Optimal. Leaf size=327 \[ -\frac{\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^3 f (a-b)^2}+\frac{\left (4 a^2 b+5 a^3-88 a b^2+64 b^3\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^4 f (a-b)^2}-\frac{\left (8 a^2 b^2+10 a^3 b+15 a^4-176 a b^3+128 b^4\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^5 f (a-b)^2}-\frac{b (11 a-8 b) \cot ^5(e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f (a-b)^{5/2}}-\frac{b \cot ^5(e+f x)}{3 a f (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.49624, antiderivative size = 327, 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, 472, 579, 583, 12, 377, 203} \[ -\frac{\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^3 f (a-b)^2}+\frac{\left (4 a^2 b+5 a^3-88 a b^2+64 b^3\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^4 f (a-b)^2}-\frac{\left (8 a^2 b^2+10 a^3 b+15 a^4-176 a b^3+128 b^4\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^5 f (a-b)^2}-\frac{b (11 a-8 b) \cot ^5(e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{f (a-b)^{5/2}}-\frac{b \cot ^5(e+f x)}{3 a f (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 472
Rule 579
Rule 583
Rule 12
Rule 377
Rule 203
Rubi steps
\begin{align*} \int \frac{\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^6 \left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{b \cot ^5(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{3 a-8 b-8 b x^2}{x^6 \left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 a (a-b) f}\\ &=-\frac{b \cot ^5(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac{(11 a-8 b) b \cot ^5(e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \tan ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{3 \left (a^2-22 a b+16 b^2\right )-6 (11 a-8 b) b x^2}{x^6 \left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{3 a^2 (a-b)^2 f}\\ &=-\frac{b \cot ^5(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac{(11 a-8 b) b \cot ^5(e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^3 (a-b)^2 f}-\frac{\operatorname{Subst}\left (\int \frac{3 \left (5 a^3+4 a^2 b-88 a b^2+64 b^3\right )+12 b \left (a^2-22 a b+16 b^2\right ) x^2}{x^4 \left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{15 a^3 (a-b)^2 f}\\ &=-\frac{b \cot ^5(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac{(11 a-8 b) b \cot ^5(e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \tan ^2(e+f x)}}+\frac{\left (5 a^3+4 a^2 b-88 a b^2+64 b^3\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^4 (a-b)^2 f}-\frac{\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^3 (a-b)^2 f}+\frac{\operatorname{Subst}\left (\int \frac{3 \left (15 a^4+10 a^3 b+8 a^2 b^2-176 a b^3+128 b^4\right )+6 b \left (5 a^3+4 a^2 b-88 a b^2+64 b^3\right ) x^2}{x^2 \left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{45 a^4 (a-b)^2 f}\\ &=-\frac{b \cot ^5(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac{(11 a-8 b) b \cot ^5(e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left (15 a^4+10 a^3 b+8 a^2 b^2-176 a b^3+128 b^4\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^5 (a-b)^2 f}+\frac{\left (5 a^3+4 a^2 b-88 a b^2+64 b^3\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^4 (a-b)^2 f}-\frac{\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^3 (a-b)^2 f}-\frac{\operatorname{Subst}\left (\int \frac{45 a^5}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{45 a^5 (a-b)^2 f}\\ &=-\frac{b \cot ^5(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac{(11 a-8 b) b \cot ^5(e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left (15 a^4+10 a^3 b+8 a^2 b^2-176 a b^3+128 b^4\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^5 (a-b)^2 f}+\frac{\left (5 a^3+4 a^2 b-88 a b^2+64 b^3\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^4 (a-b)^2 f}-\frac{\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^3 (a-b)^2 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)^2 f}\\ &=-\frac{b \cot ^5(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac{(11 a-8 b) b \cot ^5(e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left (15 a^4+10 a^3 b+8 a^2 b^2-176 a b^3+128 b^4\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^5 (a-b)^2 f}+\frac{\left (5 a^3+4 a^2 b-88 a b^2+64 b^3\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^4 (a-b)^2 f}-\frac{\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^3 (a-b)^2 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)^2 f}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{(a-b)^{5/2} f}-\frac{b \cot ^5(e+f x)}{3 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac{(11 a-8 b) b \cot ^5(e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left (15 a^4+10 a^3 b+8 a^2 b^2-176 a b^3+128 b^4\right ) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^5 (a-b)^2 f}+\frac{\left (5 a^3+4 a^2 b-88 a b^2+64 b^3\right ) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^4 (a-b)^2 f}-\frac{\left (a^2-22 a b+16 b^2\right ) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^3 (a-b)^2 f}\\ \end{align*}
Mathematica [C] time = 16.3189, size = 441, normalized size = 1.35 \[ \frac{\sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left (-\frac{15 a^5 b \sin ^2(e+f x) \sin (2 (e+f x)) \left (\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}\right )^{3/2} \left (2 (a-b) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}}}{\sqrt{2}}\right ),1\right )-2 a \Pi \left (-\frac{b}{a-b};\left .\sin ^{-1}\left (\frac{\sqrt{\frac{(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt{2}}\right )\right |1\right )\right )}{2 \sqrt{2}}-(a-b) \left ((a-b)^2 \left (23 a^2+54 a b+73 b^2\right ) \cot (e+f x) ((a-b) \cos (2 (e+f x))+a+b)^2+3 a^2 (a-b)^2 \cot (e+f x) \csc ^4(e+f x) ((a-b) \cos (2 (e+f x))+a+b)^2+10 a b^5 \sin (2 (e+f x))-5 b^4 (15 a-11 b) \sin (2 (e+f x)) ((a-b) \cos (2 (e+f x))+a+b)-a (a-b)^2 (11 a+14 b) \cot (e+f x) \csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)^2\right )\right )}{15 \sqrt{2} a^5 f (a-b)^3 ((a-b) \cos (2 (e+f x))+a+b)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.255, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cot \left ( fx+e \right ) \right ) ^{6} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{-{\frac{5}{2}}}}\, dx \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 = 3.35924, size = 2286, normalized size = 6.99 \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 \frac{\cot \left (f x + e\right )^{6}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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