Optimal. Leaf size=146 \[ -\frac {8 b (a+2 b) \sec (e+f x)}{3 a^4 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {4 b (a+2 b) \sec (e+f x)}{3 a^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(a+2 b) \cos (e+f x)}{a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\cos ^3(e+f x)}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4134, 453, 271, 192, 191} \[ -\frac {8 b (a+2 b) \sec (e+f x)}{3 a^4 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {4 b (a+2 b) \sec (e+f x)}{3 a^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(a+2 b) \cos (e+f x)}{a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\cos ^3(e+f x)}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
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 )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {-1+x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\cos ^3(e+f x)}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {(a+2 b) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{a f}\\ &=-\frac {(a+2 b) \cos (e+f x)}{a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\cos ^3(e+f x)}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(4 b (a+2 b)) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{a^2 f}\\ &=-\frac {(a+2 b) \cos (e+f x)}{a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\cos ^3(e+f x)}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {4 b (a+2 b) \sec (e+f x)}{3 a^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(8 b (a+2 b)) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{3 a^3 f}\\ &=-\frac {(a+2 b) \cos (e+f x)}{a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\cos ^3(e+f x)}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {4 b (a+2 b) \sec (e+f x)}{3 a^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {8 b (a+2 b) \sec (e+f x)}{3 a^4 f \sqrt {a+b \sec ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 2.31, size = 129, normalized size = 0.88 \[ -\frac {\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (a^3 (-\cos (6 (e+f x)))+26 a^3+3 a \left (11 a^2+96 a b+128 b^2\right ) \cos (2 (e+f x))+6 a^2 (a+4 b) \cos (4 (e+f x))+264 a^2 b+640 a b^2+512 b^3\right )}{192 a^4 f \left (a+b \sec ^2(e+f x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 138, normalized size = 0.95 \[ \frac {{\left (a^{3} \cos \left (f x + e\right )^{7} - 3 \, {\left (a^{3} + 2 \, a^{2} b\right )} \cos \left (f x + e\right )^{5} - 12 \, {\left (a^{2} b + 2 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} - 8 \, {\left (a b^{2} + 2 \, b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \, {\left (a^{6} f \cos \left (f x + e\right )^{4} + 2 \, a^{5} b f \cos \left (f x + e\right )^{2} + a^{4} b^{2} f\right )}} \]
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 \[ \int \frac {\sin \left (f x + e\right )^{3}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.10, size = 159, normalized size = 1.09 \[ -\frac {\left (a +b \right )^{5} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right ) \left (\left (\cos ^{6}\left (f x +e \right )\right ) a^{3}-3 \left (\cos ^{4}\left (f x +e \right )\right ) a^{3}-6 \left (\cos ^{4}\left (f x +e \right )\right ) a^{2} b -12 a^{2} \left (\cos ^{2}\left (f x +e \right )\right ) b -24 \left (\cos ^{2}\left (f x +e \right )\right ) a \,b^{2}-8 b^{2} a -16 b^{3}\right ) \sqrt {4}\, a}{6 f \left (\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\cos \left (f x +e \right )^{2}}\right )^{\frac {5}{2}} \cos \left (f x +e \right )^{5} \left (\sqrt {-a b}-a \right )^{5} \left (\sqrt {-a b}+a \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 195, normalized size = 1.34 \[ -\frac {\frac {3 \, \sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a^{3}} - \frac {{\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3} - 9 \, \sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} b \cos \left (f x + e\right )}{a^{4}} + \frac {6 \, {\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )} b \cos \left (f x + e\right )^{2} - b^{2}}{{\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} a^{3} \cos \left (f x + e\right )^{3}} + \frac {9 \, {\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )} b^{2} \cos \left (f x + e\right )^{2} - b^{3}}{{\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} a^{4} \cos \left (f x + e\right )^{3}}}{3 \, f} \]
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 \[ \int \frac {{\sin \left (e+f\,x\right )}^3}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{5/2}} \,d x \]
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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