Optimal. Leaf size=42 \[ -\frac{(a+b) \cot (e+f x)}{f}-\frac{a \cot ^3(e+f x)}{3 f}+\frac{b \tan (e+f x)}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0432361, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3663, 448} \[ -\frac{(a+b) \cot (e+f x)}{f}-\frac{a \cot ^3(e+f x)}{3 f}+\frac{b \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3663
Rule 448
Rubi steps
\begin{align*} \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right ) \left (a+b x^2\right )}{x^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (b+\frac{a}{x^4}+\frac{a+b}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{(a+b) \cot (e+f x)}{f}-\frac{a \cot ^3(e+f x)}{3 f}+\frac{b \tan (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.0685022, size = 60, normalized size = 1.43 \[ -\frac{2 a \cot (e+f x)}{3 f}-\frac{a \cot (e+f x) \csc ^2(e+f x)}{3 f}+\frac{b \tan (e+f x)}{f}-\frac{b \cot (e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.077, size = 54, normalized size = 1.3 \begin{align*}{\frac{1}{f} \left ( b \left ({\frac{1}{\cos \left ( fx+e \right ) \sin \left ( fx+e \right ) }}-2\,\cot \left ( fx+e \right ) \right ) +a \left ( -{\frac{2}{3}}-{\frac{ \left ( \csc \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) \cot \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05946, size = 54, normalized size = 1.29 \begin{align*} \frac{3 \, b \tan \left (f x + e\right ) - \frac{3 \,{\left (a + b\right )} \tan \left (f x + e\right )^{2} + a}{\tan \left (f x + e\right )^{3}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.79929, size = 163, normalized size = 3.88 \begin{align*} -\frac{2 \,{\left (a + 3 \, b\right )} \cos \left (f x + e\right )^{4} - 3 \,{\left (a + 3 \, b\right )} \cos \left (f x + e\right )^{2} + 3 \, b}{3 \,{\left (f \cos \left (f x + e\right )^{3} - f \cos \left (f x + e\right )\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*} \int \left (a + b \tan ^{2}{\left (e + f x \right )}\right ) \csc ^{4}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.51136, size = 72, normalized size = 1.71 \begin{align*} \frac{3 \, b \tan \left (f x + e\right ) - \frac{3 \, a \tan \left (f x + e\right )^{2} + 3 \, b \tan \left (f x + e\right )^{2} + a}{\tan \left (f x + e\right )^{3}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]