ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{\sqrt {2} (a+b)^{5/4}}+\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\sqrt {2} \left (\sqrt {2} \cot (x)+\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}\right )}{\cot ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [4]{a+b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{\cot ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{\sqrt {2} (a+b)^{5/4}}+\frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}-\sqrt {2} \cot (x)}{\cot ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{\sqrt {2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}-\frac {\sqrt [4]{a+b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{\cot ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{\sqrt {2} (a+b)^{5/4}}+\frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\sqrt {2} \cot (x)+\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}}{\cot ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{\sqrt {2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\sqrt [4]{a+b} \left (\frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}-\sqrt {2} \cot (x)}{\cot ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{\sqrt {2}}-\frac {\sqrt {2} \sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{-\left (2 \cot (x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}\right )^2-\frac {2 \sqrt {a} \left (a+\sqrt {a+b} \sqrt {a}+b\right )}{(a+b)^{3/2}}}d\left (2 \cot (x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}\right )}{(a+b)^{5/4}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}-\frac {\sqrt [4]{a+b} \left (\frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\sqrt {2} \cot (x)+\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}}{\cot ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{\sqrt {2}}-\frac {\sqrt {2} \sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{-\left (2 \cot (x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}\right )^2-\frac {2 \sqrt {a} \left (a+\sqrt {a+b} \sqrt {a}+b\right )}{(a+b)^{3/2}}}d\left (2 \cot (x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}\right )}{(a+b)^{5/4}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\sqrt [4]{a+b} \left (\frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}-\sqrt {2} \cot (x)}{\cot ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{\sqrt {2}}+\frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \arctan \left (\frac {(a+b)^{3/4} \left (2 \cot (x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}{(a+b)^{3/4}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{\sqrt {a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}-\frac {\sqrt [4]{a+b} \left (\frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\sqrt {2} \cot (x)+\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}}{\cot ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \cot (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\cot (x)}{\sqrt {2}}+\frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \arctan \left (\frac {(a+b)^{3/4} \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}{(a+b)^{3/4}}+2 \cot (x)\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{\sqrt {a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {\sqrt [4]{a+b} \left (\frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \arctan \left (\frac {(a+b)^{3/4} \left (2 \cot (x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}{(a+b)^{3/4}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{\sqrt {a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}-\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \log \left ((a+b)^{3/4} \cot ^2(x)-\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \cot (x)+\sqrt {a} \sqrt [4]{a+b}\right )\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}-\frac {\sqrt [4]{a+b} \left (\frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \arctan \left (\frac {(a+b)^{3/4} \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}{(a+b)^{3/4}}+2 \cot (x)\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{\sqrt {a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}+\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \log \left ((a+b)^{3/4} \cot ^2(x)+\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \cot (x)+\sqrt {a} \sqrt [4]{a+b}\right )\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
-1/2*((a + b)^(1/4)*(((Sqrt[a] + Sqrt[a + b])*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*ArcTan[((a + b)^(3/4)*(-((Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[ a + b]])/(a + b)^(3/4)) + 2*Cot[x]))/(Sqrt[2]*a^(1/4)*Sqrt[a + b + Sqrt[a] *Sqrt[a + b]])])/(Sqrt[a + b]*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]) - ((1 - S qrt[a]/Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) - Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*Cot[x] + (a + b)^(3/4)*Cot[x]^2])/2))/(Sqrt[2]*a^( 3/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]) - ((a + b)^(1/4)*(((Sqrt[a] + Sqrt [a + b])*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*ArcTan[((a + b)^(3/4)*((Sqrt[2] *a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])/(a + b)^(3/4) + 2*Cot[x]))/(Sq rt[2]*a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])])/(Sqrt[a + b]*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]) + ((1 - Sqrt[a]/Sqrt[a + b])*Log[Sqrt[a]*(a + b)^ (1/4) + Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*Cot[x] + (a + b) ^(3/4)*Cot[x]^2])/2))/(2*Sqrt[2]*a^(3/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]] )
3.1.70.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(a + 2*a*ff^2*x^2 + ( a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x]] / ; FreeQ[{a, b, e, f}, x] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.74 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.21
| method | result | size |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (256 a^{4}+256 a^{3} b \right ) \textit {\_Z}^{4}+32 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (\frac {128 i a^{4}}{b}+128 i a^{3}\right ) \textit {\_R}^{3}+\left (-\frac {32 a^{3}}{b}-32 a^{2}\right ) \textit {\_R}^{2}+\left (\frac {8 i a^{2}}{b}-8 i a \right ) \textit {\_R} -\frac {2 a}{b}+1\right )\) | \(101\) |
| default | \(\text {Expression too large to display}\) | \(1233\) |
sum(_R*ln(exp(2*I*x)+(128*I/b*a^4+128*I*a^3)*_R^3+(-32/b*a^3-32*a^2)*_R^2+ (8*I/b*a^2-8*I*a)*_R-2/b*a+1),_R=RootOf(1+(256*a^4+256*a^3*b)*_Z^4+32*a^2* _Z^2))
Leaf count of result is larger than twice the leaf count of optimal. 809 vs. \(2 (344) = 688\).
