Optimal. Leaf size=114 \[ -\frac {2 b (3 a+4 b) \sec (e+f x)}{3 a^3 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {(3 a+4 b) \cos (e+f x)}{3 a^2 f \sqrt {a+b \sec ^2(e+f x)}}+\frac {\cos ^3(e+f x)}{3 a f \sqrt {a+b \sec ^2(e+f x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 114, 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 = {4134, 453, 271, 191} \[ -\frac {2 b (3 a+4 b) \sec (e+f x)}{3 a^3 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {(3 a+4 b) \cos (e+f x)}{3 a^2 f \sqrt {a+b \sec ^2(e+f x)}}+\frac {\cos ^3(e+f x)}{3 a f \sqrt {a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 271
Rule 453
Rule 4134
Rubi steps
\begin {align*} \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {-1+x^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\cos ^3(e+f x)}{3 a f \sqrt {a+b \sec ^2(e+f x)}}+\frac {(3 a+4 b) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{3 a f}\\ &=-\frac {(3 a+4 b) \cos (e+f x)}{3 a^2 f \sqrt {a+b \sec ^2(e+f x)}}+\frac {\cos ^3(e+f x)}{3 a f \sqrt {a+b \sec ^2(e+f x)}}-\frac {(2 b (3 a+4 b)) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{3 a^2 f}\\ &=-\frac {(3 a+4 b) \cos (e+f x)}{3 a^2 f \sqrt {a+b \sec ^2(e+f x)}}+\frac {\cos ^3(e+f x)}{3 a f \sqrt {a+b \sec ^2(e+f x)}}-\frac {2 b (3 a+4 b) \sec (e+f x)}{3 a^3 f \sqrt {a+b \sec ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 3.40, size = 93, normalized size = 0.82 \[ -\frac {\sec ^3(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (a^2 (-\cos (4 (e+f x)))+9 a^2+8 a (a+2 b) \cos (2 (e+f x))+64 a b+64 b^2\right )}{48 a^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 98, normalized size = 0.86 \[ \frac {{\left (a^{2} \cos \left (f x + e\right )^{5} - {\left (3 \, a^{2} + 4 \, a b\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \, {\left (a^{4} f \cos \left (f x + e\right )^{2} + a^{3} b f\right )}} \]
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 \[ \int \frac {\sin \left (f x + e\right )^{3}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.90, size = 12782, normalized size = 112.12 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 141, normalized size = 1.24 \[ -\frac {\frac {3 \, \sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a^{2}} - \frac {{\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3} - 6 \, \sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} b \cos \left (f x + e\right )}{a^{3}} + \frac {3 \, b}{\sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} a^{2} \cos \left (f x + e\right )} + \frac {3 \, b^{2}}{\sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} a^{3} \cos \left (f x + e\right )}}{3 \, f} \]
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 \[ \int \frac {{\sin \left (e+f\,x\right )}^3}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2}} \,d x \]
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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