3.133 \(\int \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=55 \[ \frac {8 a^2 \sec (c+d x) \sqrt {a \sin (c+d x)+a}}{d}-\frac {2 a \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{d} \]

[Out]

-2*a*sec(d*x+c)*(a+a*sin(d*x+c))^(3/2)/d+8*a^2*sec(d*x+c)*(a+a*sin(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.12, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2674, 2673} \[ \frac {8 a^2 \sec (c+d x) \sqrt {a \sin (c+d x)+a}}{d}-\frac {2 a \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(8*a^2*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d - (2*a*Sec[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/d

Rule 2673

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m - 1)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

Rule 2674

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m + p)), x] + Dist[(a*(2*m + p - 1))/(m + p), Int[(
g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0]
&& IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]

Rubi steps

\begin {align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=-\frac {2 a \sec (c+d x) (a+a \sin (c+d x))^{3/2}}{d}+(4 a) \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac {8 a^2 \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{d}-\frac {2 a \sec (c+d x) (a+a \sin (c+d x))^{3/2}}{d}\\ \end {align*}

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Mathematica [A]  time = 4.62, size = 36, normalized size = 0.65 \[ -\frac {2 a^2 (\sin (c+d x)-3) \sec (c+d x) \sqrt {a (\sin (c+d x)+1)}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2),x]

[Out]

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

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fricas [A]  time = 0.73, size = 41, normalized size = 0.75 \[ -\frac {2 \, {\left (a^{2} \sin \left (d x + c\right ) - 3 \, a^{2}\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{d \cos \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

-2*(a^2*sin(d*x + c) - 3*a^2)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c))

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giac [B]  time = 54.37, size = 6622, normalized size = 120.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

