Optimal. Leaf size=214 \[ -\frac {b (a+b) (3 a+11 b) \cos (e+f x)}{8 a^5 f \left (a \cos ^2(e+f x)+b\right )}+\frac {(a+3 b) (3 a+5 b) \cos ^3(e+f x)}{12 a^4 b f}-\frac {\cos ^5(e+f x)}{5 a^3 f}-\frac {(a+b)^2 \cos ^7(e+f x)}{4 a^2 b f \left (a \cos ^2(e+f x)+b\right )^2}+\frac {\sqrt {b} \left (15 a^2+70 a b+63 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 a^{11/2} f}-\frac {\left (3 a^2+14 a b+13 b^2\right ) \cos (e+f x)}{2 a^5 f} \]
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Rubi [A] time = 0.25, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4133, 463, 455, 1810, 205} \[ -\frac {\left (3 a^2+14 a b+13 b^2\right ) \cos (e+f x)}{2 a^5 f}+\frac {\sqrt {b} \left (15 a^2+70 a b+63 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 a^{11/2} f}-\frac {(a+b)^2 \cos ^7(e+f x)}{4 a^2 b f \left (a \cos ^2(e+f x)+b\right )^2}+\frac {(a+3 b) (3 a+5 b) \cos ^3(e+f x)}{12 a^4 b f}-\frac {b (a+b) (3 a+11 b) \cos (e+f x)}{8 a^5 f \left (a \cos ^2(e+f x)+b\right )}-\frac {\cos ^5(e+f x)}{5 a^3 f} \]
Antiderivative was successfully verified.
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Rule 205
Rule 455
Rule 463
Rule 1810
Rule 4133
Rubi steps
\begin {align*} \int \frac {\sin ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^6 \left (1-x^2\right )^2}{\left (b+a x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {(a+b)^2 \cos ^7(e+f x)}{4 a^2 b f \left (b+a \cos ^2(e+f x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {x^6 \left (-4 a^2+7 (a+b)^2-4 a b x^2\right )}{\left (b+a x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{4 a^2 b f}\\ &=-\frac {(a+b)^2 \cos ^7(e+f x)}{4 a^2 b f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {b (a+b) (3 a+11 b) \cos (e+f x)}{8 a^5 f \left (b+a \cos ^2(e+f x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {-a b^2 (a+b) (3 a+11 b)+2 a^2 b (a+b) (3 a+11 b) x^2-2 a^3 (a+b) (3 a+11 b) x^4+8 a^4 b x^6}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{8 a^6 b f}\\ &=-\frac {(a+b)^2 \cos ^7(e+f x)}{4 a^2 b f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {b (a+b) (3 a+11 b) \cos (e+f x)}{8 a^5 f \left (b+a \cos ^2(e+f x)\right )}-\frac {\operatorname {Subst}\left (\int \left (4 a b \left (3 a^2+14 a b+13 b^2\right )-2 a^2 (a+3 b) (3 a+5 b) x^2+8 a^3 b x^4+\frac {-15 a^3 b^2-70 a^2 b^3-63 a b^4}{b+a x^2}\right ) \, dx,x,\cos (e+f x)\right )}{8 a^6 b f}\\ &=-\frac {\left (3 a^2+14 a b+13 b^2\right ) \cos (e+f x)}{2 a^5 f}+\frac {(a+3 b) (3 a+5 b) \cos ^3(e+f x)}{12 a^4 b f}-\frac {\cos ^5(e+f x)}{5 a^3 f}-\frac {(a+b)^2 \cos ^7(e+f x)}{4 a^2 b f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {b (a+b) (3 a+11 b) \cos (e+f x)}{8 a^5 f \left (b+a \cos ^2(e+f x)\right )}+\frac {\left (b \left (15 a^2+70 a b+63 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{8 a^5 f}\\ &=\frac {\sqrt {b} \left (15 a^2+70 a b+63 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{8 a^{11/2} f}-\frac {\left (3 a^2+14 a b+13 b^2\right ) \cos (e+f x)}{2 a^5 f}+\frac {(a+3 b) (3 a+5 b) \cos ^3(e+f x)}{12 a^4 b f}-\frac {\cos ^5(e+f x)}{5 a^3 f}-\frac {(a+b)^2 \cos ^7(e+f x)}{4 a^2 b f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {b (a+b) (3 a+11 b) \cos (e+f x)}{8 a^5 f \left (b+a \cos ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [C] time = 10.56, size = 1641, normalized size = 7.67 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 579, normalized size = 2.71 \[ \left [-\frac {48 \, a^{4} \cos \left (f x + e\right )^{9} - 16 \, {\left (10 \, a^{4} + 9 \, a^{3} b\right )} \cos \left (f x + e\right )^{7} + 16 \, {\left (15 \, a^{4} + 70 \, a^{3} b + 63 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{5} + 50 \, {\left (15 \, a^{3} b + 70 \, a^{2} b^{2} + 63 \, a b^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left ({\left (15 \, a^{4} + 70 \, a^{3} b + 63 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 15 \, a^{2} b^{2} + 70 \, a b^{3} + 63 \, b^{4} + 2 \, {\left (15 \, a^{3} b + 70 \, a^{2} b^{2} + 63 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, a \sqrt {-\frac {b}{a}} \cos \left (f x + e\right ) - b}{a \cos \left (f x + e\right )^{2} + b}\right ) + 30 \, {\left (15 \, a^{2} b^{2} + 70 \, a b^{3} + 63 \, b^{4}\right )} \cos \left (f x + e\right )}{240 \, {\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 )}}, -\frac {24 \, a^{4} \cos \left (f x + e\right )^{9} - 8 \, {\left (10 \, a^{4} + 9 \, a^{3} b\right )} \cos \left (f x + e\right )^{7} + 8 \, {\left (15 \, a^{4} + 70 \, a^{3} b + 63 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{5} + 25 \, {\left (15 \, a^{3} b + 70 \, a^{2} b^{2} + 63 \, a b^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left ({\left (15 \, a^{4} + 70 \, a^{3} b + 63 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 15 \, a^{2} b^{2} + 70 \, a b^{3} + 63 \, b^{4} + 2 \, {\left (15 \, a^{3} b + 70 \, a^{2} b^{2} + 63 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}} \cos \left (f x + e\right )}{b}\right ) + 15 \, {\left (15 \, a^{2} b^{2} + 70 \, a b^{3} + 63 \, b^{4}\right )} \cos \left (f x + e\right )}{120 \, {\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 )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.58, size = 837, normalized size = 3.91 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.02, size = 374, normalized size = 1.75 \[ -\frac {\cos ^{5}\left (f x +e \right )}{5 a^{3} f}+\frac {2 \left (\cos ^{3}\left (f x +e \right )\right )}{3 a^{3} f}+\frac {\left (\cos ^{3}\left (f x +e \right )\right ) b}{f \,a^{4}}-\frac {\cos \left (f x +e \right )}{a^{3} f}-\frac {6 b \cos \left (f x +e \right )}{f \,a^{4}}-\frac {6 \cos \left (f x +e \right ) b^{2}}{f \,a^{5}}-\frac {9 b \left (\cos ^{3}\left (f x +e \right )\right )}{8 f \,a^{2} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {13 b^{2} \left (\cos ^{3}\left (f x +e \right )\right )}{4 f \,a^{3} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {17 b^{3} \left (\cos ^{3}\left (f x +e \right )\right )}{8 f \,a^{4} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {7 b^{2} \cos \left (f x +e \right )}{8 f \,a^{3} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {11 b^{3} \cos \left (f x +e \right )}{4 f \,a^{4} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {15 b^{4} \cos \left (f x +e \right )}{8 f \,a^{5} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {15 b \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{8 f \,a^{3} \sqrt {a b}}+\frac {35 b^{2} \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{4 f \,a^{4} \sqrt {a b}}+\frac {63 b^{3} \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{8 f \,a^{5} \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 204, normalized size = 0.95 \[ -\frac {\frac {15 \, {\left ({\left (9 \, a^{3} b + 26 \, a^{2} b^{2} + 17 \, a b^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (7 \, a^{2} b^{2} + 22 \, a b^{3} + 15 \, b^{4}\right )} \cos \left (f x + e\right )\right )}}{a^{7} \cos \left (f x + e\right )^{4} + 2 \, a^{6} b \cos \left (f x + e\right )^{2} + a^{5} b^{2}} - \frac {15 \, {\left (15 \, a^{2} b + 70 \, a b^{2} + 63 \, b^{3}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}} + \frac {8 \, {\left (3 \, a^{2} \cos \left (f x + e\right )^{5} - 5 \, {\left (2 \, a^{2} + 3 \, a b\right )} \cos \left (f x + e\right )^{3} + 15 \, {\left (a^{2} + 6 \, a b + 6 \, b^{2}\right )} \cos \left (f x + e\right )\right )}}{a^{5}}}{120 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.51, size = 255, normalized size = 1.19 \[ \frac {{\cos \left (e+f\,x\right )}^3\,\left (\frac {b}{a^4}+\frac {2}{3\,a^3}\right )}{f}-\frac {\left (\frac {9\,a^3\,b}{8}+\frac {13\,a^2\,b^2}{4}+\frac {17\,a\,b^3}{8}\right )\,{\cos \left (e+f\,x\right )}^3+\left (\frac {7\,a^2\,b^2}{8}+\frac {11\,a\,b^3}{4}+\frac {15\,b^4}{8}\right )\,\cos \left (e+f\,x\right )}{f\,\left (a^7\,{\cos \left (e+f\,x\right )}^4+2\,a^6\,b\,{\cos \left (e+f\,x\right )}^2+a^5\,b^2\right )}-\frac {{\cos \left (e+f\,x\right )}^5}{5\,a^3\,f}-\frac {\cos \left (e+f\,x\right )\,\left (\frac {1}{a^3}-\frac {3\,b^2}{a^5}+\frac {3\,b\,\left (\frac {3\,b}{a^4}+\frac {2}{a^3}\right )}{a}\right )}{f}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,\cos \left (e+f\,x\right )\,\left (15\,a^2+70\,a\,b+63\,b^2\right )}{15\,a^2\,b+70\,a\,b^2+63\,b^3}\right )\,\left (15\,a^2+70\,a\,b+63\,b^2\right )}{8\,a^{11/2}\,f} \]
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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