Optimal. Leaf size=53 \[ \frac{2 b (a+b) \tan ^3(e+f x)}{3 f}+\frac{(a+b)^2 \tan (e+f x)}{f}+\frac{b^2 \tan ^5(e+f x)}{5 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06289, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {4146, 194} \[ \frac{2 b (a+b) \tan ^3(e+f x)}{3 f}+\frac{(a+b)^2 \tan (e+f x)}{f}+\frac{b^2 \tan ^5(e+f x)}{5 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4146
Rule 194
Rubi steps
\begin{align*} \int \sec ^2(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b+b x^2\right )^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 \left (1+\frac{b (2 a+b)}{a^2}\right )+2 a b \left (1+\frac{b}{a}\right ) x^2+b^2 x^4\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a+b)^2 \tan (e+f x)}{f}+\frac{2 b (a+b) \tan ^3(e+f x)}{3 f}+\frac{b^2 \tan ^5(e+f x)}{5 f}\\ \end{align*}
Mathematica [A] time = 0.265406, size = 48, normalized size = 0.91 \[ \frac{10 b (a+b) \tan ^3(e+f x)+15 (a+b)^2 \tan (e+f x)+3 b^2 \tan ^5(e+f x)}{15 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.032, size = 71, normalized size = 1.3 \begin{align*}{\frac{1}{f} \left ({a}^{2}\tan \left ( fx+e \right ) -2\,ab \left ( -2/3-1/3\, \left ( \sec \left ( fx+e \right ) \right ) ^{2} \right ) \tan \left ( fx+e \right ) -{b}^{2} \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( fx+e \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{15}} \right ) \tan \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.986876, size = 96, normalized size = 1.81 \begin{align*} \frac{10 \,{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a b +{\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} b^{2} + 15 \, a^{2} \tan \left (f x + e\right )}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.477477, size = 167, normalized size = 3.15 \begin{align*} \frac{{\left ({\left (15 \, a^{2} + 20 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \,{\left (5 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 3 \, b^{2}\right )} \sin \left (f x + e\right )}{15 \, f \cos \left (f x + e\right )^{5}} \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 \sec ^{2}{\left (e + f x \right )}\right )^{2} \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 = 1.35357, size = 111, normalized size = 2.09 \begin{align*} \frac{3 \, b^{2} \tan \left (f x + e\right )^{5} + 10 \, a b \tan \left (f x + e\right )^{3} + 10 \, b^{2} \tan \left (f x + e\right )^{3} + 15 \, a^{2} \tan \left (f x + e\right ) + 30 \, a b \tan \left (f x + e\right ) + 15 \, b^{2} \tan \left (f x + e\right )}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]