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 = 0.13, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, 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 191
Rule 271
Rule 453
Rule 3664
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*}
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Mathematica [A] time = 1.19, 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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fricas [A] time = 0.65, size = 158, normalized size = 1.21 \[ \frac {{\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{5} - {\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (3 \, a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \, {\left ({\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (f x + e\right )^{3}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.99, size = 14991, normalized size = 114.44 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.76, size = 216, normalized size = 1.65 \[ -\frac {\frac {3 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {{\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3} - 6 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} b \cos \left (f x + e\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {3 \, b^{2}}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )} + \frac {3 \, b}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (e+f\,x\right )}^3}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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