Optimal. Leaf size=163 \[ -\frac{\left (8 a^2-4 a b-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right )}{8 a^{3/2} f}+\frac{\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{f}-\frac{\cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(4 a-b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 a f} \]
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Rubi [A] time = 0.211899, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3670, 446, 99, 151, 156, 63, 208} \[ -\frac{\left (8 a^2-4 a b-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right )}{8 a^{3/2} f}+\frac{\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{f}-\frac{\cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(4 a-b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 a f} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 446
Rule 99
Rule 151
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{x^5 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^3 (1+x)} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac{\cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} (-4 a+b)-\frac{3 b x}{2}}{x^2 (1+x) \sqrt{a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{4 f}\\ &=\frac{(4 a-b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 a f}-\frac{\cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (-8 a^2+4 a b+b^2\right )-\frac{1}{4} (4 a-b) b x}{x (1+x) \sqrt{a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{4 a f}\\ &=\frac{(4 a-b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 a f}-\frac{\cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}-\frac{(a-b) \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}+\frac{\left (8 a^2-4 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{16 a f}\\ &=\frac{(4 a-b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 a f}-\frac{\cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}-\frac{(a-b) \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 )}{b f}+\frac{\left (8 a^2-4 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \tan ^2(e+f x)}\right )}{8 a b f}\\ &=-\frac{\left (8 a^2-4 a b-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right )}{8 a^{3/2} f}+\frac{\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{f}+\frac{(4 a-b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 a f}-\frac{\cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}\\ \end{align*}
Mathematica [A] time = 1.31372, size = 138, normalized size = 0.85 \[ \frac{\left (-8 a^2+4 a b+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right )+\sqrt{a} \left (8 a \sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )-\cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)} \left (2 a \cot ^2(e+f x)-4 a+b\right )\right )}{8 a^{3/2} f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.31, size = 5676, normalized size = 34.8 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \tan \left (f x + e\right )^{2} + a} \cot \left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90633, size = 1756, normalized size = 10.77 \begin{align*} \text{result too large to display} \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 \sqrt{a + b \tan ^{2}{\left (e + f x \right )}} \cot ^{5}{\left (e + f x \right )}\, 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 \sqrt{b \tan \left (f x + e\right )^{2} + a} \cot \left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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