-sqrt(2)*sqrt(a)*(sqrt(2)*(sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^6 - 15*sqrt
(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^4 + 18*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d
*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^5 - 3*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*
c)^6 + 15*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^2 - 60*sqrt(2)*a^2*sgn(cos(-
1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^3 + 45*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(
1/2*c)*tan(1/4*c)^4 - 6*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^5 - sqrt(2)*a^2*sgn(cos(-1/
4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^3 + 18*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/
4*c) - 45*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^2 + 20*sqrt(2)*a^2*sgn(cos(-1/
4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^3 + 3*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c) - 6*sqrt(
2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c))*log(abs(-2*tan(1/4*d*x + c)*tan(1/2*c)^3 - 6*tan(1/4*d*
x + c)*tan(1/2*c)^2 + 2*tan(1/2*c)^3 - 2*sqrt(2)*(tan(1/2*c)^2 + 1)^(3/2) + 6*tan(1/4*d*x + c)*tan(1/2*c) - 6*
tan(1/2*c)^2 + 2*tan(1/4*d*x + c) - 6*tan(1/2*c) + 2)/abs(-2*tan(1/4*d*x + c)*tan(1/2*c)^3 - 6*tan(1/4*d*x + c
)*tan(1/2*c)^2 + 2*tan(1/2*c)^3 + 2*sqrt(2)*(tan(1/2*c)^2 + 1)^(3/2) + 6*tan(1/4*d*x + c)*tan(1/2*c) - 6*tan(1
/2*c)^2 + 2*tan(1/4*d*x + c) - 6*tan(1/2*c) + 2))/((tan(1/4*c)^6 + 3*tan(1/4*c)^4 + 3*tan(1/4*c)^2 + 1)*(tan(1
/2*c)^2 + 1)^(3/2)) - 2*(sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^6*tan(1
/4*c)^6 - 6*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^6*tan(1/4*c)^5 + 12*
sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^5*tan(1/4*c)^6 - 3*sqrt(2)*a^2*s
gn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^6*tan(1/4*c)^6 - 15*sqrt(2)*a^2*sgn(cos(-1/4*
pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^6*tan(1/4*c)^4 + 72*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x
+ 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^5*tan(1/4*c)^5 - 18*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*t
an(1/4*d*x + c)^2*tan(1/2*c)^6*tan(1/4*c)^5 - 15*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x +
c)^3*tan(1/2*c)^4*tan(1/4*c)^6 + sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^
6*tan(1/4*c)^6 + 20*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^6*tan(1/4*c)
^3 - 180*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^5*tan(1/4*c)^4 + 45*sqr
t(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^6*tan(1/4*c)^4 + 90*sqrt(2)*a^2*sgn
(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^4*tan(1/4*c)^5 - 30*sqrt(2)*a^2*sgn(cos(-1/4*pi
+ 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^6*tan(1/4*c)^5 - 40*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1
/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^3*tan(1/4*c)^6 + 45*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1
/4*d*x + c)^2*tan(1/2*c)^4*tan(1/4*c)^6 + sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^6*tan(1/4
*c)^6 + 15*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^6*tan(1/4*c)^2 - 240*
sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^5*tan(1/4*c)^3 + 60*sqrt(2)*a^2*
sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^6*tan(1/4*c)^3 + 225*sqrt(2)*a^2*sgn(cos(-1/
4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^4*tan(1/4*c)^4 - 15*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d
*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^6*tan(1/4*c)^4 - 240*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*
tan(1/4*d*x + c)^3*tan(1/2*c)^3*tan(1/4*c)^5 + 270*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x
+ c)^2*tan(1/2*c)^4*tan(1/4*c)^5 - 18*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^6*tan(1/4*c)
^5 + 15*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^2*tan(1/4*c)^6 - 51*sqrt
(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^4*tan(1/4*c)^6 + 12*sqrt(2)*a^2*sgn(co
s(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^5*tan(1/4*c)^6 - 6*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*ta
n(1/4*d*x + c)^3*tan(1/2*c)^6*tan(1/4*c) + 180*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c
)^3*tan(1/2*c)^5*tan(1/4*c)^2 - 45*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*
c)^6*tan(1/4*c)^2 - 300*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^4*tan(1/
4*c)^3 + 100*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^6*tan(1/4*c)^3 + 600*
sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^3*tan(1/4*c)^4 - 675*sqrt(2)*a^2
*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^4*tan(1/4*c)^4 - 15*sqrt(2)*a^2*sgn(cos(-1/
4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^6*tan(1/4*c)^4 - 90*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/
4*d*x + c)^3*tan(1/2*c)^2*tan(1/4*c)^5 + 234*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*
tan(1/2*c)^4*tan(1/4*c)^5 - 72*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^5*tan(1/4*c)^5 + 12*
sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)*tan(1/4*c)^6 - 45*sqrt(2)*a^2*sg
n(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^2*tan(1/4*c)^6 + 21*sqrt(2)*a^2*sgn(cos(-1/4*p
i + 1/2*d*x + 1/2*c))*tan(1/2*c)^4*tan(1/4*c)^6 - sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x
+ c)^3*tan(1/2*c)^6 + 72*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^5*tan(1
/4*c) - 18*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^6*tan(1/4*c) - 225*sq
rt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^4*tan(1/4*c)^2 + 15*sqrt(2)*a^2*sg
n(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^6*tan(1/4*c)^2 + 800*sqrt(2)*a^2*sgn(cos(-1/4*pi
+ 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^3*tan(1/4*c)^3 - 900*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x
+ 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^4*tan(1/4*c)^3 + 60*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*ta
n(1/2*c)^6*tan(1/4*c)^3 - 225*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^2*
tan(1/4*c)^4 + 765*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^4*tan(1/4*c)^4
- 180*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^5*tan(1/4*c)^4 + 72*sqrt(2)*a^2*sgn(cos(-1/4*
pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)*tan(1/4*c)^5 - 270*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x
+ 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^2*tan(1/4*c)^5 + 54*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*ta
n(1/2*c)^4*tan(1/4*c)^5 - sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/4*c)^6 + 39
*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^2*tan(1/4*c)^6 - 40*sqrt(2)*a^2*s
gn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^6 - 12*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*
c))*tan(1/4*d*x + c)^3*tan(1/2*c)^5 + 3*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan
(1/2*c)^6 + 90*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^4*tan(1/4*c) - 30
*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^6*tan(1/4*c) - 600*sqrt(2)*a^2*sg
n(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^3*tan(1/4*c)^2 + 675*sqrt(2)*a^2*sgn(cos(-1/4*
pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^4*tan(1/4*c)^2 + 15*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x
+ 1/2*c))*tan(1/2*c)^6*tan(1/4*c)^2 + 300*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*
tan(1/2*c)^2*tan(1/4*c)^3 - 780*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^4*
tan(1/4*c)^3 + 240*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^5*tan(1/4*c)^3 - 180*sqrt(2)*a^2
