3.57 \(\int \frac {1}{(a+a \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=57 \[ -\frac {4 \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac {x}{a^2}-\frac {\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]

[Out]

x/a^2-4/3*tan(d*x+c)/a^2/d/(1+sec(d*x+c))-1/3*tan(d*x+c)/d/(a+a*sec(d*x+c))^2

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Rubi [A]  time = 0.07, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3777, 3919, 3794} \[ -\frac {4 \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac {x}{a^2}-\frac {\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^(-2),x]

[Out]

x/a^2 - (4*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - Tan[c + d*x]/(3*d*(a + a*Sec[c + d*x])^2)

Rule 3777

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Simp[(Cot[c + d*x]*(a + b*Csc[c + d*x])^n)/(d*(
2*n + 1)), x] + Dist[1/(a^2*(2*n + 1)), Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*x]
), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]

Rule 3794

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rubi steps

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

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Mathematica [A]  time = 0.28, size = 112, normalized size = 1.96 \[ \frac {\sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \left (12 \sin \left (c+\frac {d x}{2}\right )-10 \sin \left (c+\frac {3 d x}{2}\right )+9 d x \cos \left (c+\frac {d x}{2}\right )+3 d x \cos \left (c+\frac {3 d x}{2}\right )+3 d x \cos \left (2 c+\frac {3 d x}{2}\right )-18 \sin \left (\frac {d x}{2}\right )+9 d x \cos \left (\frac {d x}{2}\right )\right )}{24 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^3*(9*d*x*Cos[(d*x)/2] + 9*d*x*Cos[c + (d*x)/2] + 3*d*x*Cos[c + (3*d*x)/2] + 3*d*x*C
os[2*c + (3*d*x)/2] - 18*Sin[(d*x)/2] + 12*Sin[c + (d*x)/2] - 10*Sin[c + (3*d*x)/2]))/(24*a^2*d)

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fricas [A]  time = 1.37, size = 80, normalized size = 1.40 \[ \frac {3 \, d x \cos \left (d x + c\right )^{2} + 6 \, d x \cos \left (d x + c\right ) + 3 \, d x - {\left (5 \, \cos \left (d x + c\right ) + 4\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

1/3*(3*d*x*cos(d*x + c)^2 + 6*d*x*cos(d*x + c) + 3*d*x - (5*cos(d*x + c) + 4)*sin(d*x + c))/(a^2*d*cos(d*x + c
)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)

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giac [A]  time = 0.37, size = 50, normalized size = 0.88 \[ \frac {\frac {6 \, {\left (d x + c\right )}}{a^{2}} + \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

1/6*(6*(d*x + c)/a^2 + (a^4*tan(1/2*d*x + 1/2*c)^3 - 9*a^4*tan(1/2*d*x + 1/2*c))/a^6)/d

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maple [A]  time = 0.46, size = 56, normalized size = 0.98 \[ \frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{6 a^{2} d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2} d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sec(d*x+c))^2,x)

[Out]

1/6/a^2/d*tan(1/2*d*x+1/2*c)^3-3/2/a^2/d*tan(1/2*d*x+1/2*c)+2/a^2/d*arctan(tan(1/2*d*x+1/2*c))

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maxima [A]  time = 1.42, size = 72, normalized size = 1.26 \[ -\frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/6*((9*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2 - 12*arctan(sin(d*x + c)/(
cos(d*x + c) + 1))/a^2)/d

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mupad [B]  time = 0.63, size = 35, normalized size = 0.61 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6\,d\,x}{6\,a^2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a/cos(c + d*x))^2,x)

[Out]

(tan(c/2 + (d*x)/2)^3 - 9*tan(c/2 + (d*x)/2) + 6*d*x)/(6*a^2*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sec(d*x+c))**2,x)

[Out]

Integral(1/(sec(c + d*x)**2 + 2*sec(c + d*x) + 1), x)/a**2

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