Integrand size = 16, antiderivative size = 162 \[ \int \frac {1}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\frac {\cos ^2(c+d x) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a+b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right )\right ) \left (\sqrt {a}+\sqrt {a+b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+2 a \tan ^2(c+d x)+(a+b) \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {a+b} \tan ^2(c+d x)\right )^2}}}{2 \sqrt [4]{a} \sqrt [4]{a+b} d \sqrt {a+b \sin ^4(c+d x)}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3289, 1117} \[ \int \frac {1}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\frac {\cos ^2(c+d x) \left (\sqrt {a+b} \tan ^2(c+d x)+\sqrt {a}\right ) \sqrt {\frac {(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left (\sqrt {a+b} \tan ^2(c+d x)+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a+b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right )\right )}{2 \sqrt [4]{a} d \sqrt [4]{a+b} \sqrt {a+b \sin ^4(c+d x)}} \]
[In]
[Out]
Rule 1117
Rule 3289
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\cos ^2(c+d x) \sqrt {a+2 a \tan ^2(c+d x)+(a+b) \tan ^4(c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+2 a x^2+(a+b) x^4}} \, dx,x,\tan (c+d x)\right )}{d \sqrt {a+b \sin ^4(c+d x)}} \\ & = \frac {\cos ^2(c+d x) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a+b} \tan (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right )\right ) \left (\sqrt {a}+\sqrt {a+b} \tan ^2(c+d x)\right ) \sqrt {\frac {a+2 a \tan ^2(c+d x)+(a+b) \tan ^4(c+d x)}{\left (\sqrt {a}+\sqrt {a+b} \tan ^2(c+d x)\right )^2}}}{2 \sqrt [4]{a} \sqrt [4]{a+b} d \sqrt {a+b \sin ^4(c+d x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.36 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {2 i \cos ^2(c+d x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {1-\frac {i \sqrt {b}}{\sqrt {a}}} \tan (c+d x)\right ),\frac {\sqrt {a}+i \sqrt {b}}{\sqrt {a}-i \sqrt {b}}\right ) \sqrt {1+\left (1+\frac {i \sqrt {b}}{\sqrt {a}}\right ) \tan ^2(c+d x)} \sqrt {2+\left (2-\frac {2 i \sqrt {b}}{\sqrt {a}}\right ) \tan ^2(c+d x)}}{\sqrt {1-\frac {i \sqrt {b}}{\sqrt {a}}} d \sqrt {8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(395\) vs. \(2(181)=362\).
Time = 2.88 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.44
method | result | size |
default | \(-\frac {\sqrt {\left (\left (\cos ^{2}\left (2 d x +2 c \right )\right ) b +b -2 b \cos \left (2 d x +2 c \right )+4 a \right ) \left (\sin ^{2}\left (2 d x +2 c \right )\right )}\, \sqrt {-a b}\, \sqrt {\frac {\left (-b +\sqrt {-a b}\right ) \left (-1+\cos \left (2 d x +2 c \right )\right )}{\sqrt {-a b}\, \left (1+\cos \left (2 d x +2 c \right )\right )}}\, \left (1+\cos \left (2 d x +2 c \right )\right )^{2} \sqrt {\frac {-b \cos \left (2 d x +2 c \right )+2 \sqrt {-a b}+b}{\sqrt {-a b}\, \left (1+\cos \left (2 d x +2 c \right )\right )}}\, \sqrt {\frac {b \cos \left (2 d x +2 c \right )+2 \sqrt {-a b}-b}{\sqrt {-a b}\, \left (1+\cos \left (2 d x +2 c \right )\right )}}\, F\left (\sqrt {\frac {\left (-b +\sqrt {-a b}\right ) \left (-1+\cos \left (2 d x +2 c \right )\right )}{\sqrt {-a b}\, \left (1+\cos \left (2 d x +2 c \right )\right )}}, \sqrt {\frac {b +\sqrt {-a b}}{-b +\sqrt {-a b}}}\right )}{\left (-b +\sqrt {-a b}\right ) \sqrt {\frac {\left (-1+\cos \left (2 d x +2 c \right )\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \left (-b \cos \left (2 d x +2 c \right )+2 \sqrt {-a b}+b \right ) \left (b \cos \left (2 d x +2 c \right )+2 \sqrt {-a b}-b \right )}{b}}\, \sin \left (2 d x +2 c \right ) \sqrt {\left (\cos ^{2}\left (2 d x +2 c \right )\right ) b +b -2 b \cos \left (2 d x +2 c \right )+4 a}\, d}\) | \(396\) |
[In]
[Out]
\[ \int \frac {1}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {1}{\sqrt {a + b \sin ^{4}{\left (c + d x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {1}{\sqrt {b\,{\sin \left (c+d\,x\right )}^4+a}} \,d x \]
[In]
[Out]