3.324 \(\int (-\cos (x)+\sec (x))^2 \, dx\)

Optimal. Leaf size=22 \[ -\frac{3 x}{2}+\frac{3 \tan (x)}{2}-\frac{1}{2} \sin ^2(x) \tan (x) \]

[Out]

(-3*x)/2 + (3*Tan[x])/2 - (Sin[x]^2*Tan[x])/2

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Rubi [A]  time = 0.0205319, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {288, 321, 203} \[ -\frac{3 x}{2}+\frac{3 \tan (x)}{2}-\frac{1}{2} \sin ^2(x) \tan (x) \]

Antiderivative was successfully verified.

[In]

Int[(-Cos[x] + Sec[x])^2,x]

[Out]

(-3*x)/2 + (3*Tan[x])/2 - (Sin[x]^2*Tan[x])/2

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int (-\cos (x)+\sec (x))^2 \, dx &=\operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (x)\right )\\ &=-\frac{1}{2} \sin ^2(x) \tan (x)+\frac{3}{2} \operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\frac{3 \tan (x)}{2}-\frac{1}{2} \sin ^2(x) \tan (x)-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (x)\right )\\ &=-\frac{3 x}{2}+\frac{3 \tan (x)}{2}-\frac{1}{2} \sin ^2(x) \tan (x)\\ \end{align*}

Mathematica [A]  time = 0.0177295, size = 16, normalized size = 0.73 \[ -\frac{3 x}{2}+\frac{1}{4} \sin (2 x)+\tan (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Cos[x] + Sec[x])^2,x]

[Out]

(-3*x)/2 + Sin[2*x]/4 + Tan[x]

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Maple [A]  time = 0.016, size = 13, normalized size = 0.6 \begin{align*} \tan \left ( x \right ) -{\frac{3\,x}{2}}+{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-cos(x)+sec(x))^2,x)

[Out]

tan(x)-3/2*x+1/2*cos(x)*sin(x)

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Maxima [A]  time = 0.973503, size = 16, normalized size = 0.73 \begin{align*} -\frac{3}{2} \, x + \frac{1}{4} \, \sin \left (2 \, x\right ) + \tan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cos(x)+sec(x))^2,x, algorithm="maxima")

[Out]

-3/2*x + 1/4*sin(2*x) + tan(x)

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Fricas [A]  time = 2.06675, size = 68, normalized size = 3.09 \begin{align*} -\frac{3 \, x \cos \left (x\right ) -{\left (\cos \left (x\right )^{2} + 2\right )} \sin \left (x\right )}{2 \, \cos \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cos(x)+sec(x))^2,x, algorithm="fricas")

[Out]

-1/2*(3*x*cos(x) - (cos(x)^2 + 2)*sin(x))/cos(x)

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Sympy [A]  time = 2.5935, size = 14, normalized size = 0.64 \begin{align*} - \frac{3 x}{2} + \frac{\sin{\left (2 x \right )}}{4} + \tan{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cos(x)+sec(x))**2,x)

[Out]

-3*x/2 + sin(2*x)/4 + tan(x)

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Giac [A]  time = 1.15956, size = 24, normalized size = 1.09 \begin{align*} -\frac{3}{2} \, x + \frac{\tan \left (x\right )}{2 \,{\left (\tan \left (x\right )^{2} + 1\right )}} + \tan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cos(x)+sec(x))^2,x, algorithm="giac")

[Out]

-3/2*x + 1/2*tan(x)/(tan(x)^2 + 1) + tan(x)