Optimal. Leaf size=126 \[ \frac {b^2 (6 a+5 b) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 a^{7/2} f (a+b)^{3/2}}-\frac {b^3 \sin (e+f x)}{2 a^3 f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )}+\frac {(a-2 b) \sin (e+f x)}{a^3 f}-\frac {\sin ^3(e+f x)}{3 a^2 f} \]
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Rubi [A] time = 0.16, antiderivative size = 126, 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 = {4147, 390, 385, 208} \[ -\frac {b^3 \sin (e+f x)}{2 a^3 f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )}+\frac {b^2 (6 a+5 b) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 a^{7/2} f (a+b)^{3/2}}+\frac {(a-2 b) \sin (e+f x)}{a^3 f}-\frac {\sin ^3(e+f x)}{3 a^2 f} \]
Antiderivative was successfully verified.
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Rule 208
Rule 385
Rule 390
Rule 4147
Rubi steps
\begin {align*} \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{\left (a+b-a x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a-2 b}{a^3}-\frac {x^2}{a^2}+\frac {b^2 (3 a+2 b)-3 a b^2 x^2}{a^3 \left (a+b-a x^2\right )^2}\right ) \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {(a-2 b) \sin (e+f x)}{a^3 f}-\frac {\sin ^3(e+f x)}{3 a^2 f}+\frac {\operatorname {Subst}\left (\int \frac {b^2 (3 a+2 b)-3 a b^2 x^2}{\left (a+b-a x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{a^3 f}\\ &=\frac {(a-2 b) \sin (e+f x)}{a^3 f}-\frac {\sin ^3(e+f x)}{3 a^2 f}-\frac {b^3 \sin (e+f x)}{2 a^3 (a+b) f \left (a+b-a \sin ^2(e+f x)\right )}+\frac {\left (b^2 (6 a+5 b)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{2 a^3 (a+b) f}\\ &=\frac {b^2 (6 a+5 b) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 a^{7/2} (a+b)^{3/2} f}+\frac {(a-2 b) \sin (e+f x)}{a^3 f}-\frac {\sin ^3(e+f x)}{3 a^2 f}-\frac {b^3 \sin (e+f x)}{2 a^3 (a+b) f \left (a+b-a \sin ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [A] time = 1.11, size = 139, normalized size = 1.10 \[ \frac {a^{3/2} \sin (3 (e+f x))+3 \sqrt {a} \sin (e+f x) \left (-\frac {4 b^3}{(a+b) (a \cos (2 (e+f x))+a+2 b)}+3 a-8 b\right )-\frac {3 b^2 (6 a+5 b) \left (\log \left (\sqrt {a+b}-\sqrt {a} \sin (e+f x)\right )-\log \left (\sqrt {a+b}+\sqrt {a} \sin (e+f x)\right )\right )}{(a+b)^{3/2}}}{12 a^{7/2} f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 490, normalized size = 3.89 \[ \left [\frac {3 \, {\left (6 \, a b^{3} + 5 \, b^{4} + {\left (6 \, a^{2} b^{2} + 5 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a^{2} + a b} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a^{2} + a b} \sin \left (f x + e\right ) - 2 \, a - b}{a \cos \left (f x + e\right )^{2} + b}\right ) + 2 \, {\left (4 \, a^{4} b - 4 \, a^{3} b^{2} - 23 \, a^{2} b^{3} - 15 \, a b^{4} + 2 \, {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (2 \, a^{5} - a^{4} b - 8 \, a^{3} b^{2} - 5 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{12 \, {\left ({\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f\right )}}, -\frac {3 \, {\left (6 \, a b^{3} + 5 \, b^{4} + {\left (6 \, a^{2} b^{2} + 5 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a^{2} - a b} \arctan \left (\frac {\sqrt {-a^{2} - a b} \sin \left (f x + e\right )}{a + b}\right ) - {\left (4 \, a^{4} b - 4 \, a^{3} b^{2} - 23 \, a^{2} b^{3} - 15 \, a b^{4} + 2 \, {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (2 \, a^{5} - a^{4} b - 8 \, a^{3} b^{2} - 5 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{6 \, {\left ({\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 152, normalized size = 1.21 \[ \frac {\frac {3 \, b^{3} \sin \left (f x + e\right )}{{\left (a^{4} + a^{3} b\right )} {\left (a \sin \left (f x + e\right )^{2} - a - b\right )}} - \frac {3 \, {\left (6 \, a b^{2} + 5 \, b^{3}\right )} \arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{{\left (a^{4} + a^{3} b\right )} \sqrt {-a^{2} - a b}} - \frac {2 \, {\left (a^{4} \sin \left (f x + e\right )^{3} - 3 \, a^{4} \sin \left (f x + e\right ) + 6 \, a^{3} b \sin \left (f x + e\right )\right )}}{a^{6}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.40, size = 120, normalized size = 0.95 \[ \frac {-\frac {\frac {a \left (\sin ^{3}\left (f x +e \right )\right )}{3}-a \sin \left (f x +e \right )+2 b \sin \left (f x +e \right )}{a^{3}}-\frac {b^{2} \left (-\frac {b \sin \left (f x +e \right )}{2 \left (a +b \right ) \left (-a -b +a \left (\sin ^{2}\left (f x +e \right )\right )\right )}-\frac {\left (6 a +5 b \right ) \arctanh \left (\frac {a \sin \left (f x +e \right )}{\sqrt {\left (a +b \right ) a}}\right )}{2 \left (a +b \right ) \sqrt {\left (a +b \right ) a}}\right )}{a^{3}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 154, normalized size = 1.22 \[ -\frac {\frac {6 \, b^{3} \sin \left (f x + e\right )}{a^{5} + 2 \, a^{4} b + a^{3} b^{2} - {\left (a^{5} + a^{4} b\right )} \sin \left (f x + e\right )^{2}} + \frac {3 \, {\left (6 \, a b^{2} + 5 \, b^{3}\right )} \log \left (\frac {a \sin \left (f x + e\right ) - \sqrt {{\left (a + b\right )} a}}{a \sin \left (f x + e\right ) + \sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{4} + a^{3} b\right )} \sqrt {{\left (a + b\right )} a}} + \frac {4 \, {\left (a \sin \left (f x + e\right )^{3} - 3 \, {\left (a - 2 \, b\right )} \sin \left (f x + e\right )\right )}}{a^{3}}}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.62, size = 124, normalized size = 0.98 \[ \frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {a}\,\sin \left (e+f\,x\right )}{\sqrt {a+b}}\right )\,\left (6\,a+5\,b\right )}{2\,a^{7/2}\,f\,{\left (a+b\right )}^{3/2}}-\frac {{\sin \left (e+f\,x\right )}^3}{3\,a^2\,f}-\frac {b^3\,\sin \left (e+f\,x\right )}{2\,f\,\left (a+b\right )\,\left (-a^4\,{\sin \left (e+f\,x\right )}^2+a^4+b\,a^3\right )}-\frac {\sin \left (e+f\,x\right )\,\left (\frac {2\,\left (a+b\right )}{a^3}-\frac {3}{a^2}\right )}{f} \]
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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