Optimal. Leaf size=186 \[ \frac{b (3 a-5 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f (a-b)}+\frac{\cos ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{5/2}}{3 f (a-b)}-\frac{(3 a-5 b) \cos (e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{3 f (a-b)}+\frac{\sqrt{b} (3 a-5 b) \tanh ^{-1}\left (\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right )}{2 f} \]
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Rubi [A] time = 0.164599, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3664, 453, 277, 195, 217, 206} \[ \frac{b (3 a-5 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f (a-b)}+\frac{\cos ^3(e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{5/2}}{3 f (a-b)}-\frac{(3 a-5 b) \cos (e+f x) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}{3 f (a-b)}+\frac{\sqrt{b} (3 a-5 b) \tanh ^{-1}\left (\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right )}{2 f} \]
Antiderivative was successfully verified.
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Rule 3664
Rule 453
Rule 277
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \sin ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right ) \left (a-b+b x^2\right )^{3/2}}{x^4} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac{\cos ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{5/2}}{3 (a-b) f}+\frac{(3 a-5 b) \operatorname{Subst}\left (\int \frac{\left (a-b+b x^2\right )^{3/2}}{x^2} \, dx,x,\sec (e+f x)\right )}{3 (a-b) f}\\ &=-\frac{(3 a-5 b) \cos (e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{3 (a-b) f}+\frac{\cos ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{5/2}}{3 (a-b) f}+\frac{((3 a-5 b) b) \operatorname{Subst}\left (\int \sqrt{a-b+b x^2} \, dx,x,\sec (e+f x)\right )}{(a-b) f}\\ &=\frac{(3 a-5 b) b \sec (e+f x) \sqrt{a-b+b \sec ^2(e+f x)}}{2 (a-b) f}-\frac{(3 a-5 b) \cos (e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{3 (a-b) f}+\frac{\cos ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{5/2}}{3 (a-b) f}+\frac{((3 a-5 b) b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{2 f}\\ &=\frac{(3 a-5 b) b \sec (e+f x) \sqrt{a-b+b \sec ^2(e+f x)}}{2 (a-b) f}-\frac{(3 a-5 b) \cos (e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{3 (a-b) f}+\frac{\cos ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{5/2}}{3 (a-b) f}+\frac{((3 a-5 b) b) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sec (e+f x)}{\sqrt{a-b+b \sec ^2(e+f x)}}\right )}{2 f}\\ &=\frac{(3 a-5 b) \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b+b \sec ^2(e+f x)}}\right )}{2 f}+\frac{(3 a-5 b) b \sec (e+f x) \sqrt{a-b+b \sec ^2(e+f x)}}{2 (a-b) f}-\frac{(3 a-5 b) \cos (e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}{3 (a-b) f}+\frac{\cos ^3(e+f x) \left (a-b+b \sec ^2(e+f x)\right )^{5/2}}{3 (a-b) f}\\ \end{align*}
Mathematica [A] time = 1.78038, size = 188, normalized size = 1.01 \[ \frac{\sec (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left (\sqrt{(a-b) \cos (2 (e+f x))+a+b} (-8 (a-3 b) \cos (2 (e+f x))+(a-b) \cos (4 (e+f x))-9 a+37 b)+12 \sqrt{2} \sqrt{b} (3 a-5 b) \cos ^2(e+f x) \tanh ^{-1}\left (\frac{\sqrt{(a-b) \cos (2 (e+f x))+a+b}}{\sqrt{2} \sqrt{b}}\right )\right )}{24 \sqrt{2} f \sqrt{(a-b) \cos (2 (e+f x))+a+b}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.183, size = 1104, normalized size = 5.9 \begin{align*} \text{result too large to display} \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 = 4.51812, size = 768, normalized size = 4.13 \begin{align*} \left [-\frac{3 \,{\left (3 \, a - 5 \, b\right )} \sqrt{b} \cos \left (f x + e\right ) \log \left (-\frac{{\left (a - b\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt{b} \sqrt{\frac{{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) + 2 \, b}{\cos \left (f x + e\right )^{2}}\right ) - 2 \,{\left (2 \,{\left (a - b\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (3 \, a - 7 \, b\right )} \cos \left (f x + e\right )^{2} + 3 \, b\right )} \sqrt{\frac{{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{12 \, f \cos \left (f x + e\right )}, -\frac{3 \,{\left (3 \, a - 5 \, b\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{\frac{{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{b}\right ) \cos \left (f x + e\right ) -{\left (2 \,{\left (a - b\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (3 \, a - 7 \, b\right )} \cos \left (f x + e\right )^{2} + 3 \, b\right )} \sqrt{\frac{{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{6 \, f \cos \left (f x + e\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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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