Optimal. Leaf size=84 \[ \frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right )}{2 a^{5/2} f}-\frac{3 \cos (e+f x)}{2 a^2 f}+\frac{\cos ^3(e+f x)}{2 a f \left (a \cos ^2(e+f x)+b\right )} \]
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Rubi [A] time = 0.0505626, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4133, 288, 321, 205} \[ \frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right )}{2 a^{5/2} f}-\frac{3 \cos (e+f x)}{2 a^2 f}+\frac{\cos ^3(e+f x)}{2 a f \left (a \cos ^2(e+f x)+b\right )} \]
Antiderivative was successfully verified.
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Rule 4133
Rule 288
Rule 321
Rule 205
Rubi steps
\begin{align*} \int \frac{\sin (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (b+a x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=\frac{\cos ^3(e+f x)}{2 a f \left (b+a \cos ^2(e+f x)\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{2 a f}\\ &=-\frac{3 \cos (e+f x)}{2 a^2 f}+\frac{\cos ^3(e+f x)}{2 a f \left (b+a \cos ^2(e+f x)\right )}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{2 a^2 f}\\ &=\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right )}{2 a^{5/2} f}-\frac{3 \cos (e+f x)}{2 a^2 f}+\frac{\cos ^3(e+f x)}{2 a f \left (b+a \cos ^2(e+f x)\right )}\\ \end{align*}
Mathematica [C] time = 3.87624, size = 393, normalized size = 4.68 \[ \frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b)^2 \left (-\frac{a^2 \tan ^{-1}\left (\frac{\sqrt{a}-\sqrt{a+b} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{b}}\right )}{b^{3/2}}-\frac{a^2 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan \left (\frac{1}{2} (e+f x)\right )+\sqrt{a}}{\sqrt{b}}\right )}{b^{3/2}}+\frac{\left (a^2+24 b^2\right ) \tan ^{-1}\left (\frac{\sin (e) \tan \left (\frac{f x}{2}\right ) \left (-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left (\frac{f x}{2}\right )\right )}{\sqrt{b}}\right )}{b^{3/2}}+\frac{\left (a^2+24 b^2\right ) \tan ^{-1}\left (\frac{\sin (e) \tan \left (\frac{f x}{2}\right ) \left (-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left (\frac{f x}{2}\right )\right )}{\sqrt{b}}\right )}{b^{3/2}}-\frac{16 \sqrt{a} \cos (e+f x) (a \cos (2 (e+f x))+a+3 b)}{a \cos (2 (e+f x))+a+2 b}\right )}{64 a^{5/2} f \left (a+b \sec ^2(e+f x)\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 75, normalized size = 0.9 \begin{align*} -{\frac{b\sec \left ( fx+e \right ) }{2\,f{a}^{2} \left ( a+b \left ( \sec \left ( fx+e \right ) \right ) ^{2} \right ) }}-{\frac{3\,b}{2\,f{a}^{2}}\arctan \left ({b\sec \left ( fx+e \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{f{a}^{2}\sec \left ( fx+e \right ) }} \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 [A] time = 0.552978, size = 478, normalized size = 5.69 \begin{align*} \left [-\frac{4 \, a \cos \left (f x + e\right )^{3} - 3 \,{\left (a \cos \left (f x + e\right )^{2} + b\right )} \sqrt{-\frac{b}{a}} \log \left (-\frac{a \cos \left (f x + e\right )^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos \left (f x + e\right ) - b}{a \cos \left (f x + e\right )^{2} + b}\right ) + 6 \, b \cos \left (f x + e\right )}{4 \,{\left (a^{3} f \cos \left (f x + e\right )^{2} + a^{2} b f\right )}}, -\frac{2 \, a \cos \left (f x + e\right )^{3} - 3 \,{\left (a \cos \left (f x + e\right )^{2} + b\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}} \cos \left (f x + e\right )}{b}\right ) + 3 \, b \cos \left (f x + e\right )}{2 \,{\left (a^{3} f \cos \left (f x + e\right )^{2} + a^{2} b f\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20922, size = 103, normalized size = 1.23 \begin{align*} \frac{3 \, b \arctan \left (\frac{a \cos \left (f x + e\right )}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2} f} - \frac{\cos \left (f x + e\right )}{a^{2} f} - \frac{b \cos \left (f x + e\right )}{2 \,{\left (a \cos \left (f x + e\right )^{2} + b\right )} a^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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