Optimal. Leaf size=82 \[ \frac {\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f} \]
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Rubi [A] time = 0.09, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3190, 402, 217, 206, 377} \[ \frac {\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 377
Rule 402
Rule 3190
Rubi steps
\begin {align*} \int \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{1-x^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{f}+\frac {(a+b) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac {b \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f}+\frac {(a+b) \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 )}{f}\\ &=-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f}+\frac {\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 129, normalized size = 1.57 \[ \frac {\frac {\sqrt {a} \sqrt {-b} \sin ^{-1}\left (\frac {\sqrt {-b} \sin (e+f x)}{\sqrt {a}}\right ) \sqrt {\frac {2 a-b \cos (2 (e+f x))+b}{a}}}{\sqrt {2 a-b \cos (2 (e+f x))+b}}+\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {2 a+2 b} \sin (e+f x)}{\sqrt {2 a-b \cos (2 (e+f x))+b}}\right )}{f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 1246, normalized size = 15.20 \[ \text {result too large to display} \]
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 \[ \int \sqrt {b \sin \left (f x + e\right )^{2} + a} \sec \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.14, size = 155, normalized size = 1.89 \[ -\frac {\sqrt {b}\, \ln \left (\frac {\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, \sqrt {b}+b \sin \left (f x +e \right )}{\sqrt {b}}\right )}{f}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right )}{2 f}-\frac {\sqrt {a +b}\, \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right )}{2 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 126, normalized size = 1.54 \[ -\frac {2 \, \sqrt {b} \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b}}\right ) - \sqrt {a + b} \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b} {\left (\sin \left (f x + e\right ) + 1\right )}} - \frac {a}{\sqrt {a b} {\left (\sin \left (f x + e\right ) + 1\right )}}\right ) - \sqrt {a + b} \operatorname {arsinh}\left (-\frac {b \sin \left (f x + e\right )}{\sqrt {a b} {\left (\sin \left (f x + e\right ) - 1\right )}} - \frac {a}{\sqrt {a b} {\left (\sin \left (f x + e\right ) - 1\right )}}\right )}{2 \, f} \]
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 \[ \int \frac {\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}}{\cos \left (e+f\,x\right )} \,d x \]
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 \[ \int \sqrt {a + b \sin ^{2}{\left (e + f x \right )}} \sec {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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