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]

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

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Rubi [A]  time = 0.06, antiderivative size = 30, 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} \[ \frac {2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d} \]

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*}

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Mathematica [B]  time = 5.16, size = 69, normalized size = 2.30 \[ \frac {2 (a (\sin (c+d x)+1))^{5/2}}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5} \]

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)

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fricas [A]  time = 0.84, size = 43, normalized size = 1.43 \[ -\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{2}}{3 \, {\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\right )}} \]

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))

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

[Out]

Timed out

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maple [A]  time = 0.16, size = 47, normalized size = 1.57 \[ -\frac {2 a^{3} \left (1+\sin \left (d x +c \right )\right )}{3 \left (\sin \left (d x +c \right )-1\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^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

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maxima [B]  time = 0.52, size = 184, normalized size = 6.13 \[ -\frac {2 \, {\left (a^{\frac {5}{2}} + \frac {4 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{\frac {5}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )}}{3 \, d {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac {5}{2}}} \]

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))

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mupad [B]  time = 7.59, size = 225, normalized size = 7.50 \[ \frac {4\,a^2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left ({\sin \left (c+d\,x\right )}^2\,4{}\mathrm {i}+\sin \left (c+d\,x\right )\,1{}\mathrm {i}-2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2-2\,\sin \left (2\,c+2\,d\,x\right )+\sin \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}-4{}\mathrm {i}\right )}{3\,d\,\left (8\,{\sin \left (c+d\,x\right )}^2+4\,\sin \left (c+d\,x\right )-2\,{\sin \left (2\,c+2\,d\,x\right )}^2+4\,\sin \left (3\,c+3\,d\,x\right )-8\right )}+\frac {4\,a^2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (\sin \left (2\,c+2\,d\,x\right )+4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-{\sin \left (c+d\,x\right )}^2\,2{}\mathrm {i}-2+2{}\mathrm {i}\right )}{3\,d\,\left (4\,{\sin \left (c+d\,x\right )}^2+\sin \left (c+d\,x\right )+\sin \left (3\,c+3\,d\,x\right )-4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*a^2*(a*(sin(c + d*x) + 1))^(1/2)*(sin(c + d*x)*1i - 2*sin(2*c + 2*d*x) + sin(3*c + 3*d*x)*1i - 2*sin(c/2 +
(d*x)/2)^2 + 2*sin((3*c)/2 + (3*d*x)/2)^2 + sin(c + d*x)^2*4i - 4i))/(3*d*(4*sin(c + d*x) + 4*sin(3*c + 3*d*x)
- 2*sin(2*c + 2*d*x)^2 + 8*sin(c + d*x)^2 - 8)) + (4*a^2*(a*(sin(c + d*x) + 1))^(1/2)*(sin(2*c + 2*d*x) + 4*s
in(c/2 + (d*x)/2)^2 - sin(c + d*x)^2*2i - (2 - 2i)))/(3*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)**4*(a+a*sin(d*x+c))**(5/2),x)

[Out]

Timed out

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