Optimal. Leaf size=45 \[ \frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (e+f x)}{\sqrt {b}}\right )}{a f \sqrt {a+b}}+\frac {x}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4127, 3181, 205} \[ \frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (e+f x)}{\sqrt {b}}\right )}{a f \sqrt {a+b}}+\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 205
Rule 3181
Rule 4127
Rubi steps
\begin {align*} \int \frac {1}{a+b \sec ^2(e+f x)} \, dx &=\frac {x}{a}-\frac {b \int \frac {1}{b+a \cos ^2(e+f x)} \, dx}{a}\\ &=\frac {x}{a}+\frac {b \operatorname {Subst}\left (\int \frac {1}{b+(a+b) x^2} \, dx,x,\cot (e+f x)\right )}{a f}\\ &=\frac {x}{a}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (e+f x)}{\sqrt {b}}\right )}{a \sqrt {a+b} f}\\ \end {align*}
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Mathematica [C] time = 0.23, size = 182, normalized size = 4.04 \[ \frac {\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (f x \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}+b (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left (\frac {(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}\right )\right )}{2 a f \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4} \left (a+b \sec ^2(e+f x)\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 231, normalized size = 5.13 \[ \left [\frac {4 \, f x + \sqrt {-\frac {b}{a + b}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {b}{a + b}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right )}{4 \, a f}, \frac {2 \, f x + \sqrt {\frac {b}{a + b}} \arctan \left (\frac {{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {b}{a + b}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right )}{2 \, a f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 68, normalized size = 1.51 \[ -\frac {\frac {{\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )} b}{\sqrt {a b + b^{2}} a} - \frac {f x + e}{a}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 48, normalized size = 1.07 \[ -\frac {b \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{f a \sqrt {\left (a +b \right ) b}}+\frac {\arctan \left (\tan \left (f x +e \right )\right )}{f a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 44, normalized size = 0.98 \[ -\frac {\frac {b \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} a} - \frac {f x + e}{a}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.86, size = 460, normalized size = 10.22 \[ \frac {x}{a}-\frac {\mathrm {atan}\left (\frac {\frac {\left (2\,b^3\,\mathrm {tan}\left (e+f\,x\right )-\frac {\left (2\,a^2\,b^2-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (8\,a^3\,b^2+16\,a^2\,b^3\right )\,\sqrt {-b\,\left (a+b\right )}}{4\,\left (a^2+b\,a\right )}\right )\,\sqrt {-b\,\left (a+b\right )}}{2\,\left (a^2+b\,a\right )}\right )\,\sqrt {-b\,\left (a+b\right )}\,1{}\mathrm {i}}{a^2+b\,a}+\frac {\left (2\,b^3\,\mathrm {tan}\left (e+f\,x\right )+\frac {\left (2\,a^2\,b^2+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (8\,a^3\,b^2+16\,a^2\,b^3\right )\,\sqrt {-b\,\left (a+b\right )}}{4\,\left (a^2+b\,a\right )}\right )\,\sqrt {-b\,\left (a+b\right )}}{2\,\left (a^2+b\,a\right )}\right )\,\sqrt {-b\,\left (a+b\right )}\,1{}\mathrm {i}}{a^2+b\,a}}{\frac {\left (2\,b^3\,\mathrm {tan}\left (e+f\,x\right )-\frac {\left (2\,a^2\,b^2-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (8\,a^3\,b^2+16\,a^2\,b^3\right )\,\sqrt {-b\,\left (a+b\right )}}{4\,\left (a^2+b\,a\right )}\right )\,\sqrt {-b\,\left (a+b\right )}}{2\,\left (a^2+b\,a\right )}\right )\,\sqrt {-b\,\left (a+b\right )}}{a^2+b\,a}-\frac {\left (2\,b^3\,\mathrm {tan}\left (e+f\,x\right )+\frac {\left (2\,a^2\,b^2+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (8\,a^3\,b^2+16\,a^2\,b^3\right )\,\sqrt {-b\,\left (a+b\right )}}{4\,\left (a^2+b\,a\right )}\right )\,\sqrt {-b\,\left (a+b\right )}}{2\,\left (a^2+b\,a\right )}\right )\,\sqrt {-b\,\left (a+b\right )}}{a^2+b\,a}}\right )\,\sqrt {-b\,\left (a+b\right )}\,1{}\mathrm {i}}{f\,\left (a^2+b\,a\right )} \]
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 \[ \int \frac {1}{a + b \sec ^{2}{\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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