Optimal. Leaf size=217 \[ \frac {(a+2 b) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b) \sqrt {a+b \sin ^2(e+f x)}}+\frac {\sin (e+f x) \cos (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (e+f x\left |-\frac {b}{a}\right .\right )}{3 a b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(a+2 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{3 a b f (a+b) \sqrt {a+b \sin ^2(e+f x)}} \]
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Rubi [A] time = 0.24, antiderivative size = 257, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3192, 412, 527, 524, 426, 424, 421, 419} \[ \frac {(a+2 b) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b) \sqrt {a+b \sin ^2(e+f x)}}+\frac {(a+2 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{3 a^2 b f (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {\sin (e+f x) \cos (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{3 a b f \sqrt {a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 412
Rule 419
Rule 421
Rule 424
Rule 426
Rule 524
Rule 527
Rule 3192
Rubi steps
\begin {align*} \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-x^2}}{\left (a+b x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\cos (e+f x) \sin (e+f x)}{3 a 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 ) \operatorname {Subst}\left (\int \frac {-2+x^2}{\sqrt {1-x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 a f}\\ &=\frac {\cos (e+f x) \sin (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(a+2 b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {a+(a+2 b) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 (a+b) f}\\ &=\frac {\cos (e+f x) \sin (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(a+2 b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a b f}+\frac {\left ((a+2 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 b (a+b) f}\\ &=\frac {\cos (e+f x) \sin (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(a+2 b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left ((a+2 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right ) \operatorname {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 (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}\right ) \operatorname {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 f \sqrt {a+b \sin ^2(e+f x)}}\\ &=\frac {\cos (e+f x) \sin (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(a+2 b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(a+2 b) \sqrt {\cos ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\sqrt {\cos ^2(e+f x)} F\left (\sin ^{-1}(\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 f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 1.43, size = 175, normalized size = 0.81 \[ \frac {-\sqrt {2} b \sin (2 (e+f x)) \left (-4 a^2+b (a+2 b) \cos (2 (e+f x))-7 a b-2 b^2\right )-2 a^2 (a+b) \left (\frac {2 a-b \cos (2 (e+f x))+b}{a}\right )^{3/2} F\left (e+f x\left |-\frac {b}{a}\right .\right )+2 a^2 (a+2 b) \left (\frac {2 a-b \cos (2 (e+f x))+b}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{6 a^2 b f (a+b) (2 a-b \cos (2 (e+f x))+b)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \cos \left (f x + e\right )^{2}}{b^{3} \cos \left (f x + e\right )^{6} - 3 \, {\left (a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.85, size = 552, normalized size = 2.54 \[ \frac {\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b \left (\sin ^{2}\left (f x +e \right )\right )+2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2} \left (\sin ^{2}\left (f x +e \right )\right )-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b \left (\sin ^{2}\left (f x +e \right )\right )-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2} \left (\sin ^{2}\left (f x +e \right )\right )-a \,b^{2} \left (\sin ^{5}\left (f x +e \right )\right )-2 b^{3} \left (\sin ^{5}\left (f x +e \right )\right )+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticE \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 {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticE \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 {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}-a^{2} \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -2 a^{2} b \left (\sin ^{3}\left (f x +e \right )\right )-2 a \,b^{2} \left (\sin ^{3}\left (f x +e \right )\right )+2 b^{3} \left (\sin ^{3}\left (f x +e \right )\right )+2 a^{2} b \sin \left (f x +e \right )+3 a \,b^{2} \sin \left (f x +e \right )}{3 a^{2} \left (a +b \right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} b \cos \left (f x +e \right ) f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (e+f\,x\right )}^2}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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