Optimal. Leaf size=65 \[ \frac{\left (a^2-b^2\right ) \cot ^3(e+f x)}{3 f}-\frac{a^2 \cot (e+f x)}{f}-a^2 x-\frac{(a+b)^2 \cot ^5(e+f x)}{5 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0933106, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4141, 1802, 203} \[ \frac{\left (a^2-b^2\right ) \cot ^3(e+f x)}{3 f}-\frac{a^2 \cot (e+f x)}{f}-a^2 x-\frac{(a+b)^2 \cot ^5(e+f x)}{5 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4141
Rule 1802
Rule 203
Rubi steps
\begin{align*} \int \cot ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \left (1+x^2\right )\right )^2}{x^6 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{(a+b)^2}{x^6}+\frac{-a^2+b^2}{x^4}+\frac{a^2}{x^2}-\frac{a^2}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{a^2 \cot (e+f x)}{f}+\frac{\left (a^2-b^2\right ) \cot ^3(e+f x)}{3 f}-\frac{(a+b)^2 \cot ^5(e+f x)}{5 f}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-a^2 x-\frac{a^2 \cot (e+f x)}{f}+\frac{\left (a^2-b^2\right ) \cot ^3(e+f x)}{3 f}-\frac{(a+b)^2 \cot ^5(e+f x)}{5 f}\\ \end{align*}
Mathematica [B] time = 1.06115, size = 256, normalized size = 3.94 \[ \frac{\csc (e) \csc ^5(e+f x) \left (180 a^2 \sin (2 e+f x)-140 a^2 \sin (2 e+3 f x)-90 a^2 \sin (4 e+3 f x)+46 a^2 \sin (4 e+5 f x)+150 a^2 f x \cos (2 e+f x)+75 a^2 f x \cos (2 e+3 f x)-75 a^2 f x \cos (4 e+3 f x)-15 a^2 f x \cos (4 e+5 f x)+15 a^2 f x \cos (6 e+5 f x)+280 a^2 \sin (f x)-150 a^2 f x \cos (f x)-60 a b \sin (4 e+3 f x)+12 a b \sin (4 e+5 f x)+120 a b \sin (f x)-60 b^2 \sin (2 e+f x)+20 b^2 \sin (2 e+3 f x)-4 b^2 \sin (4 e+5 f x)+20 b^2 \sin (f x)\right )}{480 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.059, size = 107, normalized size = 1.7 \begin{align*}{\frac{1}{f} \left ({a}^{2} \left ( -{\frac{ \left ( \cot \left ( fx+e \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \cot \left ( fx+e \right ) \right ) ^{3}}{3}}-\cot \left ( fx+e \right ) -fx-e \right ) -{\frac{2\,ab \left ( \cos \left ( fx+e \right ) \right ) ^{5}}{5\, \left ( \sin \left ( fx+e \right ) \right ) ^{5}}}+{b}^{2} \left ( -{\frac{ \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{5\, \left ( \sin \left ( fx+e \right ) \right ) ^{5}}}-{\frac{2\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{15\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.47225, size = 97, normalized size = 1.49 \begin{align*} -\frac{15 \,{\left (f x + e\right )} a^{2} + \frac{15 \, a^{2} \tan \left (f x + e\right )^{4} - 5 \,{\left (a^{2} - b^{2}\right )} \tan \left (f x + e\right )^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}}{\tan \left (f x + e\right )^{5}}}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.499277, size = 328, normalized size = 5.05 \begin{align*} -\frac{{\left (23 \, a^{2} + 6 \, a b - 2 \, b^{2}\right )} \cos \left (f x + e\right )^{5} - 5 \,{\left (7 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{3} + 15 \, a^{2} \cos \left (f x + e\right ) + 15 \,{\left (a^{2} f x \cos \left (f x + e\right )^{4} - 2 \, a^{2} f x \cos \left (f x + e\right )^{2} + a^{2} f x\right )} \sin \left (f x + e\right )}{15 \,{\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )} \sin \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.50462, size = 392, normalized size = 6.03 \begin{align*} \frac{3 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 6 \, a b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 3 \, b^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 35 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 30 \, a b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 5 \, b^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 480 \,{\left (f x + e\right )} a^{2} + 330 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 60 \, a b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 30 \, b^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - \frac{330 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 60 \, a b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 30 \, b^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 35 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 30 \, a b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 5 \, b^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}}{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5}}}{480 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]