Optimal. Leaf size=84 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{a^{5/2} f (a-b)}+\frac{(a+b) \cot (e+f x)}{a^2 f}+\frac{x}{a-b}-\frac{\cot ^3(e+f x)}{3 a f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.172941, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3670, 480, 583, 522, 203, 205} \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{a^{5/2} f (a-b)}+\frac{(a+b) \cot (e+f x)}{a^2 f}+\frac{x}{a-b}-\frac{\cot ^3(e+f x)}{3 a f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3670
Rule 480
Rule 583
Rule 522
Rule 203
Rule 205
Rubi steps
\begin{align*} \int \frac{\cot ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cot ^3(e+f x)}{3 a f}+\frac{\operatorname{Subst}\left (\int \frac{-3 (a+b)-3 b x^2}{x^2 \left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{3 a f}\\ &=\frac{(a+b) \cot (e+f x)}{a^2 f}-\frac{\cot ^3(e+f x)}{3 a f}-\frac{\operatorname{Subst}\left (\int \frac{-3 \left (a^2+a b+b^2\right )-3 b (a+b) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{3 a^2 f}\\ &=\frac{(a+b) \cot (e+f x)}{a^2 f}-\frac{\cot ^3(e+f x)}{3 a f}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{(a-b) f}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{a^2 (a-b) f}\\ &=\frac{x}{a-b}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{a^{5/2} (a-b) f}+\frac{(a+b) \cot (e+f x)}{a^2 f}-\frac{\cot ^3(e+f x)}{3 a f}\\ \end{align*}
Mathematica [A] time = 0.647017, size = 92, normalized size = 1.1 \[ \frac{\sqrt{a} \left (3 a^2 (e+f x)-(a-b) \cot (e+f x) \left (a \csc ^2(e+f x)-4 a-3 b\right )\right )-3 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{3 a^{5/2} f (a-b)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.079, size = 104, normalized size = 1.2 \begin{align*} -{\frac{1}{3\,fa \left ( \tan \left ( fx+e \right ) \right ) ^{3}}}+{\frac{1}{fa\tan \left ( fx+e \right ) }}+{\frac{b}{f{a}^{2}\tan \left ( fx+e \right ) }}-{\frac{{b}^{3}}{f{a}^{2} \left ( a-b \right ) }\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) }{f \left ( a-b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.19186, size = 702, normalized size = 8.36 \begin{align*} \left [\frac{12 \, a^{2} f x \tan \left (f x + e\right )^{3} - 3 \, b^{2} \sqrt{-\frac{b}{a}} \log \left (\frac{b^{2} \tan \left (f x + e\right )^{4} - 6 \, a b \tan \left (f x + e\right )^{2} + a^{2} + 4 \,{\left (a b \tan \left (f x + e\right )^{3} - a^{2} \tan \left (f x + e\right )\right )} \sqrt{-\frac{b}{a}}}{b^{2} \tan \left (f x + e\right )^{4} + 2 \, a b \tan \left (f x + e\right )^{2} + a^{2}}\right ) \tan \left (f x + e\right )^{3} + 12 \,{\left (a^{2} - b^{2}\right )} \tan \left (f x + e\right )^{2} - 4 \, a^{2} + 4 \, a b}{12 \,{\left (a^{3} - a^{2} b\right )} f \tan \left (f x + e\right )^{3}}, \frac{6 \, a^{2} f x \tan \left (f x + e\right )^{3} - 3 \, b^{2} \sqrt{\frac{b}{a}} \arctan \left (\frac{{\left (b \tan \left (f x + e\right )^{2} - a\right )} \sqrt{\frac{b}{a}}}{2 \, b \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{3} + 6 \,{\left (a^{2} - b^{2}\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2} + 2 \, a b}{6 \,{\left (a^{3} - a^{2} b\right )} f \tan \left (f x + e\right )^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 89.7141, size = 823, normalized size = 9.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.60999, size = 504, normalized size = 6. \begin{align*} -\frac{\frac{3 \,{\left (a^{4} b + a^{2} b^{3} - a b{\left | -a^{3} + a^{2} b \right |} - b^{2}{\left | -a^{3} + a^{2} b \right |}\right )}{\left (\pi \left \lfloor \frac{f x + e}{\pi } + \frac{1}{2} \right \rfloor + \arctan \left (\frac{2 \, \tan \left (f x + e\right )}{\sqrt{\frac{2 \, a^{3} + 2 \, a^{2} b + \sqrt{-16 \, a^{5} b + 4 \,{\left (a^{3} + a^{2} b\right )}^{2}}}{a^{2} b}}}\right )\right )}}{a^{3}{\left | -a^{3} + a^{2} b \right |} + a^{2} b{\left | -a^{3} + a^{2} b \right |} +{\left (a^{3} - a^{2} b\right )}^{2}} + \frac{3 \,{\left (\sqrt{a b}{\left (a + b\right )}{\left | -a^{3} + a^{2} b \right |}{\left | b \right |} +{\left (a^{4} + a^{2} b^{2}\right )} \sqrt{a b}{\left | b \right |}\right )}{\left (\pi \left \lfloor \frac{f x + e}{\pi } + \frac{1}{2} \right \rfloor + \arctan \left (\frac{2 \, \tan \left (f x + e\right )}{\sqrt{\frac{2 \, a^{3} + 2 \, a^{2} b - \sqrt{-16 \, a^{5} b + 4 \,{\left (a^{3} + a^{2} b\right )}^{2}}}{a^{2} b}}}\right )\right )}}{{\left (a^{3} - a^{2} b\right )}^{2} b -{\left (a^{3} b + a^{2} b^{2}\right )}{\left | -a^{3} + a^{2} b \right |}} - \frac{3 \, a \tan \left (f x + e\right )^{2} + 3 \, b \tan \left (f x + e\right )^{2} - a}{a^{2} \tan \left (f x + e\right )^{3}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]