Optimal. Leaf size=167 \[ -\frac{\left (15 a^2+25 a b+8 b^2\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^2}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right )}{f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 f}-\frac{(b-5 (a+b)) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)} \]
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Rubi [A] time = 0.333206, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {4141, 1975, 475, 583, 12, 377, 203} \[ -\frac{\left (15 a^2+25 a b+8 b^2\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)^2}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)+b}}\right )}{f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{5 f}-\frac{(b-5 (a+b)) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{15 f (a+b)} \]
Antiderivative was successfully verified.
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Rule 4141
Rule 1975
Rule 475
Rule 583
Rule 12
Rule 377
Rule 203
Rubi steps
\begin{align*} \int \cot ^6(e+f x) \sqrt{a+b \sec ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b \left (1+x^2\right )}}{x^6 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b+b x^2}}{x^6 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cot ^5(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{5 f}+\frac{\operatorname{Subst}\left (\int \frac{b-5 (a+b)-4 b x^2}{x^4 \left (1+x^2\right ) \sqrt{a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=-\frac{(b-5 (a+b)) \cot ^3(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{15 (a+b) f}-\frac{\cot ^5(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{5 f}-\frac{\operatorname{Subst}\left (\int \frac{-15 a^2-25 a b-8 b^2+2 b (b-5 (a+b)) x^2}{x^2 \left (1+x^2\right ) \sqrt{a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{15 (a+b) f}\\ &=-\frac{\left (15 a^2+25 a b+8 b^2\right ) \cot (e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{15 (a+b)^2 f}-\frac{(b-5 (a+b)) \cot ^3(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{15 (a+b) f}-\frac{\cot ^5(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{5 f}+\frac{\operatorname{Subst}\left (\int -\frac{15 a (a+b)^2}{\left (1+x^2\right ) \sqrt{a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{15 (a+b)^2 f}\\ &=-\frac{\left (15 a^2+25 a b+8 b^2\right ) \cot (e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{15 (a+b)^2 f}-\frac{(b-5 (a+b)) \cot ^3(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{15 (a+b) f}-\frac{\cot ^5(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{5 f}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\left (15 a^2+25 a b+8 b^2\right ) \cot (e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{15 (a+b)^2 f}-\frac{(b-5 (a+b)) \cot ^3(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{15 (a+b) f}-\frac{\cot ^5(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{5 f}-\frac{a \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,\frac{\tan (e+f x)}{\sqrt{a+b+b \tan ^2(e+f x)}}\right )}{f}\\ &=-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+b+b \tan ^2(e+f x)}}\right )}{f}-\frac{\left (15 a^2+25 a b+8 b^2\right ) \cot (e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{15 (a+b)^2 f}-\frac{(b-5 (a+b)) \cot ^3(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{15 (a+b) f}-\frac{\cot ^5(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{5 f}\\ \end{align*}
Mathematica [A] time = 1.76971, size = 178, normalized size = 1.07 \[ -\frac{\cot (e+f x) \left (-\left (11 a^2+21 a b+10 b^2\right ) \csc ^2(e+f x)+23 a^2+3 (a+b)^2 \csc ^4(e+f x)+40 a b+15 b^2\right ) \sqrt{a+b \sec ^2(e+f x)}}{15 f (a+b)^2}-\frac{\sqrt{2} \sqrt{a} \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (e+f x)}{\sqrt{-a \sin ^2(e+f x)+a+b}}\right )}{f \sqrt{a \cos (2 e+2 f x)+a+2 b}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.556, size = 8605, normalized size = 51.5 \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 \sec \left (f x + e\right )^{2} + a} \cot \left (f x + e\right )^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 7.95205, size = 2071, normalized size = 12.4 \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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sec \left (f x + e\right )^{2} + a} \cot \left (f x + e\right )^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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