Optimal. Leaf size=114 \[ \frac {\sqrt {b} (3 a+5 b) \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{2 a^{7/2} f}-\frac {b (a+b) \cos (e+f x)}{2 a^3 f \left (a \cos ^2(e+f x)+b\right )}-\frac {(a+2 b) \cos (e+f x)}{a^3 f}+\frac {\cos ^3(e+f x)}{3 a^2 f} \]
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Rubi [A] time = 0.11, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4133, 455, 1153, 205} \[ -\frac {b (a+b) \cos (e+f x)}{2 a^3 f \left (a \cos ^2(e+f x)+b\right )}-\frac {(a+2 b) \cos (e+f x)}{a^3 f}+\frac {\sqrt {b} (3 a+5 b) \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{2 a^{7/2} f}+\frac {\cos ^3(e+f x)}{3 a^2 f} \]
Antiderivative was successfully verified.
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Rule 205
Rule 455
Rule 1153
Rule 4133
Rubi steps
\begin {align*} \int \frac {\sin ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (1-x^2\right )}{\left (b+a x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {b (a+b) \cos (e+f x)}{2 a^3 f \left (b+a \cos ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {b (a+b)-2 a (a+b) x^2+2 a^2 x^4}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{2 a^3 f}\\ &=-\frac {b (a+b) \cos (e+f x)}{2 a^3 f \left (b+a \cos ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \left (-2 (a+2 b)+2 a x^2+\frac {3 a b+5 b^2}{b+a x^2}\right ) \, dx,x,\cos (e+f x)\right )}{2 a^3 f}\\ &=-\frac {(a+2 b) \cos (e+f x)}{a^3 f}+\frac {\cos ^3(e+f x)}{3 a^2 f}-\frac {b (a+b) \cos (e+f x)}{2 a^3 f \left (b+a \cos ^2(e+f x)\right )}+\frac {(b (3 a+5 b)) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{2 a^3 f}\\ &=\frac {\sqrt {b} (3 a+5 b) \tan ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{2 a^{7/2} f}-\frac {(a+2 b) \cos (e+f x)}{a^3 f}+\frac {\cos ^3(e+f x)}{3 a^2 f}-\frac {b (a+b) \cos (e+f x)}{2 a^3 f \left (b+a \cos ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [C] time = 3.47, size = 403, normalized size = 3.54 \[ \frac {-\frac {9 a^3 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b}}\right )}{b^{3/2}}-\frac {9 a^3 \tan ^{-1}\left (\frac {\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )+\sqrt {a}}{\sqrt {b}}\right )}{b^{3/2}}+\frac {3 \left (3 a^3+192 a b^2+320 b^3\right ) \tan ^{-1}\left (\frac {\sin (e) \tan \left (\frac {f x}{2}\right ) \left (-\sqrt {a}-i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt {a}-\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right )}{b^{3/2}}+\frac {3 \left (3 a^3+192 a b^2+320 b^3\right ) \tan ^{-1}\left (\frac {\sin (e) \tan \left (\frac {f x}{2}\right ) \left (-\sqrt {a}+i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right )+\cos (e) \left (\sqrt {a}+\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right )}{b^{3/2}}-\frac {32 \sqrt {a} \cos (e+f x) \left (a^2 (-\cos (4 (e+f x)))+9 a^2+4 a (2 a+5 b) \cos (2 (e+f x))+56 a b+60 b^2\right )}{a \cos (2 (e+f x))+a+2 b}}{384 a^{7/2} f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 297, normalized size = 2.61 \[ \left [\frac {4 \, a^{2} \cos \left (f x + e\right )^{5} - 4 \, {\left (3 \, a^{2} + 5 \, a b\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left ({\left (3 \, a^{2} + 5 \, a b\right )} \cos \left (f x + e\right )^{2} + 3 \, a b + 5 \, b^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, a \sqrt {-\frac {b}{a}} \cos \left (f x + e\right ) - b}{a \cos \left (f x + e\right )^{2} + b}\right ) - 6 \, {\left (3 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )}{12 \, {\left (a^{4} f \cos \left (f x + e\right )^{2} + a^{3} b f\right )}}, \frac {2 \, a^{2} \cos \left (f x + e\right )^{5} - 2 \, {\left (3 \, a^{2} + 5 \, a b\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left ({\left (3 \, a^{2} + 5 \, a b\right )} \cos \left (f x + e\right )^{2} + 3 \, a b + 5 \, b^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}} \cos \left (f x + e\right )}{b}\right ) - 3 \, {\left (3 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )}{6 \, {\left (a^{4} f \cos \left (f x + e\right )^{2} + a^{3} b f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 143, normalized size = 1.25 \[ \frac {{\left (3 \, a b + 5 \, b^{2}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3} f} - \frac {\frac {a b \cos \left (f x + e\right )}{f} + \frac {b^{2} \cos \left (f x + e\right )}{f}}{2 \, {\left (a \cos \left (f x + e\right )^{2} + b\right )} a^{3}} + \frac {a^{4} f^{11} \cos \left (f x + e\right )^{3} - 3 \, a^{4} f^{11} \cos \left (f x + e\right ) - 6 \, a^{3} b f^{11} \cos \left (f x + e\right )}{3 \, a^{6} f^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.91, size = 165, normalized size = 1.45 \[ \frac {\cos ^{3}\left (f x +e \right )}{3 a^{2} f}-\frac {\cos \left (f x +e \right )}{a^{2} f}-\frac {2 \cos \left (f x +e \right ) b}{f \,a^{3}}-\frac {b \cos \left (f x +e \right )}{2 f \,a^{2} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}-\frac {b^{2} \cos \left (f x +e \right )}{2 f \,a^{3} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}+\frac {3 b \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{2 f \,a^{2} \sqrt {a b}}+\frac {5 b^{2} \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{2 f \,a^{3} \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 104, normalized size = 0.91 \[ -\frac {\frac {3 \, {\left (a b + b^{2}\right )} \cos \left (f x + e\right )}{a^{4} \cos \left (f x + e\right )^{2} + a^{3} b} - \frac {3 \, {\left (3 \, a b + 5 \, b^{2}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {2 \, {\left (a \cos \left (f x + e\right )^{3} - 3 \, {\left (a + 2 \, b\right )} \cos \left (f x + e\right )\right )}}{a^{3}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 130, normalized size = 1.14 \[ \frac {{\cos \left (e+f\,x\right )}^3}{3\,a^2\,f}-\frac {\cos \left (e+f\,x\right )\,\left (\frac {2\,b}{a^3}+\frac {1}{a^2}\right )}{f}-\frac {\cos \left (e+f\,x\right )\,\left (\frac {b^2}{2}+\frac {a\,b}{2}\right )}{f\,\left (a^4\,{\cos \left (e+f\,x\right )}^2+b\,a^3\right )}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,\cos \left (e+f\,x\right )\,\left (3\,a+5\,b\right )}{5\,b^2+3\,a\,b}\right )\,\left (3\,a+5\,b\right )}{2\,a^{7/2}\,f} \]
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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