Optimal. Leaf size=103 \[ -\frac {a}{3 b f (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {1}{f (a-b)^2 \sqrt {a+b \tan ^2(e+f x)}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f (a-b)^{5/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3670, 446, 78, 51, 63, 208} \[ -\frac {a}{3 b f (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {1}{f (a-b)^2 \sqrt {a+b \tan ^2(e+f x)}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f (a-b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rule 446
Rule 3670
Rubi steps
\begin {align*} \int \frac {\tan ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{(1+x) (a+b x)^{5/2}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac {a}{3 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{(1+x) (a+b x)^{3/2}} \, dx,x,\tan ^2(e+f x)\right )}{2 (a-b) f}\\ &=-\frac {a}{3 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {1}{(a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 (a-b)^2 f}\\ &=-\frac {a}{3 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {1}{(a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}-\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)^2 b f}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2} f}-\frac {a}{3 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {1}{(a-b)^2 f \sqrt {a+b \tan ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.30, size = 84, normalized size = 0.82 \[ \frac {a (b-a)-3 b \left (a+b \tan ^2(e+f x)\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b \tan ^2(e+f x)+a}{a-b}\right )}{3 b f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 572, normalized size = 5.55 \[ \left [\frac {3 \, {\left (b^{3} \tan \left (f x + e\right )^{4} + 2 \, a b^{2} \tan \left (f x + e\right )^{2} + a^{2} b\right )} \sqrt {a - b} \log \left (-\frac {b^{2} \tan \left (f x + e\right )^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + 4 \, {\left (b \tan \left (f x + e\right )^{2} + 2 \, a - b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) - 4 \, {\left (a^{3} + a^{2} b - 2 \, a b^{2} + 3 \, {\left (a b^{2} - b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{12 \, {\left ({\left (a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right )} f \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right )} f\right )}}, -\frac {3 \, {\left (b^{3} \tan \left (f x + e\right )^{4} + 2 \, a b^{2} \tan \left (f x + e\right )^{2} + a^{2} b\right )} \sqrt {-a + b} \arctan \left (\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{b \tan \left (f x + e\right )^{2} + 2 \, a - b}\right ) + 2 \, {\left (a^{3} + a^{2} b - 2 \, a b^{2} + 3 \, {\left (a b^{2} - b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{6 \, {\left ({\left (a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right )} f \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right )} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.35, size = 116, normalized size = 1.13 \[ -\frac {\frac {3 \, b \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{\sqrt {-a + b}}\right )}{{\left (a^{2} f - 2 \, a b f + b^{2} f\right )} \sqrt {-a + b}} + \frac {a^{2} + 3 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )} b - a b}{{\left (a^{2} f - 2 \, a b f + b^{2} f\right )} {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 118, normalized size = 1.15 \[ -\frac {1}{3 f b \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}-\frac {1}{3 \left (a -b \right ) f \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}-\frac {1}{\left (a -b \right )^{2} f \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}-\frac {\arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{f \left (a -b \right )^{2} \sqrt {-a +b}} \]
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 [B] time = 15.72, size = 138, normalized size = 1.34 \[ -\frac {\frac {a}{3\,\left (a-b\right )}+\frac {b\,\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}{{\left (a-b\right )}^2}}{b\,f\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}}-\frac {\mathrm {atan}\left (\frac {a^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,1{}\mathrm {i}+b^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,1{}\mathrm {i}-a\,b\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,2{}\mathrm {i}}{{\left (a-b\right )}^{5/2}}\right )\,1{}\mathrm {i}}{f\,{\left (a-b\right )}^{5/2}} \]
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 \[ \int \frac {\tan ^{3}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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