Optimal. Leaf size=98 \[ \frac{(2 a+b) \cos ^3(e+f x)}{3 a^2 f}-\frac{(a+b)^2 \cos (e+f x)}{a^3 f}+\frac{\sqrt{b} (a+b)^2 \tan ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right )}{a^{7/2} f}-\frac{\cos ^5(e+f x)}{5 a f} \]
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Rubi [A] time = 0.105306, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4133, 461, 205} \[ \frac{(2 a+b) \cos ^3(e+f x)}{3 a^2 f}-\frac{(a+b)^2 \cos (e+f x)}{a^3 f}+\frac{\sqrt{b} (a+b)^2 \tan ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right )}{a^{7/2} f}-\frac{\cos ^5(e+f x)}{5 a f} \]
Antiderivative was successfully verified.
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Rule 4133
Rule 461
Rule 205
Rubi steps
\begin{align*} \int \frac{\sin ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (1-x^2\right )^2}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{(a+b)^2}{a^3}-\frac{(2 a+b) x^2}{a^2}+\frac{x^4}{a}+\frac{-a^2 b-2 a b^2-b^3}{a^3 \left (b+a x^2\right )}\right ) \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{(a+b)^2 \cos (e+f x)}{a^3 f}+\frac{(2 a+b) \cos ^3(e+f x)}{3 a^2 f}-\frac{\cos ^5(e+f x)}{5 a f}+\frac{\left (b (a+b)^2\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{a^3 f}\\ &=\frac{\sqrt{b} (a+b)^2 \tan ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{b}}\right )}{a^{7/2} f}-\frac{(a+b)^2 \cos (e+f x)}{a^3 f}+\frac{(2 a+b) \cos ^3(e+f x)}{3 a^2 f}-\frac{\cos ^5(e+f x)}{5 a f}\\ \end{align*}
Mathematica [C] time = 3.51777, size = 425, normalized size = 4.34 \[ \frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (-8 \sqrt{a} \sqrt{b} \cos (e+f x) \left (3 a^2 \cos (4 (e+f x))+89 a^2-4 a (7 a+5 b) \cos (2 (e+f x))+220 a b+120 b^2\right )+15 \left (64 a^2 b+5 a^3+128 a b^2+64 b^3\right ) \tan ^{-1}\left (\frac{\sin (e) \tan \left (\frac{f x}{2}\right ) \left (-\sqrt{a}-i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt{a}-\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left (\frac{f x}{2}\right )\right )}{\sqrt{b}}\right )+15 \left (64 a^2 b+5 a^3+128 a b^2+64 b^3\right ) \tan ^{-1}\left (\frac{\sin (e) \tan \left (\frac{f x}{2}\right ) \left (-\sqrt{a}+i \sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt{a}+\sqrt{a+b} \sqrt{(\cos (e)-i \sin (e))^2} \tan \left (\frac{f x}{2}\right )\right )}{\sqrt{b}}\right )-75 a^3 \tan ^{-1}\left (\frac{\sqrt{a}-\sqrt{a+b} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{b}}\right )-75 a^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan \left (\frac{1}{2} (e+f x)\right )+\sqrt{a}}{\sqrt{b}}\right )\right )}{1920 a^{7/2} \sqrt{b} f \left (a+b \sec ^2(e+f x)\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.066, size = 183, normalized size = 1.9 \begin{align*} -{\frac{ \left ( \cos \left ( fx+e \right ) \right ) ^{5}}{5\,af}}+{\frac{2\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{3\,af}}+{\frac{ \left ( \cos \left ( fx+e \right ) \right ) ^{3}b}{3\,f{a}^{2}}}-{\frac{\cos \left ( fx+e \right ) }{af}}-2\,{\frac{b\cos \left ( fx+e \right ) }{f{a}^{2}}}-{\frac{{b}^{2}\cos \left ( fx+e \right ) }{f{a}^{3}}}+{\frac{b}{af}\arctan \left ({a\cos \left ( fx+e \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+2\,{\frac{{b}^{2}}{f{a}^{2}\sqrt{ab}}\arctan \left ({\frac{a\cos \left ( fx+e \right ) }{\sqrt{ab}}} \right ) }+{\frac{{b}^{3}}{f{a}^{3}}\arctan \left ({a\cos \left ( fx+e \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 0.569868, size = 555, normalized size = 5.66 \begin{align*} \left [-\frac{6 \, a^{2} \cos \left (f x + e\right )^{5} - 10 \,{\left (2 \, a^{2} + a b\right )} \cos \left (f x + e\right )^{3} - 15 \,{\left (a^{2} + 2 \, a b + b^{2}\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 ) + 30 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )}{30 \, a^{3} f}, -\frac{3 \, a^{2} \cos \left (f x + e\right )^{5} - 5 \,{\left (2 \, a^{2} + a b\right )} \cos \left (f x + e\right )^{3} - 15 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}} \cos \left (f x + e\right )}{b}\right ) + 15 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )}{15 \, a^{3} f}\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 [B] time = 1.15467, size = 504, normalized size = 5.14 \begin{align*} -\frac{\frac{15 \,{\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac{a \cos \left (f x + e\right ) - b}{\sqrt{a b} \cos \left (f x + e\right ) + \sqrt{a b}}\right )}{\sqrt{a b} a^{3}} - \frac{2 \,{\left (8 \, a^{2} + 25 \, a b + 15 \, b^{2} - \frac{40 \, a^{2}{\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac{110 \, a b{\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac{60 \, b^{2}{\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac{80 \, a^{2}{\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{160 \, a b{\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{90 \, b^{2}{\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{90 \, a b{\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac{60 \, b^{2}{\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{15 \, a b{\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{15 \, b^{2}{\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )}}{a^{3}{\left (\frac{\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} - 1\right )}^{5}}}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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