Optimal. Leaf size=82 \[ \frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 f \sqrt {a+b}}+\frac {\tan (e+f x) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 f} \]
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Rubi [A] time = 0.09, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3190, 378, 377, 206} \[ \frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 f \sqrt {a+b}}+\frac {\tan (e+f x) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 f} \]
Antiderivative was successfully verified.
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Rule 206
Rule 377
Rule 378
Rule 3190
Rubi steps
\begin {align*} \int \sec ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\left (1-x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{2 f}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{2 f}\\ &=\frac {\sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{2 f}+\frac {a \operatorname {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 f}\\ &=\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 \sqrt {a+b} f}+\frac {\sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 2.17, size = 164, normalized size = 2.00 \[ \frac {\sin (e+f x) \left (\sqrt {2} a \sqrt {\frac {2 a-b \cos (2 (e+f x))+b}{a}} \tanh ^{-1}\left (\frac {\sqrt {\frac {(a+b) \sin ^2(e+f x)}{a}}}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}\right )+\sec ^2(e+f x) \sqrt {\frac {(a+b) \sin ^2(e+f x)}{a}} (2 a-b \cos (2 (e+f x))+b)\right )}{4 f \sqrt {\frac {(a+b) \sin ^2(e+f x)}{a}} \sqrt {a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 337, normalized size = 4.11 \[ \left [\frac {\sqrt {a + b} a \cos \left (f x + e\right )^{2} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 8 \, {\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - 2 \, a - 2 \, b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a + b} \sin \left (f x + e\right ) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos \left (f x + e\right )^{4}}\right ) + 4 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (a + b\right )} \sin \left (f x + e\right )}{8 \, {\left (a + b\right )} f \cos \left (f x + e\right )^{2}}, -\frac {a \sqrt {-a - b} \arctan \left (\frac {{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - 2 \, a - 2 \, b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a - b}}{2 \, {\left ({\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} - a^{2} - 2 \, a b - b^{2}\right )} \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (a + b\right )} \sin \left (f x + e\right )}{4 \, {\left (a + b\right )} f \cos \left (f x + e\right )^{2}}\right ] \]
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 \[ \int \sqrt {b \sin \left (f x + e\right )^{2} + a} \sec \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.43, size = 291, normalized size = 3.55 \[ \frac {2 \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, \sqrt {a +b}\, b \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \left (a +b -b \left (\cos ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \sqrt {a +b}\, \sin \left (f x +e \right )-a \left (\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) a +\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) b -\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) a -\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}{4 \left (a +b \right )^{\frac {3}{2}} \cos \left (f x +e \right )^{2} f} \]
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 \[ \int \sqrt {b \sin \left (f x + e\right )^{2} + a} \sec \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}}{{\cos \left (e+f\,x\right )}^3} \,d x \]
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 \[ \int \sqrt {a + b \sin ^{2}{\left (e + f x \right )}} \sec ^{3}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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