3.135 \(\int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=30 \[ \frac{2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.058527, antiderivative size = 30, normalized size of antiderivative = 1., 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} \[ \frac{2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(2*a*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(3*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 ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac{2 a \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}\\ \end{align*}

Mathematica [B]  time = 5.14894, size = 69, normalized size = 2.3 \[ \frac{2 (a (\sin (c+d x)+1))^{5/2}}{3 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(2*(a*(1 + Sin[c + d*x]))^(5/2))/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x
)/2])^5)

________________________________________________________________________________________

Maple [A]  time = 0.082, size = 47, normalized size = 1.6 \begin{align*} -{\frac{2\,{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) }{ \left ( 3\,\sin \left ( dx+c \right ) -3 \right ) \cos \left ( dx+c \right ) d}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4*(a+a*sin(d*x+c))^(5/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.65006, size = 248, normalized size = 8.27 \begin{align*} -\frac{2 \,{\left (a^{\frac{5}{2}} + \frac{4 \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a^{\frac{5}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )}}{3 \, d{\left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

-2/3*(a^(5/2) + 4*a^(5/2)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*a^(5/2)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4
+ 4*a^(5/2)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a^(5/2)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)/(d*(3*sin(d*x +
c)/(cos(d*x + c) + 1) - 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 1)*(sin
(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^(5/2))

________________________________________________________________________________________

Fricas [A]  time = 1.62536, size = 111, normalized size = 3.7 \begin{align*} -\frac{2 \, \sqrt{a \sin \left (d x + c\right ) + a} a^{2}}{3 \,{\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**4*(a+a*sin(d*x+c))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out