Optimal. Leaf size=128 \[ -\frac{(a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a^2 f (a-b)}-\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f (a-b)^{3/2}}-\frac{b \cot (e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}} \]
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Rubi [A] time = 0.184107, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3670, 472, 583, 12, 377, 203} \[ -\frac{(a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a^2 f (a-b)}-\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f (a-b)^{3/2}}-\frac{b \cot (e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 472
Rule 583
Rule 12
Rule 377
Rule 203
Rubi steps
\begin{align*} \int \frac{\cot ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{b \cot (e+f x)}{a (a-b) f \sqrt{a+b \tan ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{a-2 b-2 b x^2}{x^2 \left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{a (a-b) f}\\ &=-\frac{b \cot (e+f x)}{a (a-b) f \sqrt{a+b \tan ^2(e+f x)}}-\frac{(a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a^2 (a-b) f}-\frac{\operatorname{Subst}\left (\int \frac{a^2}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{a^2 (a-b) f}\\ &=-\frac{b \cot (e+f x)}{a (a-b) f \sqrt{a+b \tan ^2(e+f x)}}-\frac{(a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a^2 (a-b) f}-\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{b \cot (e+f x)}{a (a-b) f \sqrt{a+b \tan ^2(e+f x)}}-\frac{(a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a^2 (a-b) f}-\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{b \cot (e+f x)}{a (a-b) f \sqrt{a+b \tan ^2(e+f x)}}-\frac{(a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a^2 (a-b) f}\\ \end{align*}
Mathematica [C] time = 13.503, size = 882, normalized size = 6.89 \[ -\frac{\cos ^2(e+f x) \cot (e+f x) \left (\frac{8 (a-b) b^2 \text{Hypergeometric2F1}\left (2,2,\frac{7}{2},\frac{(a-b) \sin ^2(e+f x)}{a}\right ) \sin ^2(e+f x) \tan ^4(e+f x)}{5 a^3}+\frac{8 (a-b) b^2 \text{HypergeometricPFQ}\left (\{2,2,2\},\left \{1,\frac{7}{2}\right \},\frac{(a-b) \sin ^2(e+f x)}{a}\right ) \sin ^2(e+f x) \tan ^4(e+f x)}{15 a^3}-\frac{8 b^2 \sin ^{-1}\left (\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right ) \tan ^4(e+f x)}{a^2 \left (\frac{(a-b) \sin ^2(e+f x)}{a}\right )^{3/2} \sqrt{\frac{\cos ^2(e+f x) \left (b \tan ^2(e+f x)+a\right )}{a}}}+\frac{8 b^2 \sin ^{-1}\left (\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right ) \tan ^4(e+f x)}{a^2 \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left (b \tan ^2(e+f x)+a\right )}{a^2}}}+\frac{8 b^2 \sec ^2(e+f x) \tan ^2(e+f x)}{a (a-b)}+\frac{8 (a-b) b \text{Hypergeometric2F1}\left (2,2,\frac{7}{2},\frac{(a-b) \sin ^2(e+f x)}{a}\right ) \sin ^2(e+f x) \tan ^2(e+f x)}{3 a^2}+\frac{16 (a-b) b \text{HypergeometricPFQ}\left (\{2,2,2\},\left \{1,\frac{7}{2}\right \},\frac{(a-b) \sin ^2(e+f x)}{a}\right ) \sin ^2(e+f x) \tan ^2(e+f x)}{15 a^2}-\frac{12 b \sin ^{-1}\left (\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right ) \tan ^2(e+f x)}{a \left (\frac{(a-b) \sin ^2(e+f x)}{a}\right )^{3/2} \sqrt{\frac{\cos ^2(e+f x) \left (b \tan ^2(e+f x)+a\right )}{a}}}+\frac{12 b \sin ^{-1}\left (\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right ) \tan ^2(e+f x)}{a \sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left (b \tan ^2(e+f x)+a\right )}{a^2}}}+\frac{3 a \csc ^2(e+f x)}{a-b}+\frac{12 b \sec ^2(e+f x)}{a-b}+\frac{16 (a-b) \text{Hypergeometric2F1}\left (2,2,\frac{7}{2},\frac{(a-b) \sin ^2(e+f x)}{a}\right ) \sin ^2(e+f x)}{15 a}+\frac{8 (a-b) \text{HypergeometricPFQ}\left (\{2,2,2\},\left \{1,\frac{7}{2}\right \},\frac{(a-b) \sin ^2(e+f x)}{a}\right ) \sin ^2(e+f x)}{15 a}-\frac{3 \sin ^{-1}\left (\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right )}{\left (\frac{(a-b) \sin ^2(e+f x)}{a}\right )^{3/2} \sqrt{\frac{\cos ^2(e+f x) \left (b \tan ^2(e+f x)+a\right )}{a}}}+\frac{3 \sin ^{-1}\left (\sqrt{\frac{(a-b) \sin ^2(e+f x)}{a}}\right )}{\sqrt{\frac{(a-b) \cos ^2(e+f x) \sin ^2(e+f x) \left (b \tan ^2(e+f x)+a\right )}{a^2}}}\right )}{a f \sqrt{b \tan ^2(e+f x)+a}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.304, size = 1305, normalized size = 10.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.99376, size = 1102, normalized size = 8.61 \begin{align*} \left [\frac{{\left (a^{2} b \tan \left (f x + e\right )^{3} + a^{3} \tan \left (f x + e\right )\right )} \sqrt{-a + b} \log \left (-\frac{{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - 2 \,{\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (f x + e\right )^{2} + a^{2} - 4 \,{\left ({\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{3} - a \tan \left (f x + e\right )\right )} \sqrt{b \tan \left (f x + e\right )^{2} + a} \sqrt{-a + b}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) - 4 \,{\left (a^{3} - 2 \, a^{2} b + a b^{2} +{\left (a^{2} b - 3 \, a b^{2} + 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^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{3} +{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f \tan \left (f x + e\right )\right )}}, -\frac{{\left (a^{2} b \tan \left (f x + e\right )^{3} + a^{3} \tan \left (f x + e\right )\right )} \sqrt{a - b} \arctan \left (-\frac{2 \, \sqrt{b \tan \left (f x + e\right )^{2} + a} \sqrt{a - b} \tan \left (f x + e\right )}{{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - a}\right ) + 2 \,{\left (a^{3} - 2 \, a^{2} b + a b^{2} +{\left (a^{2} b - 3 \, a b^{2} + 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^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{3} +{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f \tan \left (f x + e\right )\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{2}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (f x + e\right )^{2}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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