Optimal. Leaf size=57 \[ \frac {1}{a f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f} \]
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Rubi [A] time = 0.07, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4139, 266, 51, 63, 208} \[ \frac {1}{a f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rule 4139
Rubi steps
\begin {align*} \int \frac {\tan (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac {1}{a f \sqrt {a+b \sec ^2(e+f x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sec ^2(e+f x)\right )}{2 a f}\\ &=\frac {1}{a f \sqrt {a+b \sec ^2(e+f x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sec ^2(e+f x)}\right )}{a b f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}+\frac {1}{a f \sqrt {a+b \sec ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] time = 6.36, size = 382, normalized size = 6.70 \[ \frac {\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)^{3/2} \left (-\frac {2}{b \sqrt {a \cos (2 (e+f x))+a+2 b}}+\frac {\sqrt {2} e^{i (e+f x)} \sec (e+f x) \sqrt {4 b+a e^{-2 i (e+f x)} \left (1+e^{2 i (e+f x)}\right )^2} \left (\frac {\sqrt {a} (a+4 b) \left (1+e^{2 i (e+f x)}\right )}{b \left (a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}\right )}+\frac {-2 \log \left (\sqrt {a} \sqrt {a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right )-2 \log \left (\sqrt {a} \sqrt {a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right )+4 i f x}{\sqrt {a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}}}\right )}{a^{3/2}}\right )}{16 f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 392, normalized size = 6.88 \[ \left [\frac {8 \, a \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )^{2} + {\left (a \cos \left (f x + e\right )^{2} + b\right )} \sqrt {a} \log \left (128 \, a^{4} \cos \left (f x + e\right )^{8} + 256 \, a^{3} b \cos \left (f x + e\right )^{6} + 160 \, a^{2} b^{2} \cos \left (f x + e\right )^{4} + 32 \, a b^{3} \cos \left (f x + e\right )^{2} + b^{4} - 8 \, {\left (16 \, a^{3} \cos \left (f x + e\right )^{8} + 24 \, a^{2} b \cos \left (f x + e\right )^{6} + 10 \, a b^{2} \cos \left (f x + e\right )^{4} + b^{3} \cos \left (f x + e\right )^{2}\right )} \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}\right )}{8 \, {\left (a^{3} f \cos \left (f x + e\right )^{2} + a^{2} b f\right )}}, \frac {4 \, a \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )^{2} + {\left (a \cos \left (f x + e\right )^{2} + b\right )} \sqrt {-a} \arctan \left (\frac {{\left (8 \, a^{2} \cos \left (f x + e\right )^{4} + 8 \, a b \cos \left (f x + e\right )^{2} + b^{2}\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{4 \, {\left (2 \, a^{3} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b \cos \left (f x + e\right )^{2} + a b^{2}\right )}}\right )}{4 \, {\left (a^{3} f \cos \left (f x + e\right )^{2} + a^{2} b f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 64, normalized size = 1.12 \[ \frac {1}{a f \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{\sec \left (f x +e \right )}\right )}{f \,a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan \left (f x + e\right )}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.56, size = 49, normalized size = 0.86 \[ \frac {1}{a\,f\,\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}}}{\sqrt {a}}\right )}{a^{3/2}\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.81, size = 53, normalized size = 0.93 \[ \frac {1}{a f \sqrt {a + b \sec ^{2}{\left (e + f x \right )}}} + \frac {\operatorname {atan}{\left (\frac {\sqrt {a + b \sec ^{2}{\left (e + f x \right )}}}{\sqrt {- a}} \right )}}{a f \sqrt {- a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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