Optimal. Leaf size=81 \[ \frac {\tan (e+f x)}{f (a-b) \sqrt {a+b \tan ^2(e+f x)}}-\frac {\tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f (a-b)^{3/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3670, 471, 377, 203} \[ \frac {\tan (e+f x)}{f (a-b) \sqrt {a+b \tan ^2(e+f x)}}-\frac {\tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f (a-b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 377
Rule 471
Rule 3670
Rubi steps
\begin {align*} \int \frac {\tan ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\tan (e+f x)}{(a-b) f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{(a-b) f}\\ &=\frac {\tan (e+f x)}{(a-b) f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{(a-b) f}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{(a-b)^{3/2} f}+\frac {\tan (e+f x)}{(a-b) f \sqrt {a+b \tan ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 3.23, size = 154, normalized size = 1.90 \[ \frac {\tan (e+f x) \left ((a-b) \sqrt {\frac {b \tan ^2(e+f x)}{a}+1}+\sqrt {\frac {(b-a) \tan ^2(e+f x)}{a}} \left (a \cot ^2(e+f x)+b\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {(b-a) \tan ^2(e+f x)}{a}}}{\sqrt {\frac {b \tan ^2(e+f x)}{a}+1}}\right )\right )}{f (a-b)^2 \sqrt {a+b \tan ^2(e+f x)} \sqrt {\frac {b \tan ^2(e+f x)}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 285, normalized size = 3.52 \[ \left [\frac {{\left (b \tan \left (f x + e\right )^{2} + a\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b} \tan \left (f x + e\right ) - a}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} {\left (a - b\right )} \tan \left (f x + e\right )}{2 \, {\left ({\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} f\right )}}, -\frac {{\left (b \tan \left (f x + e\right )^{2} + a\right )} \sqrt {a - b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{\sqrt {a - b} \tan \left (f x + e\right )}\right ) - \sqrt {b \tan \left (f x + e\right )^{2} + a} {\left (a - b\right )} \tan \left (f x + e\right )}{{\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan \left (f x + e\right )^{2}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 131, normalized size = 1.62 \[ \frac {\tan \left (f x +e \right )}{f a \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}+\frac {b \tan \left (f x +e \right )}{a \left (a -b \right ) f \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {\left (a -b \right ) b^{2} \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}\right )}{f \left (a -b \right )^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tan}\left (e+f\,x\right )}^2}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\tan ^{2}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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