Integrand size = 25, antiderivative size = 223 \[ \int \frac {\cos ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {(a+b) \cos (e+f x) \sin (e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (a-b) \cos (e+f x) \sin (e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a-b) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a b^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(2 a-b) \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a b^2 f \sqrt {a+b \sin ^2(e+f x)}} \]
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Time = 0.30 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3271, 424, 541, 538, 437, 435, 432, 430} \[ \int \frac {\cos ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {2 (a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 a^2 b^2 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {2 (a-b) \sin (e+f x) \cos (e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(2 a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{3 a b^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(a+b) \sin (e+f x) \cos (e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
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Rule 424
Rule 430
Rule 432
Rule 435
Rule 437
Rule 538
Rule 541
Rule 3271
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {(a+b) \cos (e+f x) \sin (e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-a+2 b+(2 a-b) x^2}{\sqrt {1-x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 a b f} \\ & = \frac {(a+b) \cos (e+f x) \sin (e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (a-b) \cos (e+f x) \sin (e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-a (a+b)+2 \left (a^2-b^2\right ) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 b (a+b) f} \\ & = \frac {(a+b) \cos (e+f x) \sin (e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (a-b) \cos (e+f x) \sin (e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left ((2 a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a b^2 f}-\frac {\left (2 \left (a^2-b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 b^2 (a+b) f} \\ & = \frac {(a+b) \cos (e+f x) \sin (e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (a-b) \cos (e+f x) \sin (e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (2 \left (a^2-b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {b x^2}{a}}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 b^2 (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\left ((2 a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {b x^2}{a}}} \, dx,x,\sin (e+f x)\right )}{3 a b^2 f \sqrt {a+b \sin ^2(e+f x)}} \\ & = \frac {(a+b) \cos (e+f x) \sin (e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (a-b) \cos (e+f x) \sin (e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a-b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {(2 a-b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a b^2 f \sqrt {a+b \sin ^2(e+f x)}} \\ \end{align*}
Time = 1.81 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.77 \[ \int \frac {\cos ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {-2 a^2 (a-b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )+a^2 (2 a-b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )-\sqrt {2} b \left (a^2-2 a b-b^2+b (-a+b) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{3 a^2 b^2 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \]
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Time = 2.79 (sec) , antiderivative size = 485, normalized size of antiderivative = 2.17
method | result | size |
default | \(\frac {\left (2 a \,b^{2}-2 b^{3}\right ) \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-a^{2} b +a \,b^{2}+2 b^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, a b \left (2 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a -F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -2 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +2 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b -\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}-2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}}{3 a^{2} {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} b^{2} \cos \left (f x +e \right ) f}\) | \(485\) |
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Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 1343, normalized size of antiderivative = 6.02 \[ \int \frac {\cos ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cos ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\cos ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^4}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
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