*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)*tan(1/4*c)^4 + 675*sqrt(2)*a^2*sgn(cos(-1/4
*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^2*tan(1/4*c)^4 - 315*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d
*x + 1/2*c))*tan(1/2*c)^4*tan(1/4*c)^4 + 6*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*
tan(1/4*c)^5 - 306*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^2*tan(1/4*c)^5
+ 240*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^5 + 3*sqrt(2)*a^2*sgn(cos(-1/4*p
i + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/4*c)^6 - 9*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(
1/2*c)^2*tan(1/4*c)^6 + 15*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^4 - s
qrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^6 - 240*sqrt(2)*a^2*sgn(cos(-1/4*pi
+ 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^3*tan(1/4*c) + 270*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x +
1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^4*tan(1/4*c) - 18*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/
2*c)^6*tan(1/4*c) + 225*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^2*tan(1/
4*c)^2 - 765*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^4*tan(1/4*c)^2 + 180*
sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^5*tan(1/4*c)^2 - 240*sqrt(2)*a^2*sgn(cos(-1/4*pi +
1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)*tan(1/4*c)^3 + 900*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2
*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^2*tan(1/4*c)^3 - 180*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/
2*c)^4*tan(1/4*c)^3 + 15*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/4*c)^4 - 585
*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^2*tan(1/4*c)^4 + 600*sqrt(2)*a^2*
sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^4 + 18*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2
*c))*tan(1/4*d*x + c)^2*tan(1/4*c)^5 - 126*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/
4*c)^5 - 5*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/4*c)^6 + 12*sqrt(2)*a^2*sgn(
cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^6 + 40*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*t
an(1/4*d*x + c)^3*tan(1/2*c)^3 - 45*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2
*c)^4 - sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^6 - 90*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*
x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)^2*tan(1/4*c) + 234*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*t
an(1/4*d*x + c)*tan(1/2*c)^4*tan(1/4*c) - 72*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^5*tan(
1/4*c) + 180*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)*tan(1/4*c)^2 - 675*
sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^2*tan(1/4*c)^2 + 315*sqrt(2)*a^2
*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^4*tan(1/4*c)^2 - 20*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/
2*c))*tan(1/4*d*x + c)^3*tan(1/4*c)^3 + 1020*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*
tan(1/2*c)^2*tan(1/4*c)^3 - 800*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^3 - 45
*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/4*c)^4 + 135*sqrt(2)*a^2*sgn(cos(-1/
4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^4 + 6*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4
*d*x + c)*tan(1/4*c)^5 - 72*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^5 + 3*sqrt(2
)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^6 - 15*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*ta
n(1/4*d*x + c)^3*tan(1/2*c)^2 + 51*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)
^4 - 12*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^5 + 72*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*
x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c)*tan(1/4*c) - 270*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan
(1/4*d*x + c)^2*tan(1/2*c)^2*tan(1/4*c) + 54*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^4*tan(
1/4*c) - 15*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/4*c)^2 + 585*sqrt(2)*a^2*
sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^2*tan(1/4*c)^2 - 600*sqrt(2)*a^2*sgn(cos(-1/4*
pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^2 - 60*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*
d*x + c)^2*tan(1/4*c)^3 + 420*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^3 + 75*s
qrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/4*c)^4 - 180*sqrt(2)*a^2*sgn(cos(-1/4*pi
+ 1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^4 - 6*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^5
- 12*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/2*c) + 45*sqrt(2)*a^2*sgn(cos(-1
/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/2*c)^2 - 21*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))
*tan(1/2*c)^4 + 6*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3*tan(1/4*c) - 306*sqrt(2)*
a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^2*tan(1/4*c) + 240*sqrt(2)*a^2*sgn(cos(-1/
4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c) + 45*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*
d*x + c)^2*tan(1/4*c)^2 - 135*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^2 - 20*s
qrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/4*c)^3 + 240*sqrt(2)*a^2*sgn(cos(-1/4*pi
+ 1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^3 - 45*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^4
+ sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^3 - 39*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d
*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^2 + 40*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^3 +
18*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)^2*tan(1/4*c) - 126*sqrt(2)*a^2*sgn(cos(-1
/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c) - 75*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4
*d*x + c)*tan(1/4*c)^2 + 180*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^2 + 20*sqrt
(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^3 - 3*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*t
an(1/4*d*x + c)^2 + 9*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2 + 6*sqrt(2)*a^2*sgn(cos(-1/
4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/4*c) - 72*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(
1/2*c)*tan(1/4*c) + 45*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^2 + 5*sqrt(2)*a^2*sgn(cos(-1
/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c) - 12*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c) - 6
*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c) - 3*sqrt(2)*a^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)
))/((tan(1/2*c)^3*tan(1/4*c)^6 + 3*tan(1/2*c)^2*tan(1/4*c)^6 + 3*tan(1/2*c)^3*tan(1/4*c)^4 - 3*tan(1/2*c)*tan(
1/4*c)^6 + 9*tan(1/2*c)^2*tan(1/4*c)^4 - tan(1/4*c)^6 + 3*tan(1/2*c)^3*tan(1/4*c)^2 - 9*tan(1/2*c)*tan(1/4*c)^
4 + 9*tan(1/2*c)^2*tan(1/4*c)^2 - 3*tan(1/4*c)^4 + tan(1/2*c)^3 - 9*tan(1/2*c)*tan(1/4*c)^2 + 3*tan(1/2*c)^2 -
3*tan(1/4*c)^2 - 3*tan(1/2*c) - 1)*(tan(1/4*d*x + c)^4*tan(1/2*c)^3 + 3*tan(1/4*d*x + c)^4*tan(1/2*c)^2 - 2*t
an(1/4*d*x + c)^3*tan(1/2*c)^3 - 3*tan(1/4*d*x + c)^4*tan(1/2*c) + 6*tan(1/4*d*x + c)^3*tan(1/2*c)^2 - tan(1/4
*d*x + c)^4 + 6*tan(1/4*d*x + c)^3*tan(1/2*c) - 2*tan(1/4*d*x + c)*tan(1/2*c)^3 - 2*tan(1/4*d*x + c)^3 + 6*tan
(1/4*d*x + c)*tan(1/2*c)^2 - tan(1/2*c)^3 + 6*tan(1/4*d*x + c)*tan(1/2*c) - 3*tan(1/2*c)^2 - 2*tan(1/4*d*x + c
) + 3*tan(1/2*c) + 1)))/d

