Optimal. Leaf size=98 \[ \frac {a^2}{b^2 f (a-b) \sqrt {a+b \tan ^2(e+f x)}}+\frac {\sqrt {a+b \tan ^2(e+f x)}}{b^2 f}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f (a-b)^{3/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3670, 446, 87, 63, 208} \[ \frac {a^2}{b^2 f (a-b) \sqrt {a+b \tan ^2(e+f x)}}+\frac {\sqrt {a+b \tan ^2(e+f x)}}{b^2 f}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f (a-b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 87
Rule 208
Rule 446
Rule 3670
Rubi steps
\begin {align*} \int \frac {\tan ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^5}{\left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{(1+x) (a+b x)^{3/2}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a^2}{(a-b) b (a+b x)^{3/2}}+\frac {1}{b \sqrt {a+b x}}+\frac {1}{(a-b) (1+x) \sqrt {a+b x}}\right ) \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {a^2}{(a-b) b^2 f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\sqrt {a+b \tan ^2(e+f x)}}{b^2 f}+\frac {\operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 (a-b) f}\\ &=\frac {a^2}{(a-b) b^2 f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\sqrt {a+b \tan ^2(e+f x)}}{b^2 f}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan ^2(e+f x)}\right )}{(a-b) b f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} f}+\frac {a^2}{(a-b) b^2 f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\sqrt {a+b \tan ^2(e+f x)}}{b^2 f}\\ \end {align*}
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Mathematica [C] time = 0.38, size = 84, normalized size = 0.86 \[ \frac {b^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b \tan ^2(e+f x)+a}{a-b}\right )+(a-b) \left (2 a+b \tan ^2(e+f x)+b\right )}{b^2 f (a-b) \sqrt {a+b \tan ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 432, normalized size = 4.41 \[ \left [-\frac {{\left (b^{3} \tan \left (f x + e\right )^{2} + a b^{2}\right )} \sqrt {a - b} \log \left (-\frac {b^{2} \tan \left (f x + e\right )^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + 4 \, {\left (b \tan \left (f x + e\right )^{2} + 2 \, a - b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) - 4 \, {\left (2 \, a^{3} - 3 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{4 \, {\left ({\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )} f\right )}}, \frac {{\left (b^{3} \tan \left (f x + e\right )^{2} + a b^{2}\right )} \sqrt {-a + b} \arctan \left (\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{b \tan \left (f x + e\right )^{2} + 2 \, a - b}\right ) + 2 \, {\left (2 \, a^{3} - 3 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{2 \, {\left ({\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.15, size = 99, normalized size = 1.01 \[ \frac {a^{2}}{{\left (a b^{2} f - b^{3} f\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}} + \frac {\arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{\sqrt {-a + b}}\right )}{{\left (a f - b f\right )} \sqrt {-a + b}} + \frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{b^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 141, normalized size = 1.44 \[ \frac {\tan ^{2}\left (f x +e \right )}{f b \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}+\frac {2 a}{f \,b^{2} \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}+\frac {1}{f b \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}+\frac {1}{\left (a -b \right ) f \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}+\frac {\arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{f \left (a -b \right ) \sqrt {-a +b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [B] time = 13.73, size = 112, normalized size = 1.14 \[ \frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{b^2\,f}+\frac {a^2}{b^2\,f\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (a-b\right )}+\frac {\mathrm {atan}\left (\frac {a\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,1{}\mathrm {i}-b\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,1{}\mathrm {i}}{{\left (a-b\right )}^{3/2}}\right )\,1{}\mathrm {i}}{f\,{\left (a-b\right )}^{3/2}} \]
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 \[ \int \frac {\tan ^{5}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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