Optimal. Leaf size=226 \[ -\frac {8 b \left (5 a^2-10 a b+b^2\right ) \tan (e+f x)}{15 f (a+b)^5 \sqrt {a+b \tan ^2(e+f x)+b}}-\frac {4 b \left (5 a^2-10 a b+b^2\right ) \tan (e+f x)}{15 f (a+b)^4 \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}-\frac {\left (5 a^2-10 a b+b^2\right ) \cot (e+f x)}{5 f (a+b)^3 \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}-\frac {\cot ^5(e+f x)}{5 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}-\frac {2 (5 a+b) \cot ^3(e+f x)}{15 f (a+b)^2 \left (a+b \tan ^2(e+f x)+b\right )^{3/2}} \]
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Rubi [A] time = 0.24, antiderivative size = 226, 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 = {4132, 462, 453, 271, 192, 191} \[ -\frac {8 b \left (5 a^2-10 a b+b^2\right ) \tan (e+f x)}{15 f (a+b)^5 \sqrt {a+b \tan ^2(e+f x)+b}}-\frac {4 b \left (5 a^2-10 a b+b^2\right ) \tan (e+f x)}{15 f (a+b)^4 \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}-\frac {\left (5 a^2-10 a b+b^2\right ) \cot (e+f x)}{5 f (a+b)^3 \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}-\frac {\cot ^5(e+f x)}{5 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}-\frac {2 (5 a+b) \cot ^3(e+f x)}{15 f (a+b)^2 \left (a+b \tan ^2(e+f x)+b\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 271
Rule 453
Rule 462
Rule 4132
Rubi steps
\begin {align*} \int \frac {\csc ^6(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+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot ^5(e+f x)}{5 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {2 (5 a+b)+5 (a+b) x^2}{x^4 \left (a+b+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{5 (a+b) f}\\ &=-\frac {2 (5 a+b) \cot ^3(e+f x)}{15 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^5(e+f x)}{5 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\left (5 a^2-10 a b+b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{5 (a+b)^2 f}\\ &=-\frac {\left (5 a^2-10 a b+b^2\right ) \cot (e+f x)}{5 (a+b)^3 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {2 (5 a+b) \cot ^3(e+f x)}{15 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^5(e+f x)}{5 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\left (4 b \left (5 a^2-10 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+b+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{5 (a+b)^3 f}\\ &=-\frac {\left (5 a^2-10 a b+b^2\right ) \cot (e+f x)}{5 (a+b)^3 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {2 (5 a+b) \cot ^3(e+f x)}{15 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^5(e+f x)}{5 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {4 b \left (5 a^2-10 a b+b^2\right ) \tan (e+f x)}{15 (a+b)^4 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\left (8 b \left (5 a^2-10 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{15 (a+b)^4 f}\\ &=-\frac {\left (5 a^2-10 a b+b^2\right ) \cot (e+f x)}{5 (a+b)^3 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {2 (5 a+b) \cot ^3(e+f x)}{15 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^5(e+f x)}{5 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {4 b \left (5 a^2-10 a b+b^2\right ) \tan (e+f x)}{15 (a+b)^4 f \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}-\frac {8 b \left (5 a^2-10 a b+b^2\right ) \tan (e+f x)}{15 (a+b)^5 f \sqrt {a+b+b \tan ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 7.41, size = 173, normalized size = 0.77 \[ \frac {\tan (e+f x) \sec ^4(e+f x) (a \cos (2 (e+f x))+a+2 b)^3 \left (\left (-8 a^2+50 a b-15 b^2\right ) \csc ^2(e+f x)+\frac {20 a b^2 (a+b)}{(a \cos (2 (e+f x))+a+2 b)^2}+\frac {10 a b (5 b-6 a)}{a \cos (2 (e+f x))+a+2 b}-3 (a+b)^2 \csc ^6(e+f x)+2 (a+b) (5 b-2 a) \csc ^4(e+f x)\right )}{120 f (a+b)^5 \left (a+b \sec ^2(e+f x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 15.77, size = 460, normalized size = 2.04 \[ -\frac {{\left (8 \, {\left (a^{4} - 10 \, a^{3} b + 5 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{9} - 4 \, {\left (5 \, a^{4} - 53 \, a^{3} b + 55 \, a^{2} b^{2} - 15 \, a b^{3}\right )} \cos \left (f x + e\right )^{7} + 3 \, {\left (5 \, a^{4} - 60 \, a^{3} b + 126 \, a^{2} b^{2} - 60 \, a b^{3} + 5 \, b^{4}\right )} \cos \left (f x + e\right )^{5} + 4 \, {\left (15 \, a^{3} b - 55 \, a^{2} b^{2} + 53 \, a b^{3} - 5 \, b^{4}\right )} \cos \left (f x + e\right )^{3} + 8 \, {\left (5 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{15 \, {\left ({\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right )} f \cos \left (f x + e\right )^{8} - 2 \, {\left (a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right )} f \cos \left (f x + e\right )^{6} + {\left (a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7}\right )} f\right )} \sin \left (f x + e\right )} \]
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 \[ \int \frac {\csc \left (f x + e\right )^{6}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.74, size = 324, normalized size = 1.43 \[ -\frac {\left (8 \left (\cos ^{8}\left (f x +e \right )\right ) a^{4}-80 \left (\cos ^{8}\left (f x +e \right )\right ) a^{3} b +40 \left (\cos ^{8}\left (f x +e \right )\right ) a^{2} b^{2}-20 \left (\cos ^{6}\left (f x +e \right )\right ) a^{4}+212 \left (\cos ^{6}\left (f x +e \right )\right ) a^{3} b -220 \left (\cos ^{6}\left (f x +e \right )\right ) a^{2} b^{2}+60 \left (\cos ^{6}\left (f x +e \right )\right ) a \,b^{3}+15 \left (\cos ^{4}\left (f x +e \right )\right ) a^{4}-180 \left (\cos ^{4}\left (f x +e \right )\right ) a^{3} b +378 \left (\cos ^{4}\left (f x +e \right )\right ) a^{2} b^{2}-180 \left (\cos ^{4}\left (f x +e \right )\right ) a \,b^{3}+15 \left (\cos ^{4}\left (f x +e \right )\right ) b^{4}+60 \left (\cos ^{2}\left (f x +e \right )\right ) a^{3} b -220 \left (\cos ^{2}\left (f x +e \right )\right ) a^{2} b^{2}+212 \left (\cos ^{2}\left (f x +e \right )\right ) a \,b^{3}-20 \left (\cos ^{2}\left (f x +e \right )\right ) b^{4}+40 a^{2} b^{2}-80 a \,b^{3}+8 b^{4}\right ) \left (\cos ^{5}\left (f x +e \right )\right ) \left (\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\cos \left (f x +e \right )^{2}}\right )^{\frac {5}{2}}}{15 f \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{4} \sin \left (f x +e \right )^{5} \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 373, normalized size = 1.65 \[ -\frac {\frac {40 \, b \tan \left (f x + e\right )}{\sqrt {b \tan \left (f x + e\right )^{2} + a + b} {\left (a + b\right )}^{3}} + \frac {20 \, b \tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} {\left (a + b\right )}^{2}} - \frac {160 \, b^{2} \tan \left (f x + e\right )}{\sqrt {b \tan \left (f x + e\right )^{2} + a + b} {\left (a + b\right )}^{4}} - \frac {80 \, b^{2} \tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} {\left (a + b\right )}^{3}} + \frac {128 \, b^{3} \tan \left (f x + e\right )}{\sqrt {b \tan \left (f x + e\right )^{2} + a + b} {\left (a + b\right )}^{5}} + \frac {64 \, b^{3} \tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} {\left (a + b\right )}^{4}} + \frac {15}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} {\left (a + b\right )} \tan \left (f x + e\right )} - \frac {60 \, b}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} {\left (a + b\right )}^{2} \tan \left (f x + e\right )} + \frac {48 \, b^{2}}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} {\left (a + b\right )}^{3} \tan \left (f x + e\right )} + \frac {10}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} {\left (a + b\right )} \tan \left (f x + e\right )^{3}} - \frac {8 \, b}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} {\left (a + b\right )}^{2} \tan \left (f x + e\right )^{3}} + \frac {3}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} {\left (a + b\right )} \tan \left (f x + e\right )^{5}}}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
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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