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maple [A]  time = 0.17, size = 45, normalized size = 0.82 \[ -\frac {2 a^{3} \left (1+\sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )-3\right )}{\cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2*(a+a*sin(d*x+c))^(5/2),x)

[Out]

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

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maxima [B]  time = 1.07, size = 191, normalized size = 3.47 \[ -\frac {2 \, {\left (3 \, a^{\frac {5}{2}} - \frac {2 \, a^{\frac {5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {9 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {9 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )}}{d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

-2*(3*a^(5/2) - 2*a^(5/2)*sin(d*x + c)/(cos(d*x + c) + 1) + 9*a^(5/2)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 4*
a^(5/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 9*a^(5/2)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 2*a^(5/2)*sin(d*
x + c)^5/(cos(d*x + c) + 1)^5 + 3*a^(5/2)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)/(d*(sin(d*x + c)/(cos(d*x + c)
+ 1) - 1)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^(5/2))

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mupad [B]  time = 5.46, size = 88, normalized size = 1.60 \[ \frac {2\,a^2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (-22\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-2\,{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2+4\,\sin \left (2\,c+2\,d\,x\right )+12\right )}{d\,\left (-4\,{\sin \left (c+d\,x\right )}^2+\sin \left (c+d\,x\right )+\sin \left (3\,c+3\,d\,x\right )+4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(c + d*x))^(5/2)/cos(c + d*x)^2,x)

[Out]

(2*a^2*(a*(sin(c + d*x) + 1))^(1/2)*(4*sin(2*c + 2*d*x) - 22*sin(c/2 + (d*x)/2)^2 - 2*sin((3*c)/2 + (3*d*x)/2)
^2 + 12))/(d*(sin(c + d*x) + sin(3*c + 3*d*x) - 4*sin(c + d*x)^2 + 4))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**2*(a+a*sin(d*x+c))**(5/2),x)

[Out]

Timed out

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