Optimal. Leaf size=73 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f (a-b)^{3/2}}-\frac {a}{b f (a-b) \sqrt {a+b \tan ^2(e+f x)}} \]
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Rubi [A] time = 0.13, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3670, 446, 78, 63, 208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f (a-b)^{3/2}}-\frac {a}{b f (a-b) \sqrt {a+b \tan ^2(e+f x)}} \]
Antiderivative was successfully verified.
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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 )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{\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}{(1+x) (a+b x)^{3/2}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac {a}{(a-b) b 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) f}\\ &=-\frac {a}{(a-b) b 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) 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}{(a-b) b f \sqrt {a+b \tan ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 75, normalized size = 1.03 \[ \frac {\frac {a (b-a)}{b \sqrt {a+b \tan ^2(e+f x)}}+\sqrt {a-b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f (a-b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 358, normalized size = 4.90 \[ \left [-\frac {{\left (b^{2} \tan \left (f x + e\right )^{2} + a b\right )} \sqrt {a - b} \log \left (-\frac {b^{2} \tan \left (f x + e\right )^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2}\right )} \tan \left (f x + e\right )^{2} - 4 \, {\left (b \tan \left (f x + e\right )^{2} + 2 \, a - b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) + 4 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} {\left (a^{2} - a b\right )}}{4 \, {\left ({\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} f\right )}}, -\frac {{\left (b^{2} \tan \left (f x + e\right )^{2} + a b\right )} \sqrt {-a + b} \arctan \left (\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{b \tan \left (f x + e\right )^{2} + 2 \, a - b}\right ) + 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} {\left (a^{2} - a b\right )}}{2 \, {\left ({\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.12, size = 76, normalized size = 1.04 \[ -\frac {\frac {b \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{\sqrt {-a + b}}\right )}{{\left (a f - b f\right )} \sqrt {-a + b}} + \frac {a}{\sqrt {b \tan \left (f x + e\right )^{2} + a} {\left (a f - b f\right )}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 92, normalized size = 1.26 \[ -\frac {1}{f b \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}-\frac {1}{\left (a -b \right ) f \sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}-\frac {\arctan \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {-a +b}}\right )}{f \left (a -b \right ) \sqrt {-a +b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-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 = 13.10, size = 90, normalized size = 1.23 \[ -\frac {a}{b\,f\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (a-b\right )}-\frac {\mathrm {atan}\left (\frac {a\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,1{}\mathrm {i}-b\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,1{}\mathrm {i}}{{\left (a-b\right )}^{3/2}}\right )\,1{}\mathrm {i}}{f\,{\left (a-b\right )}^{3/2}} \]
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 \[ \int \frac {\tan ^{3}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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