3.122 \(\int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\)
Optimal. Leaf size=26 \[ \frac {2 a \sec (c+d x) \sqrt {a \sin (c+d x)+a}}{d} \]
[Out]
2*a*sec(d*x+c)*(a+a*sin(d*x+c))^(1/2)/d
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Rubi [A] time = 0.06, antiderivative size = 26, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used =
{2673} \[ \frac {2 a \sec (c+d x) \sqrt {a \sin (c+d x)+a}}{d} \]
Antiderivative was successfully verified.
[In]
Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]
[Out]
(2*a*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/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]
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac {2 a \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{d}\\ \end {align*}
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Mathematica [B] time = 0.16, size = 67, normalized size = 2.58 \[ \frac {2 (a (\sin (c+d x)+1))^{3/2}}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
[In]
Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]
[Out]
(2*(a*(1 + Sin[c + d*x]))^(3/2))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]
)^3)
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fricas [A] time = 0.65, size = 26, normalized size = 1.00 \[ \frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} a}{d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)^2*(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas")
[Out]
2*sqrt(a*sin(d*x + c) + a)*a/(d*cos(d*x + c))
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giac [B] time = 155.91, size = 3663, normalized size = 140.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)^2*(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")
[Out]
1/2*sqrt(2)*sqrt(a)*(sqrt(2)*(sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^6 - 15*sqr
t(2)*a*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*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*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*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*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*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*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^5 - sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x
+ 1/2*c))*tan(1/2*c)^3 + 18*sqrt(2)*a*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*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*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*
c))*tan(1/4*c)^3 + 3*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c) - 6*sqrt(2)*a*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*ta
n(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)) - 4*(6*
sqrt(2)*a*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 - 3*sqrt(2)*a*sgn(cos
(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^5*tan(1/4*c)^6 - 18*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d
*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^5*tan(1/4*c)^5 + 6*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(
1/2*c)^6*tan(1/4*c)^5 + 9*sqrt(2)*a*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 - 3*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^5*tan(1/4*c)^6 - 20*sqrt(2)*a*sgn(cos(-1/4*p
i + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^6*tan(1/4*c)^3 + 45*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/
2*c))*tan(1/4*d*x + c)*tan(1/2*c)^5*tan(1/4*c)^4 - 36*sqrt(2)*a*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 + 18*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^5*tan(1/4*c)^5
+ 10*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^3*tan(1/4*c)^6 - 9*sqrt(2)*a*sg
n(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^4*tan(1/4*c)^6 + 60*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))
*tan(1/4*d*x + c)*tan(1/2*c)^5*tan(1/4*c)^3 - 20*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^6*ta
n(1/4*c)^3 - 135*sqrt(2)*a*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 + 45
*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^5*tan(1/4*c)^4 + 60*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*
d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^3*tan(1/4*c)^5 - 36*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*ta
n(1/2*c)^4*tan(1/4*c)^5 - 6*sqrt(2)*a*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 + 10*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^3*tan(1/4*c)^6 + 6*sqrt(2)*a*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) - 45*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/
2*c))*tan(1/4*d*x + c)*tan(1/2*c)^5*tan(1/4*c)^2 + 120*sqrt(2)*a*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 - 60*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^5*tan(1/4*c)^3
- 150*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^3*tan(1/4*c)^4 + 135*sqrt(2)*
a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^4*tan(1/4*c)^4 + 54*sqrt(2)*a*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 - 60*sqrt(2)*a*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*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)*tan(1/4*c)^6 + 6*
sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^6 - 18*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d
*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^5*tan(1/4*c) + 6*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/
2*c)^6*tan(1/4*c) + 135*sqrt(2)*a*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 - 45*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^5*tan(1/4*c)^2 - 200*sqrt(2)*a*sgn(cos(-1/4*p
i + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^3*tan(1/4*c)^3 + 120*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1
/2*c))*tan(1/2*c)^4*tan(1/4*c)^3 + 90*sqrt(2)*a*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 - 150*sqrt(2)*a*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*
sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)*tan(1/4*c)^5 + 54*sqrt(2)*a*sgn(cos(-1/4*pi +
1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^5 + sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*t
an(1/4*c)^6 - 3*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^6 + 3*sqrt(2)*a*sgn(cos(-1
/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^5 - 36*sqrt(2)*a*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) + 18*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^5*tan(1/4*
c) + 150*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^3*tan(1/4*c)^2 - 135*sqrt(2
)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^4*tan(1/4*c)^2 - 180*sqrt(2)*a*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 + 200*sqrt(2)*a*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*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)*tan(1/4*c)^4
- 90*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^4 + 18*sqrt(2)*a*sgn(cos(-1/4*pi +
1/2*d*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^5 - sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^6 - 9*sq
rt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^4 + 3*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*
d*x + 1/2*c))*tan(1/2*c)^5 + 60*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^3*ta
n(1/4*c) - 36*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^4*tan(1/4*c) - 90*sqrt(2)*a*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 + 150*sqrt(2)*a*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*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/
2*c)*tan(1/4*c)^3 - 180*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^3 - 15*sqrt(2)*a
*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/4*c)^4 + 45*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x +
1/2*c))*tan(1/2*c)*tan(1/4*c)^4 - 10*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)
^3 + 9*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^4 + 54*sqrt(2)*a*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) - 60*sqrt(2)*a*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*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)*tan(1/4*c)^2 + 90*
sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c)^2 - 60*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d
*x + 1/2*c))*tan(1/2*c)*tan(1/4*c)^3 + 15*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^4 + 6*sqrt(
2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)^2 - 10*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*
x + 1/2*c))*tan(1/2*c)^3 - 18*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c)*tan(1/
4*c) + 54*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2*tan(1/4*c) + 15*sqrt(2)*a*sgn(cos(-1/4*pi
+ 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/4*c)^2 - 45*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*
c)*tan(1/4*c)^2 + 3*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*d*x + c)*tan(1/2*c) - 6*sqrt(2)*a*sg
n(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)^2 + 18*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c)*t
an(1/4*c) - 15*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/4*c)^2 - sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*
x + 1/2*c))*tan(1/4*d*x + c) + 3*sqrt(2)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*tan(1/2*c) + sqrt(2)*a*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)^2*tan(1/2*c)^3 + 3*tan(1/4*d*x
+ c)^2*tan(1/2*c)^2 - 2*tan(1/4*d*x + c)*tan(1/2*c)^3 - 3*tan(1/4*d*x + c)^2*tan(1/2*c) + 6*tan(1/4*d*x + c)*
tan(1/2*c)^2 - tan(1/2*c)^3 - tan(1/4*d*x + c)^2 + 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 = 0.14, size = 37, normalized size = 1.42 \[ \frac {2 a^{2} \left (1+\sin \left (d x +c \right )\right )}{\cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(d*x+c)^2*(a+a*sin(d*x+c))^(3/2),x)
[Out]
2*a^2*(1+sin(d*x+c))/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d
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maxima [B] time = 2.51, size = 98, normalized size = 3.77 \[ -\frac {2 \, {\left (a^{\frac {3}{2}} + \frac {2 \, a^{\frac {3}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{\frac {3}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )}}{d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)^2*(a+a*sin(d*x+c))^(3/2),x, algorithm="maxima")
[Out]
-2*(a^(3/2) + 2*a^(3/2)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^(3/2)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)/(d*
(sin(d*x + c)/(cos(d*x + c) + 1) - 1)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^(3/2))
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mupad [B] time = 4.77, size = 37, normalized size = 1.42 \[ \frac {4\,a\,\cos \left (c+d\,x\right )\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}}{d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + a*sin(c + d*x))^(3/2)/cos(c + d*x)^2,x)
[Out]
(4*a*cos(c + d*x)*(a*(sin(c + d*x) + 1))^(1/2))/(d*(cos(2*c + 2*d*x) + 1))
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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))**(3/2),x)
[Out]
Timed out
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