3.4 \(\int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2 \, dx\)

Optimal. Leaf size=47 \[ \frac{a^2 c^2 \tan ^3(e+f x)}{3 f}-\frac{a^2 c^2 \tan (e+f x)}{f}+a^2 c^2 x \]

[Out]

a^2*c^2*x - (a^2*c^2*Tan[e + f*x])/f + (a^2*c^2*Tan[e + f*x]^3)/(3*f)

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Rubi [A]  time = 0.0646025, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3904, 3473, 8} \[ \frac{a^2 c^2 \tan ^3(e+f x)}{3 f}-\frac{a^2 c^2 \tan (e+f x)}{f}+a^2 c^2 x \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^2,x]

[Out]

a^2*c^2*x - (a^2*c^2*Tan[e + f*x])/f + (a^2*c^2*Tan[e + f*x]^3)/(3*f)

Rule 3904

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Di
st[(-(a*c))^m, Int[Cot[e + f*x]^(2*m)*(c + d*Csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
&& EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] &&  !(IntegerQ[n] && GtQ[m - n, 0])

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2 \, dx &=\left (a^2 c^2\right ) \int \tan ^4(e+f x) \, dx\\ &=\frac{a^2 c^2 \tan ^3(e+f x)}{3 f}-\left (a^2 c^2\right ) \int \tan ^2(e+f x) \, dx\\ &=-\frac{a^2 c^2 \tan (e+f x)}{f}+\frac{a^2 c^2 \tan ^3(e+f x)}{3 f}+\left (a^2 c^2\right ) \int 1 \, dx\\ &=a^2 c^2 x-\frac{a^2 c^2 \tan (e+f x)}{f}+\frac{a^2 c^2 \tan ^3(e+f x)}{3 f}\\ \end{align*}

Mathematica [A]  time = 0.0339771, size = 45, normalized size = 0.96 \[ a^2 c^2 \left (\frac{\tan ^3(e+f x)}{3 f}+\frac{\tan ^{-1}(\tan (e+f x))}{f}-\frac{\tan (e+f x)}{f}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^2,x]

[Out]

a^2*c^2*(ArcTan[Tan[e + f*x]]/f - Tan[e + f*x]/f + Tan[e + f*x]^3/(3*f))

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Maple [A]  time = 0.021, size = 58, normalized size = 1.2 \begin{align*}{\frac{1}{f} \left ( -2\,{c}^{2}{a}^{2}\tan \left ( fx+e \right ) +{c}^{2}{a}^{2} \left ( fx+e \right ) -{c}^{2}{a}^{2} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) \tan \left ( fx+e \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^2,x)

[Out]

1/f*(-2*c^2*a^2*tan(f*x+e)+c^2*a^2*(f*x+e)-c^2*a^2*(-2/3-1/3*sec(f*x+e)^2)*tan(f*x+e))

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Maxima [A]  time = 1.01444, size = 77, normalized size = 1.64 \begin{align*} \frac{{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c^{2} + 3 \,{\left (f x + e\right )} a^{2} c^{2} - 6 \, a^{2} c^{2} \tan \left (f x + e\right )}{3 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^2,x, algorithm="maxima")

[Out]

1/3*((tan(f*x + e)^3 + 3*tan(f*x + e))*a^2*c^2 + 3*(f*x + e)*a^2*c^2 - 6*a^2*c^2*tan(f*x + e))/f

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Fricas [A]  time = 1.07713, size = 144, normalized size = 3.06 \begin{align*} \frac{3 \, a^{2} c^{2} f x \cos \left (f x + e\right )^{3} -{\left (4 \, a^{2} c^{2} \cos \left (f x + e\right )^{2} - a^{2} c^{2}\right )} \sin \left (f x + e\right )}{3 \, f \cos \left (f x + e\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^2,x, algorithm="fricas")

[Out]

1/3*(3*a^2*c^2*f*x*cos(f*x + e)^3 - (4*a^2*c^2*cos(f*x + e)^2 - a^2*c^2)*sin(f*x + e))/(f*cos(f*x + e)^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} c^{2} \left (\int 1\, dx + \int - 2 \sec ^{2}{\left (e + f x \right )}\, dx + \int \sec ^{4}{\left (e + f x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))**2*(c-c*sec(f*x+e))**2,x)

[Out]

a**2*c**2*(Integral(1, x) + Integral(-2*sec(e + f*x)**2, x) + Integral(sec(e + f*x)**4, x))

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Giac [A]  time = 1.43435, size = 69, normalized size = 1.47 \begin{align*} \frac{a^{2} c^{2} \tan \left (f x + e\right )^{3} + 3 \,{\left (f x + e\right )} a^{2} c^{2} - 3 \, a^{2} c^{2} \tan \left (f x + e\right )}{3 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^2,x, algorithm="giac")

[Out]

1/3*(a^2*c^2*tan(f*x + e)^3 + 3*(f*x + e)*a^2*c^2 - 3*a^2*c^2*tan(f*x + e))/f