Optimal. Leaf size=74 \[ \frac {\sin (e+f x)}{2 f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 \sqrt {a} f (a+b)^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4147, 199, 208} \[ \frac {\sin (e+f x)}{2 f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 \sqrt {a} f (a+b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 208
Rule 4147
Rubi steps
\begin {align*} \int \frac {\sec ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a+b-a x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\sin (e+f x)}{2 (a+b) f \left (a+b-a \sin ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{2 (a+b) f}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 \sqrt {a} (a+b)^{3/2} f}+\frac {\sin (e+f x)}{2 (a+b) f \left (a+b-a \sin ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 88, normalized size = 1.19 \[ \frac {\sqrt {a} \sqrt {a+b} \sin (e+f x)+\left (-a \sin ^2(e+f x)+a+b\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{\sqrt {a} f (a+b)^{3/2} (a \cos (2 (e+f x))+a+2 b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.34, size = 262, normalized size = 3.54 \[ \left [\frac {{\left (a \cos \left (f x + e\right )^{2} + b\right )} \sqrt {a^{2} + a b} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a^{2} + a b} \sin \left (f x + e\right ) - 2 \, a - b}{a \cos \left (f x + e\right )^{2} + b}\right ) + 2 \, {\left (a^{2} + a b\right )} \sin \left (f x + e\right )}{4 \, {\left ({\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} f\right )}}, -\frac {{\left (a \cos \left (f x + e\right )^{2} + b\right )} \sqrt {-a^{2} - a b} \arctan \left (\frac {\sqrt {-a^{2} - a b} \sin \left (f x + e\right )}{a + b}\right ) - {\left (a^{2} + a b\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 79, normalized size = 1.07 \[ -\frac {\frac {\arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{\sqrt {-a^{2} - a b} {\left (a + b\right )}} + \frac {\sin \left (f x + e\right )}{{\left (a \sin \left (f x + e\right )^{2} - a - b\right )} {\left (a + b\right )}}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 68, normalized size = 0.92 \[ \frac {-\frac {\sin \left (f x +e \right )}{2 \left (a +b \right ) \left (-a -b +a \left (\sin ^{2}\left (f x +e \right )\right )\right )}+\frac {\arctanh \left (\frac {a \sin \left (f x +e \right )}{\sqrt {\left (a +b \right ) a}}\right )}{2 \left (a +b \right ) \sqrt {\left (a +b \right ) a}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 98, normalized size = 1.32 \[ -\frac {\frac {2 \, \sin \left (f x + e\right )}{{\left (a^{2} + a b\right )} \sin \left (f x + e\right )^{2} - a^{2} - 2 \, a b - b^{2}} + \frac {\log \left (\frac {a \sin \left (f x + e\right ) - \sqrt {{\left (a + b\right )} a}}{a \sin \left (f x + e\right ) + \sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} {\left (a + b\right )}}}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 62, normalized size = 0.84 \[ \frac {\sin \left (e+f\,x\right )}{2\,f\,\left (a+b\right )\,\left (-a\,{\sin \left (e+f\,x\right )}^2+a+b\right )}+\frac {\mathrm {atanh}\left (\frac {\sqrt {a}\,\sin \left (e+f\,x\right )}{\sqrt {a+b}}\right )}{2\,\sqrt {a}\,f\,{\left (a+b\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{3}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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