Optimal. Leaf size=91 \[ \frac{(a+2 b) \tanh ^{-1}\left (\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right )}{2 f (a+b)^{3/2}}+\frac{\tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f (a+b)} \]
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Rubi [A] time = 0.111315, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3190, 382, 377, 206} \[ \frac{(a+2 b) \tanh ^{-1}\left (\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right )}{2 f (a+b)^{3/2}}+\frac{\tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f (a+b)} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 382
Rule 377
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2 \sqrt{a+b x^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 (a+b) f}+\frac{(a+2 b) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{2 (a+b) f}\\ &=\frac{\sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} \tan (e+f x)}{2 (a+b) f}+\frac{(a+2 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 )}{2 (a+b) f}\\ &=\frac{(a+2 b) \tanh ^{-1}\left (\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right )}{2 (a+b)^{3/2} f}+\frac{\sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} \tan (e+f x)}{2 (a+b) f}\\ \end{align*}
Mathematica [C] time = 9.84732, size = 436, normalized size = 4.79 \[ \frac{\tan (e+f x) \sec ^3(e+f x) \left (\frac{b \sin ^2(e+f x)}{a}+1\right ) \left (-30 b \sin ^2(e+f x) \sqrt{-\frac{\tan ^2(e+f x) \sec ^2(e+f x) \left (a^2+a b \left (\sin ^2(e+f x)+1\right )+b^2 \sin ^2(e+f x)\right )}{a^2}}-45 a \sqrt{-\frac{\tan ^2(e+f x) \sec ^2(e+f x) \left (a^2+a b \left (\sin ^2(e+f x)+1\right )+b^2 \sin ^2(e+f x)\right )}{a^2}}+16 b \sin ^2(e+f x) \left (-\frac{(a+b) \tan ^2(e+f x)}{a}\right )^{5/2} \, _2F_1\left (2,3;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right ) \sqrt{\frac{\sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )}{a}}+16 a \left (-\frac{(a+b) \tan ^2(e+f x)}{a}\right )^{5/2} \, _2F_1\left (2,3;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right ) \sqrt{\frac{\sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )}{a}}+30 b \sin ^2(e+f x) \sin ^{-1}\left (\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right )+45 a \sin ^{-1}\left (\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right )\right )}{30 a f \sqrt{a+b \sin ^2(e+f x)} \left (-\frac{(a+b) \tan ^2(e+f x)}{a}\right )^{3/2} \sqrt{\frac{\sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )}{a}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 2.79, size = 360, normalized size = 4. \begin{align*}{\frac{1}{4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}f} \left ( 2\,\sin \left ( fx+e \right ) \sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( a+b \right ) ^{3/2}- \left ( \ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}-b\sin \left ( fx+e \right ) +a}{1+\sin \left ( fx+e \right ) }} \right ){a}^{2}+3\,\ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}-b\sin \left ( fx+e \right ) +a}{1+\sin \left ( fx+e \right ) }} \right ) ab+2\,\ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}-b\sin \left ( fx+e \right ) +a}{1+\sin \left ( fx+e \right ) }} \right ){b}^{2}-\ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}+b\sin \left ( fx+e \right ) +a}{-1+\sin \left ( fx+e \right ) }} \right ){a}^{2}-3\,\ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}+b\sin \left ( fx+e \right ) +a}{-1+\sin \left ( fx+e \right ) }} \right ) ab-2\,\ln \left ( 2\,{\frac{\sqrt{a+b}\sqrt{a+b-b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}+b\sin \left ( fx+e \right ) +a}{-1+\sin \left ( fx+e \right ) }} \right ){b}^{2} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \left ( a+b \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (f x + e\right )^{3}}{\sqrt{b \sin \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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Fricas [B] time = 3.2708, size = 906, normalized size = 9.96 \begin{align*} \left [\frac{{\left (a + 2 \, b\right )} \sqrt{a + b} \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^{2} + 2 \, a b + b^{2}\right )} f \cos \left (f x + e\right )^{2}}, -\frac{{\left (a + 2 \, b\right )} \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^{2} + 2 \, a b + b^{2}\right )} f \cos \left (f x + e\right )^{2}}\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 ^{3}{\left (e + f x \right )}}{\sqrt{a + b \sin ^{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 )^{3}}{\sqrt{b \sin \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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