Optimal. Leaf size=67 \[ \frac{\cot ^9\left (\frac{1}{2} (e+f x)\right )}{36 c^7 f}-\frac{\cot ^7\left (\frac{1}{2} (e+f x)\right )}{14 c^7 f}+\frac{\cot ^5\left (\frac{1}{2} (e+f x)\right )}{20 c^7 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.298053, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {12, 270} \[ \frac{\cot ^9\left (\frac{1}{2} (e+f x)\right )}{36 c^7 f}-\frac{\cot ^7\left (\frac{1}{2} (e+f x)\right )}{14 c^7 f}+\frac{\cot ^5\left (\frac{1}{2} (e+f x)\right )}{20 c^7 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 270
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^7} \, dx &=\frac{2 \operatorname{Subst}\left (\int -\frac{\left (1-x^2\right )^2}{8 c^7 x^{10}} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{f}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^{10}} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{4 c^7 f}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x^{10}}-\frac{2}{x^8}+\frac{1}{x^6}\right ) \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{4 c^7 f}\\ &=\frac{\cot ^5\left (\frac{1}{2} (e+f x)\right )}{20 c^7 f}-\frac{\cot ^7\left (\frac{1}{2} (e+f x)\right )}{14 c^7 f}+\frac{\cot ^9\left (\frac{1}{2} (e+f x)\right )}{36 c^7 f}\\ \end{align*}
Mathematica [B] time = 0.914968, size = 151, normalized size = 2.25 \[ \frac{\csc \left (\frac{e}{2}\right ) \left (-718830 \sin \left (e+\frac{f x}{2}\right )+467208 \sin \left (e+\frac{3 f x}{2}\right )+659400 \sin \left (2 e+\frac{3 f x}{2}\right )-303192 \sin \left (2 e+\frac{5 f x}{2}\right )-179640 \sin \left (3 e+\frac{5 f x}{2}\right )+30753 \sin \left (3 e+\frac{7 f x}{2}\right )+89955 \sin \left (4 e+\frac{7 f x}{2}\right )-13427 \sin \left (4 e+\frac{9 f x}{2}\right )+15 \sin \left (5 e+\frac{9 f x}{2}\right )-971082 \sin \left (\frac{f x}{2}\right )\right ) \csc ^9\left (\frac{1}{2} (e+f x)\right )}{23063040 c^7 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.12, size = 49, normalized size = 0.7 \begin{align*}{\frac{1}{4\,f{c}^{7}} \left ({\frac{1}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}-{\frac{2}{7} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-7}}+{\frac{1}{9} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-9}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02752, size = 92, normalized size = 1.37 \begin{align*} -\frac{{\left (\frac{90 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{63 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 35\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{1260 \, c^{7} f \sin \left (f x + e\right )^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.664516, size = 309, normalized size = 4.61 \begin{align*} \frac{47 \, \cos \left (f x + e\right )^{5} + 127 \, \cos \left (f x + e\right )^{4} + 101 \, \cos \left (f x + e\right )^{3} + 11 \, \cos \left (f x + e\right )^{2} - 8 \, \cos \left (f x + e\right ) + 2}{315 \,{\left (c^{7} f \cos \left (f x + e\right )^{4} - 4 \, c^{7} f \cos \left (f x + e\right )^{3} + 6 \, c^{7} f \cos \left (f x + e\right )^{2} - 4 \, c^{7} f \cos \left (f x + e\right ) + c^{7} f\right )} \sin \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\tan ^{4}{\left (e + f x \right )} \sec{\left (e + f x \right )}}{\sec ^{7}{\left (e + f x \right )} - 7 \sec ^{6}{\left (e + f x \right )} + 21 \sec ^{5}{\left (e + f x \right )} - 35 \sec ^{4}{\left (e + f x \right )} + 35 \sec ^{3}{\left (e + f x \right )} - 21 \sec ^{2}{\left (e + f x \right )} + 7 \sec{\left (e + f x \right )} - 1}\, dx}{c^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.18048, size = 68, normalized size = 1.01 \begin{align*} \frac{63 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 90 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 35}{1260 \, c^{7} f \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]