Optimal. Leaf size=39 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right )}{\sqrt{b} f} \]
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Rubi [A] time = 0.0714095, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {4146, 217, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right )}{\sqrt{b} f} \]
Antiderivative was successfully verified.
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Rule 4146
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\tan (e+f x)}{\sqrt{a+b+b \tan ^2(e+f x)}}\right )}{f}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b+b \tan ^2(e+f x)}}\right )}{\sqrt{b} f}\\ \end{align*}
Mathematica [B] time = 0.147321, size = 87, normalized size = 2.23 \[ \frac{\sec (e+f x) \sqrt{a \cos (2 e+2 f x)+a+2 b} \tanh ^{-1}\left (\frac{\sqrt{b} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right )}{\sqrt{2} \sqrt{b} f \sqrt{a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.411, size = 379, normalized size = 9.7 \begin{align*} -{\frac{\sqrt{2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{f\cos \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) }\sqrt{{\frac{1}{ \left ( a+b \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) } \left ( i\cos \left ( fx+e \right ) \sqrt{a}\sqrt{b}-i\sqrt{a}\sqrt{b}+a\cos \left ( fx+e \right ) +b \right ) }}\sqrt{-2\,{\frac{i\cos \left ( fx+e \right ) \sqrt{a}\sqrt{b}-i\sqrt{a}\sqrt{b}-a\cos \left ( fx+e \right ) -b}{ \left ( a+b \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}} \left ({\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }\sqrt{{\frac{1}{a+b} \left ( 2\,i\sqrt{a}\sqrt{b}+a-b \right ) }}},\sqrt{-{\frac{1}{ \left ( a+b \right ) ^{2}} \left ( 4\,i{a}^{{\frac{3}{2}}}\sqrt{b}-4\,i\sqrt{a}{b}^{{\frac{3}{2}}}-{a}^{2}+6\,ab-{b}^{2} \right ) }} \right ) -2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }\sqrt{{\frac{2\,i\sqrt{a}\sqrt{b}+a-b}{a+b}}}},{\frac{a+b}{2\,i\sqrt{a}\sqrt{b}+a-b}},{\sqrt{-{\frac{2\,i\sqrt{a}\sqrt{b}-a+b}{a+b}}}{\frac{1}{\sqrt{{\frac{2\,i\sqrt{a}\sqrt{b}+a-b}{a+b}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{1}{a+b} \left ( 2\,i\sqrt{a}\sqrt{b}+a-b \right ) }}}}{\frac{1}{\sqrt{{\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.655339, size = 543, normalized size = 13.92 \begin{align*} \left [\frac{\log \left (\frac{{\left (a^{2} - 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \,{\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \,{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt{b} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right ) + 8 \, b^{2}}{\cos \left (f x + e\right )^{4}}\right )}{4 \, \sqrt{b} f}, \frac{\sqrt{-b} \arctan \left (-\frac{{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt{-b} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{2 \,{\left (a b \cos \left (f x + e\right )^{2} + b^{2}\right )} \sin \left (f x + e\right )}\right )}{2 \, b f}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (e + f x \right )}}{\sqrt{a + b \sec ^{2}{\left (e + f x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (f x + e\right )^{2}}{\sqrt{b \sec \left (f x + e\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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