Optimal. Leaf size=223 \[ \frac {2 b (2 a+b) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt {a+b \sin ^2(e+f x)}}+\frac {2 (2 a+b) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{3 a^2 f (a+b)^2 \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {b \sin (e+f x) \cos (e+f x)}{3 a f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (e+f x\left |-\frac {b}{a}\right .\right )}{3 a f (a+b) \sqrt {a+b \sin ^2(e+f x)}} \]
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Rubi [A] time = 0.26, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3184, 3173, 3172, 3178, 3177, 3183, 3182} \[ \frac {2 b (2 a+b) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt {a+b \sin ^2(e+f x)}}+\frac {2 (2 a+b) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{3 a^2 f (a+b)^2 \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {b \sin (e+f x) \cos (e+f x)}{3 a f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (e+f x\left |-\frac {b}{a}\right .\right )}{3 a f (a+b) \sqrt {a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3172
Rule 3173
Rule 3177
Rule 3178
Rule 3182
Rule 3183
Rule 3184
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx &=\frac {b \cos (e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\int \frac {-3 a-2 b+b \sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx}{3 a (a+b)}\\ &=\frac {b \cos (e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (2 a+b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\int \frac {-a (3 a+b)-2 b (2 a+b) \sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx}{3 a^2 (a+b)^2}\\ &=\frac {b \cos (e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (2 a+b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\int \frac {1}{\sqrt {a+b \sin ^2(e+f x)}} \, dx}{3 a (a+b)}+\frac {(2 (2 a+b)) \int \sqrt {a+b \sin ^2(e+f x)} \, dx}{3 a^2 (a+b)^2}\\ &=\frac {b \cos (e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (2 a+b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (2 (2 a+b) \sqrt {a+b \sin ^2(e+f x)}\right ) \int \sqrt {1+\frac {b \sin ^2(e+f x)}{a}} \, dx}{3 a^2 (a+b)^2 \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\sqrt {1+\frac {b \sin ^2(e+f x)}{a}} \int \frac {1}{\sqrt {1+\frac {b \sin ^2(e+f x)}{a}}} \, dx}{3 a (a+b) \sqrt {a+b \sin ^2(e+f x)}}\\ &=\frac {b \cos (e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (2 a+b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {2 (2 a+b) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {F\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a (a+b) f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 1.38, size = 172, normalized size = 0.77 \[ \frac {-\sqrt {2} b \sin (2 (e+f x)) \left (-5 a^2+b (2 a+b) \cos (2 (e+f x))-5 a b-b^2\right )-a^2 (a+b) \left (\frac {2 a-b \cos (2 (e+f x))+b}{a}\right )^{3/2} F\left (e+f x\left |-\frac {b}{a}\right .\right )+2 a^2 (2 a+b) \left (\frac {2 a-b \cos (2 (e+f x))+b}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{3 a^2 f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{b^{3} \cos \left (f x + e\right )^{6} - 3 \, {\left (a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.93, size = 547, normalized size = 2.45 \[ -\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}}\, \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 )-4 \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 )+4 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}}\, \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 -4 \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 +5 a^{2} b \left (\sin ^{3}\left (f x +e \right )\right )-a \,b^{2} \left (\sin ^{3}\left (f x +e \right )\right )-2 b^{3} \left (\sin ^{3}\left (f x +e \right )\right )-5 a^{2} b \sin \left (f x +e \right )-3 a \,b^{2} \sin \left (f x +e \right )}{3 \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} a^{2} \left (a +b \right )^{2} \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 \[ \int \frac {1}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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