Optimal. Leaf size=131 \[ -\frac{2 b (3 a+b) \sec (e+f x)}{3 f (a-b)^3 \sqrt{a+b \sec ^2(e+f x)-b}}+\frac{\cos ^3(e+f x)}{3 f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{(3 a+b) \cos (e+f x)}{3 f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}} \]
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Rubi [A] time = 0.134308, antiderivative size = 131, 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 = {3664, 453, 271, 191} \[ -\frac{2 b (3 a+b) \sec (e+f x)}{3 f (a-b)^3 \sqrt{a+b \sec ^2(e+f x)-b}}+\frac{\cos ^3(e+f x)}{3 f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{(3 a+b) \cos (e+f x)}{3 f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}} \]
Antiderivative was successfully verified.
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Rule 3664
Rule 453
Rule 271
Rule 191
Rubi steps
\begin{align*} \int \frac{\sin ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-1+x^2}{x^4 \left (a-b+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac{\cos ^3(e+f x)}{3 (a-b) f \sqrt{a-b+b \sec ^2(e+f x)}}+\frac{(3 a+b) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a-b+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{3 (a-b) f}\\ &=-\frac{(3 a+b) \cos (e+f x)}{3 (a-b)^2 f \sqrt{a-b+b \sec ^2(e+f x)}}+\frac{\cos ^3(e+f x)}{3 (a-b) f \sqrt{a-b+b \sec ^2(e+f x)}}-\frac{(2 b (3 a+b)) \operatorname{Subst}\left (\int \frac{1}{\left (a-b+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{3 (a-b)^2 f}\\ &=-\frac{(3 a+b) \cos (e+f x)}{3 (a-b)^2 f \sqrt{a-b+b \sec ^2(e+f x)}}+\frac{\cos ^3(e+f x)}{3 (a-b) f \sqrt{a-b+b \sec ^2(e+f x)}}-\frac{2 b (3 a+b) \sec (e+f x)}{3 (a-b)^3 f \sqrt{a-b+b \sec ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.13964, size = 106, normalized size = 0.81 \[ -\frac{\sec (e+f x) \left (8 \left (a^2-b^2\right ) \cos (2 (e+f x))+9 a^2-(a-b)^2 \cos (4 (e+f x))+46 a b+9 b^2\right )}{12 \sqrt{2} f (a-b)^3 \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.089, size = 14991, normalized size = 114.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09813, size = 292, normalized size = 2.23 \begin{align*} -\frac{\frac{3 \, \sqrt{a - b + \frac{b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a^{2} - 2 \, a b + b^{2}} - \frac{{\left (a - b + \frac{b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac{3}{2}} \cos \left (f x + e\right )^{3} - 6 \, \sqrt{a - b + \frac{b}{\cos \left (f x + e\right )^{2}}} b \cos \left (f x + e\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac{3 \, b^{2}}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \sqrt{a - b + \frac{b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )} + \frac{3 \, b}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt{a - b + \frac{b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.74309, size = 358, normalized size = 2.73 \begin{align*} \frac{{\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{5} -{\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{3} - 2 \,{\left (3 \, a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt{\frac{{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \,{\left ({\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} f \cos \left (f x + e\right )^{2} +{\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (f x + e\right )^{3}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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