3.21 \(\int \sec ^3(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=20 \[ \frac {a^3}{d (a-a \sin (c+d x))} \]

[Out]

a^3/d/(a-a*sin(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 32} \[ \frac {a^3}{d (a-a \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]

[Out]

a^3/(d*(a - a*Sin[c + d*x]))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 32, normalized size = 1.60 \[ \frac {a^2}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]

[Out]

a^2/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2)

________________________________________________________________________________________

fricas [A]  time = 0.77, size = 19, normalized size = 0.95 \[ -\frac {a^{2}}{d \sin \left (d x + c\right ) - d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-a^2/(d*sin(d*x + c) - d)

________________________________________________________________________________________

giac [A]  time = 0.74, size = 30, normalized size = 1.50 \[ \frac {2 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.21, size = 75, normalized size = 3.75 \[ \frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {a^{2} \sin \left (d x +c \right )}{2 d}+\frac {a^{2}}{d \cos \left (d x +c \right )^{2}}+\frac {a^{2} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3*(a+a*sin(d*x+c))^2,x)

[Out]

1/2/d*a^2*sin(d*x+c)^3/cos(d*x+c)^2+1/2*a^2*sin(d*x+c)/d+1/d*a^2/cos(d*x+c)^2+1/2/d*a^2*sec(d*x+c)*tan(d*x+c)

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 18, normalized size = 0.90 \[ -\frac {a^{2}}{d {\left (\sin \left (d x + c\right ) - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-a^2/(d*(sin(d*x + c) - 1))

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 18, normalized size = 0.90 \[ -\frac {a^2}{d\,\left (\sin \left (c+d\,x\right )-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(c + d*x))^2/cos(c + d*x)^3,x)

[Out]

-a^2/(d*(sin(c + d*x) - 1))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int 2 \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**3*(a+a*sin(d*x+c))**2,x)

[Out]

a**2*(Integral(2*sin(c + d*x)*sec(c + d*x)**3, x) + Integral(sin(c + d*x)**2*sec(c + d*x)**3, x) + Integral(se
c(c + d*x)**3, x))

________________________________________________________________________________________