Time = 0.35 (sec) , antiderivative size = 809, normalized size of antiderivative = 1.66 \[ \int \frac {1}{a+b \cos ^4(x)} \, dx=-\frac {1}{8} \, \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} \log \left (b \cos \left (x\right )^{2} + 2 \, {\left (a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} - {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right ) + \frac {1}{8} \, \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} \log \left (b \cos \left (x\right )^{2} - 2 \, {\left (a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} - {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right ) + \frac {1}{8} \, \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} \log \left (-b \cos \left (x\right )^{2} + 2 \, {\left (a b \cos \left (x\right ) \sin \left (x\right ) - {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} - {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right ) - \frac {1}{8} \, \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} \log \left (-b \cos \left (x\right )^{2} - 2 \, {\left (a b \cos \left (x\right ) \sin \left (x\right ) - {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} - {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right ) \]
-1/8*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1)/(a^2 + a*b ))*log(b*cos(x)^2 + 2*(a*b*cos(x)*sin(x) + (a^4 + a^3*b)*sqrt(-b/(a^5 + 2* a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1)/(a^2 + a*b)) - (a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)* sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))) + 1/8*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1)/(a^2 + a*b))*log(b*cos(x)^2 - 2*(a*b*cos(x)*sin (x) + (a^4 + a^3*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt (-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1)/(a^2 + a*b)) - (a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))) + 1/8*sqrt(((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b) )*log(-b*cos(x)^2 + 2*(a*b*cos(x)*sin(x) - (a^4 + a^3*b)*sqrt(-b/(a^5 + 2* a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b)) - (a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)*s qrt(-b/(a^5 + 2*a^4*b + a^3*b^2))) - 1/8*sqrt(((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b))*log(-b*cos(x)^2 - 2*(a*b*cos(x)*sin( x) - (a^4 + a^3*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt( ((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b)) - (a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)))
Timed out. \[ \int \frac {1}{a+b \cos ^4(x)} \, dx=\text {Timed out} \]
\[ \int \frac {1}{a+b \cos ^4(x)} \, dx=\int { \frac {1}{b \cos \left (x\right )^{4} + a} \,d x } \]
Time = 0.40 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.63 \[ \int \frac {1}{a+b \cos ^4(x)} \, dx=\frac {{\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} a^{2} + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} a b - 3 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a - 4 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} b\right )} {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \left (x\right )}{\sqrt {\frac {4 \, a + \sqrt {-16 \, {\left (a + b\right )} a + 16 \, a^{2}}}{a}}}\right )\right )} {\left | a \right |}}{2 \, {\left (3 \, a^{5} + 7 \, a^{4} b + 4 \, a^{3} b^{2}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} a^{2} + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} a b - 3 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a - 4 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} b\right )} {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \left (x\right )}{\sqrt {\frac {4 \, a - \sqrt {-16 \, {\left (a + b\right )} a + 16 \, a^{2}}}{a}}}\right )\right )} {\left | a \right |}}{2 \, {\left (3 \, a^{5} + 7 \, a^{4} b + 4 \, a^{3} b^{2}\right )}} \]
