Optimal. Leaf size=86 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{a f (a+b)^{5/2}}-\frac{\cot ^3(e+f x)}{3 f (a+b)}+\frac{(a+2 b) \cot (e+f x)}{f (a+b)^2}+\frac{x}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.248174, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4141, 1975, 480, 583, 522, 203, 205} \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{a f (a+b)^{5/2}}-\frac{\cot ^3(e+f x)}{3 f (a+b)}+\frac{(a+2 b) \cot (e+f x)}{f (a+b)^2}+\frac{x}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4141
Rule 1975
Rule 480
Rule 583
Rule 522
Rule 203
Rule 205
Rubi steps
\begin{align*} \int \frac{\cot ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+x^2\right ) \left (a+b \left (1+x^2\right )\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cot ^3(e+f x)}{3 (a+b) f}+\frac{\operatorname{Subst}\left (\int \frac{-3 (a+2 b)-3 b x^2}{x^2 \left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{3 (a+b) f}\\ &=\frac{(a+2 b) \cot (e+f x)}{(a+b)^2 f}-\frac{\cot ^3(e+f x)}{3 (a+b) f}-\frac{\operatorname{Subst}\left (\int \frac{-3 \left (a^2+3 a b+3 b^2\right )-3 b (a+2 b) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{3 (a+b)^2 f}\\ &=\frac{(a+2 b) \cot (e+f x)}{(a+b)^2 f}-\frac{\cot ^3(e+f x)}{3 (a+b) f}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{a f}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{a (a+b)^2 f}\\ &=\frac{x}{a}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{a (a+b)^{5/2} f}+\frac{(a+2 b) \cot (e+f x)}{(a+b)^2 f}-\frac{\cot ^3(e+f x)}{3 (a+b) f}\\ \end{align*}
Mathematica [C] time = 3.70735, size = 390, normalized size = 4.53 \[ \frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (\frac{1}{8} \sqrt{a+b} \csc (e) \sqrt{b (\cos (e)-i \sin (e))^4} \csc ^3(e+f x) \left (-12 a^2 \sin (2 e+f x)+8 a^2 \sin (2 e+3 f x)-3 a^2 f x \cos (2 e+3 f x)+3 a^2 f x \cos (4 e+3 f x)-12 a^2 \sin (f x)-18 a b \sin (2 e+f x)+14 a b \sin (2 e+3 f x)-6 a b f x \cos (2 e+3 f x)+6 a b f x \cos (4 e+3 f x)-9 f x (a+b)^2 \cos (2 e+f x)-24 a b \sin (f x)+9 f x (a+b)^2 \cos (f x)-3 b^2 f x \cos (2 e+3 f x)+3 b^2 f x \cos (4 e+3 f x)\right )+3 b^3 (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left (\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right )\right )}{6 a f (a+b)^{5/2} \sqrt{b (\cos (e)-i \sin (e))^4} \left (a+b \sec ^2(e+f x)\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.099, size = 110, normalized size = 1.3 \begin{align*}{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) }{fa}}-{\frac{1}{3\,f \left ( a+b \right ) \left ( \tan \left ( fx+e \right ) \right ) ^{3}}}+{\frac{a}{f \left ( a+b \right ) ^{2}\tan \left ( fx+e \right ) }}+2\,{\frac{b}{f \left ( a+b \right ) ^{2}\tan \left ( fx+e \right ) }}-{\frac{{b}^{3}}{f \left ( a+b \right ) ^{2}a}\arctan \left ({\tan \left ( fx+e \right ) b{\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \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 [B] time = 0.621233, size = 1261, normalized size = 14.66 \begin{align*} \left [\frac{4 \,{\left (4 \, a^{2} + 7 \, a b\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (b^{2} \cos \left (f x + e\right )^{2} - b^{2}\right )} \sqrt{-\frac{b}{a + b}} \log \left (\frac{{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \,{\left ({\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{3} -{\left (a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt{-\frac{b}{a + b}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right ) \sin \left (f x + e\right ) - 12 \,{\left (a^{2} + 2 \, a b\right )} \cos \left (f x + e\right ) + 12 \,{\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} f x \cos \left (f x + e\right )^{2} -{\left (a^{2} + 2 \, a b + b^{2}\right )} f x\right )} \sin \left (f x + e\right )}{12 \,{\left ({\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} f \cos \left (f x + e\right )^{2} -{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} f\right )} \sin \left (f x + e\right )}, \frac{2 \,{\left (4 \, a^{2} + 7 \, a b\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (b^{2} \cos \left (f x + e\right )^{2} - b^{2}\right )} \sqrt{\frac{b}{a + b}} \arctan \left (\frac{{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt{\frac{b}{a + b}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 6 \,{\left (a^{2} + 2 \, a b\right )} \cos \left (f x + e\right ) + 6 \,{\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} f x \cos \left (f x + e\right )^{2} -{\left (a^{2} + 2 \, a b + b^{2}\right )} f x\right )} \sin \left (f x + e\right )}{6 \,{\left ({\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} f \cos \left (f x + e\right )^{2} -{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} f\right )} \sin \left (f x + e\right )}\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*} \int \frac{\cot ^{4}{\left (e + f x \right )}}{a + b \sec ^{2}{\left (e + f x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.43611, size = 189, normalized size = 2.2 \begin{align*} -\frac{\frac{3 \,{\left (\pi \left \lfloor \frac{f x + e}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (f x + e\right )}{\sqrt{a b + b^{2}}}\right )\right )} b^{3}}{{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \sqrt{a b + b^{2}}} - \frac{3 \,{\left (f x + e\right )}}{a} - \frac{3 \, a \tan \left (f x + e\right )^{2} + 6 \, b \tan \left (f x + e\right )^{2} - a - b}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{3}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]