Optimal. Leaf size=219 \[ \frac {2 (5 a+4 b) \cos ^3(e+f x)}{15 a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {8 b \left (5 a^2+20 a b+16 b^2\right ) \sec (e+f x)}{15 a^5 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {4 b \left (5 a^2+20 a b+16 b^2\right ) \sec (e+f x)}{15 a^4 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\left (5 a^2+20 a b+16 b^2\right ) \cos (e+f x)}{5 a^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cos ^5(e+f x)}{5 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.22, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4134, 462, 453, 271, 192, 191} \[ -\frac {8 b \left (5 a^2+20 a b+16 b^2\right ) \sec (e+f x)}{15 a^5 f \sqrt {a+b \sec ^2(e+f x)}}-\frac {4 b \left (5 a^2+20 a b+16 b^2\right ) \sec (e+f x)}{15 a^4 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {2 (5 a+4 b) \cos ^3(e+f x)}{15 a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\left (\frac {4 b (5 a+4 b)}{a^2}+5\right ) \cos (e+f x)}{5 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cos ^5(e+f x)}{5 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 271
Rule 453
Rule 462
Rule 4134
Rubi steps
\begin {align*} \int \frac {\sin ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {\cos ^5(e+f x)}{5 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {-2 (5 a+4 b)+5 a x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{5 a f}\\ &=\frac {2 (5 a+4 b) \cos ^3(e+f x)}{15 a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cos ^5(e+f x)}{5 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\left (5 a^2+20 a b+16 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{5 a^2 f}\\ &=-\frac {\left (5 a^2+20 a b+16 b^2\right ) \cos (e+f x)}{5 a^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {2 (5 a+4 b) \cos ^3(e+f x)}{15 a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cos ^5(e+f x)}{5 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\left (4 b \left (5 a^2+20 a b+16 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{5 a^3 f}\\ &=-\frac {\left (5 a^2+20 a b+16 b^2\right ) \cos (e+f x)}{5 a^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {2 (5 a+4 b) \cos ^3(e+f x)}{15 a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cos ^5(e+f x)}{5 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {4 b \left (5 a^2+20 a b+16 b^2\right ) \sec (e+f x)}{15 a^4 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\left (8 b \left (5 a^2+20 a b+16 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{15 a^4 f}\\ &=-\frac {\left (5 a^2+20 a b+16 b^2\right ) \cos (e+f x)}{5 a^3 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac {2 (5 a+4 b) \cos ^3(e+f x)}{15 a^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cos ^5(e+f x)}{5 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {4 b \left (5 a^2+20 a b+16 b^2\right ) \sec (e+f x)}{15 a^4 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {8 b \left (5 a^2+20 a b+16 b^2\right ) \sec (e+f x)}{15 a^5 f \sqrt {a+b \sec ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 3.21, size = 182, normalized size = 0.83 \[ -\frac {\sec ^5(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (-16 a^4 \cos (6 (e+f x))+3 a^4 \cos (8 (e+f x))+425 a^4-32 a^3 b \cos (6 (e+f x))+6400 a^3 b+12 a^2 \left (7 a^2+64 a b+64 b^2\right ) \cos (4 (e+f x))+22784 a^2 b^2+48 a \left (11 a^3+150 a^2 b+384 a b^2+256 b^3\right ) \cos (2 (e+f x))+32768 a b^3+16384 b^4\right )}{3840 a^5 f \left (a+b \sec ^2(e+f x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 189, normalized size = 0.86 \[ -\frac {{\left (3 \, a^{4} \cos \left (f x + e\right )^{9} - 2 \, {\left (5 \, a^{4} + 4 \, a^{3} b\right )} \cos \left (f x + e\right )^{7} + 3 \, {\left (5 \, a^{4} + 20 \, a^{3} b + 16 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{5} + 12 \, {\left (5 \, a^{3} b + 20 \, a^{2} b^{2} + 16 \, a b^{3}\right )} \cos \left (f x + e\right )^{3} + 8 \, {\left (5 \, a^{2} b^{2} + 20 \, a b^{3} + 16 \, b^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{15 \, {\left (a^{7} f \cos \left (f x + e\right )^{4} + 2 \, a^{6} b f \cos \left (f x + e\right )^{2} + a^{5} b^{2} f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.41, size = 229, normalized size = 1.05 \[ \frac {\left (a +b \right )^{7} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right ) \left (3 \left (\cos ^{8}\left (f x +e \right )\right ) a^{4}-10 \left (\cos ^{6}\left (f x +e \right )\right ) a^{4}-8 \left (\cos ^{6}\left (f x +e \right )\right ) a^{3} b +15 \left (\cos ^{4}\left (f x +e \right )\right ) a^{4}+60 \left (\cos ^{4}\left (f x +e \right )\right ) a^{3} b +48 \left (\cos ^{4}\left (f x +e \right )\right ) a^{2} b^{2}+60 \left (\cos ^{2}\left (f x +e \right )\right ) a^{3} b +240 \left (\cos ^{2}\left (f x +e \right )\right ) a^{2} b^{2}+192 \left (\cos ^{2}\left (f x +e \right )\right ) a \,b^{3}+40 a^{2} b^{2}+160 a \,b^{3}+128 b^{4}\right ) \sqrt {4}\, a^{2}}{30 f \left (\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\cos \left (f x +e \right )^{2}}\right )^{\frac {5}{2}} \cos \left (f x +e \right )^{5} \left (\sqrt {-a b}+a \right )^{7} \left (\sqrt {-a b}-a \right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 334, normalized size = 1.53 \[ -\frac {\frac {15 \, \sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a^{3}} - \frac {10 \, {\left ({\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3} - 9 \, \sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} b \cos \left (f x + e\right )\right )}}{a^{4}} + \frac {3 \, {\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {5}{2}} \cos \left (f x + e\right )^{5} - 20 \, {\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} b \cos \left (f x + e\right )^{3} + 90 \, \sqrt {a + \frac {b}{\cos \left (f x + e\right )^{2}}} b^{2} \cos \left (f x + e\right )}{a^{5}} + \frac {5 \, {\left (6 \, {\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )} b \cos \left (f x + e\right )^{2} - b^{2}\right )}}{{\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} a^{3} \cos \left (f x + e\right )^{3}} + \frac {10 \, {\left (9 \, {\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )} b^{2} \cos \left (f x + e\right )^{2} - b^{3}\right )}}{{\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} a^{4} \cos \left (f x + e\right )^{3}} + \frac {5 \, {\left (12 \, {\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )} b^{3} \cos \left (f x + e\right )^{2} - b^{4}\right )}}{{\left (a + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} a^{5} \cos \left (f x + e\right )^{3}}}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\sin \left (e+f\,x\right )}^5}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{5/2}} \,d x \]
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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