1/2*(3*sqrt(a^2 + sqrt(-a*b)*a)*a^2 + 4*sqrt(a^2 + sqrt(-a*b)*a)*a*b - 3*s qrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*a - 4*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a* b)*b)*(pi*floor(x/pi + 1/2) + arctan(2*tan(x)/sqrt((4*a + sqrt(-16*(a + b) *a + 16*a^2))/a)))*abs(a)/(3*a^5 + 7*a^4*b + 4*a^3*b^2) + 1/2*(3*sqrt(a^2 - sqrt(-a*b)*a)*a^2 + 4*sqrt(a^2 - sqrt(-a*b)*a)*a*b - 3*sqrt(a^2 - sqrt(- a*b)*a)*sqrt(-a*b)*a - 4*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*b)*(pi*floor( x/pi + 1/2) + arctan(2*tan(x)/sqrt((4*a - sqrt(-16*(a + b)*a + 16*a^2))/a) ))*abs(a)/(3*a^5 + 7*a^4*b + 4*a^3*b^2)
Time = 3.56 (sec) , antiderivative size = 926, normalized size of antiderivative = 1.90 \[ \int \frac {1}{a+b \cos ^4(x)} \, dx=-2\,\mathrm {atanh}\left (\frac {8\,a^6\,b\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}-\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}}}{\frac {2\,a^9\,b}{a^4+b\,a^3}-2\,a^4\,b^2-2\,a^5\,b+\frac {2\,a^8\,b^2}{a^4+b\,a^3}+\frac {2\,a^7\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}+\frac {2\,a^6\,b^2\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}-\frac {8\,a^2\,b\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}-\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}}}{\frac {2\,a^5\,b}{a^4+b\,a^3}-2\,a\,b+\frac {2\,a^3\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}+\frac {8\,a^4\,b\,\mathrm {tan}\left (x\right )\,\sqrt {-a^3\,b}\,\sqrt {-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}-\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}}}{\frac {2\,a^9\,b}{a^4+b\,a^3}-2\,a^4\,b^2-2\,a^5\,b+\frac {2\,a^8\,b^2}{a^4+b\,a^3}+\frac {2\,a^7\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}+\frac {2\,a^6\,b^2\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}\right )\,\sqrt {-\frac {a^2+\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}}-2\,\mathrm {atanh}\left (\frac {8\,a^2\,b\,\mathrm {tan}\left (x\right )\,\sqrt {\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}}}{2\,a\,b-\frac {2\,a^5\,b}{a^4+b\,a^3}+\frac {2\,a^3\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}-\frac {8\,a^6\,b\,\mathrm {tan}\left (x\right )\,\sqrt {\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}}}{2\,a^5\,b+2\,a^4\,b^2-\frac {2\,a^9\,b}{a^4+b\,a^3}-\frac {2\,a^8\,b^2}{a^4+b\,a^3}+\frac {2\,a^7\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}+\frac {2\,a^6\,b^2\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}+\frac {8\,a^4\,b\,\mathrm {tan}\left (x\right )\,\sqrt {-a^3\,b}\,\sqrt {\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}}}{2\,a^5\,b+2\,a^4\,b^2-\frac {2\,a^9\,b}{a^4+b\,a^3}-\frac {2\,a^8\,b^2}{a^4+b\,a^3}+\frac {2\,a^7\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}+\frac {2\,a^6\,b^2\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}\right )\,\sqrt {-\frac {a^2-\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}} \]
- 2*atanh((8*a^6*b*tan(x)*(- a^2/(16*(a^3*b + a^4)) - (-a^3*b)^(1/2)/(16*( a^3*b + a^4)))^(1/2))/((2*a^9*b)/(a^3*b + a^4) - 2*a^4*b^2 - 2*a^5*b + (2* a^8*b^2)/(a^3*b + a^4) + (2*a^7*b*(-a^3*b)^(1/2))/(a^3*b + a^4) + (2*a^6*b ^2*(-a^3*b)^(1/2))/(a^3*b + a^4)) - (8*a^2*b*tan(x)*(- a^2/(16*(a^3*b + a^ 4)) - (-a^3*b)^(1/2)/(16*(a^3*b + a^4)))^(1/2))/((2*a^5*b)/(a^3*b + a^4) - 2*a*b + (2*a^3*b*(-a^3*b)^(1/2))/(a^3*b + a^4)) + (8*a^4*b*tan(x)*(-a^3*b )^(1/2)*(- a^2/(16*(a^3*b + a^4)) - (-a^3*b)^(1/2)/(16*(a^3*b + a^4)))^(1/ 2))/((2*a^9*b)/(a^3*b + a^4) - 2*a^4*b^2 - 2*a^5*b + (2*a^8*b^2)/(a^3*b + a^4) + (2*a^7*b*(-a^3*b)^(1/2))/(a^3*b + a^4) + (2*a^6*b^2*(-a^3*b)^(1/2)) /(a^3*b + a^4)))*(-(a^2 + (-a^3*b)^(1/2))/(16*(a^3*b + a^4)))^(1/2) - 2*at anh((8*a^2*b*tan(x)*((-a^3*b)^(1/2)/(16*(a^3*b + a^4)) - a^2/(16*(a^3*b + a^4)))^(1/2))/(2*a*b - (2*a^5*b)/(a^3*b + a^4) + (2*a^3*b*(-a^3*b)^(1/2))/ (a^3*b + a^4)) - (8*a^6*b*tan(x)*((-a^3*b)^(1/2)/(16*(a^3*b + a^4)) - a^2/ (16*(a^3*b + a^4)))^(1/2))/(2*a^5*b + 2*a^4*b^2 - (2*a^9*b)/(a^3*b + a^4) - (2*a^8*b^2)/(a^3*b + a^4) + (2*a^7*b*(-a^3*b)^(1/2))/(a^3*b + a^4) + (2* a^6*b^2*(-a^3*b)^(1/2))/(a^3*b + a^4)) + (8*a^4*b*tan(x)*(-a^3*b)^(1/2)*(( -a^3*b)^(1/2)/(16*(a^3*b + a^4)) - a^2/(16*(a^3*b + a^4)))^(1/2))/(2*a^5*b + 2*a^4*b^2 - (2*a^9*b)/(a^3*b + a^4) - (2*a^8*b^2)/(a^3*b + a^4) + (2*a^ 7*b*(-a^3*b)^(1/2))/(a^3*b + a^4) + (2*a^6*b^2*(-a^3*b)^(1/2))/(a^3*b + a^ 4)))*(-(a^2 - (-a^3*b)^(1/2))/(16*(a^3*b + a^4)))^(1/2)