\(\int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx\) [279]

Optimal result

Integrand size = 27, antiderivative size = 154 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=-\frac {2 \sqrt {a+a \sin (c+d x)}}{5 d e (e \cos (c+d x))^{7/2}}-\frac {12 (a+a \sin (c+d x))^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}+\frac {16 (a+a \sin (c+d x))^{5/2}}{5 a^2 d e (e \cos (c+d x))^{7/2}}-\frac {32 (a+a \sin (c+d x))^{7/2}}{35 a^3 d e (e \cos (c+d x))^{7/2}} \]

[Out]

-12/5*(a+a*sin(d*x+c))^(3/2)/a/d/e/(e*cos(d*x+c))^(7/2)+16/5*(a+a*sin(d*x+c))^(5/2)/a^2/d/e/(e*cos(d*x+c))^(7/
2)-32/35*(a+a*sin(d*x+c))^(7/2)/a^3/d/e/(e*cos(d*x+c))^(7/2)-2/5*(a+a*sin(d*x+c))^(1/2)/d/e/(e*cos(d*x+c))^(7/
2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2751, 2750} \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=-\frac {32 (a \sin (c+d x)+a)^{7/2}}{35 a^3 d e (e \cos (c+d x))^{7/2}}+\frac {16 (a \sin (c+d x)+a)^{5/2}}{5 a^2 d e (e \cos (c+d x))^{7/2}}-\frac {12 (a \sin (c+d x)+a)^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}-\frac {2 \sqrt {a \sin (c+d x)+a}}{5 d e (e \cos (c+d x))^{7/2}} \]

[In]

Int[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(9/2),x]

[Out]

(-2*Sqrt[a + a*Sin[c + d*x]])/(5*d*e*(e*Cos[c + d*x])^(7/2)) - (12*(a + a*Sin[c + d*x])^(3/2))/(5*a*d*e*(e*Cos
[c + d*x])^(7/2)) + (16*(a + a*Sin[c + d*x])^(5/2))/(5*a^2*d*e*(e*Cos[c + d*x])^(7/2)) - (32*(a + a*Sin[c + d*
x])^(7/2))/(35*a^3*d*e*(e*Cos[c + d*x])^(7/2))

Rule 2750

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

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), 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] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a+a \sin (c+d x)}}{5 d e (e \cos (c+d x))^{7/2}}+\frac {6 \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx}{5 a} \\ & = -\frac {2 \sqrt {a+a \sin (c+d x)}}{5 d e (e \cos (c+d x))^{7/2}}-\frac {12 (a+a \sin (c+d x))^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}+\frac {24 \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{9/2}} \, dx}{5 a^2} \\ & = -\frac {2 \sqrt {a+a \sin (c+d x)}}{5 d e (e \cos (c+d x))^{7/2}}-\frac {12 (a+a \sin (c+d x))^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}+\frac {16 (a+a \sin (c+d x))^{5/2}}{5 a^2 d e (e \cos (c+d x))^{7/2}}-\frac {16 \int \frac {(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{9/2}} \, dx}{5 a^3} \\ & = -\frac {2 \sqrt {a+a \sin (c+d x)}}{5 d e (e \cos (c+d x))^{7/2}}-\frac {12 (a+a \sin (c+d x))^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}+\frac {16 (a+a \sin (c+d x))^{5/2}}{5 a^2 d e (e \cos (c+d x))^{7/2}}-\frac {32 (a+a \sin (c+d x))^{7/2}}{35 a^3 d e (e \cos (c+d x))^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=\frac {2 \sqrt {e \cos (c+d x)} \sec ^4(c+d x) \sqrt {a (1+\sin (c+d x))} (-5-4 \cos (2 (c+d x))+10 \sin (c+d x)+4 \sin (3 (c+d x)))}{35 d e^5} \]

[In]

Integrate[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(9/2),x]

[Out]

(2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^4*Sqrt[a*(1 + Sin[c + d*x])]*(-5 - 4*Cos[2*(c + d*x)] + 10*Sin[c + d*x] +
4*Sin[3*(c + d*x)]))/(35*d*e^5)

Maple [A] (verified)

Time = 2.74 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.48

method	result	size
default	\(\frac {2 \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \left (16 \tan \left (d x +c \right )-8 \sec \left (d x +c \right )+6 \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )-\left (\sec ^{3}\left (d x +c \right )\right )\right )}{35 d \sqrt {e \cos \left (d x +c \right )}\, e^{4}}\)	\(74\)

[In]

int((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(9/2),x,method=_RETURNVERBOSE)

[Out]

2/35/d*(a*(1+sin(d*x+c)))^(1/2)/(e*cos(d*x+c))^(1/2)/e^4*(16*tan(d*x+c)-8*sec(d*x+c)+6*tan(d*x+c)*sec(d*x+c)^2
-sec(d*x+c)^3)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.45 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=-\frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (8 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (8 \, \cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) + 1\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{35 \, d e^{5} \cos \left (d x + c\right )^{4}} \]

[In]

integrate((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(9/2),x, algorithm="fricas")

[Out]

-2/35*sqrt(e*cos(d*x + c))*(8*cos(d*x + c)^2 - 2*(8*cos(d*x + c)^2 + 3)*sin(d*x + c) + 1)*sqrt(a*sin(d*x + c)
+ a)/(d*e^5*cos(d*x + c)^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(d*x+c))**(1/2)/(e*cos(d*x+c))**(9/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (130) = 260\).

Time = 0.32 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.32 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=-\frac {2 \, {\left (9 \, \sqrt {a} \sqrt {e} - \frac {44 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {14 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {84 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {84 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {14 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {44 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {9 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{35 \, {\left (e^{5} + \frac {4 \, e^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, e^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, e^{5} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {e^{5} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}}} \]

[In]

integrate((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(9/2),x, algorithm="maxima")

[Out]

-2/35*(9*sqrt(a)*sqrt(e) - 44*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) - 14*sqrt(a)*sqrt(e)*sin(d*x + c
)^2/(cos(d*x + c) + 1)^2 + 84*sqrt(a)*sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 84*sqrt(a)*sqrt(e)*sin(d*x
+ c)^5/(cos(d*x + c) + 1)^5 + 14*sqrt(a)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 44*sqrt(a)*sqrt(e)*sin
(d*x + c)^7/(cos(d*x + c) + 1)^7 - 9*sqrt(a)*sqrt(e)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)*(sin(d*x + c)^2/(cos
(d*x + c) + 1)^2 + 1)^4/((e^5 + 4*e^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*e^5*sin(d*x + c)^4/(cos(d*x + c)
+ 1)^4 + 4*e^5*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + e^5*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)*d*(sin(d*x + c)
/(cos(d*x + c) + 1) + 1)^(7/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2))

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(9/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 7.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx=-\frac {16\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (23\,\cos \left (c+d\,x\right )+11\,\cos \left (3\,c+3\,d\,x\right )+2\,\cos \left (5\,c+5\,d\,x\right )-16\,\sin \left (2\,c+2\,d\,x\right )-11\,\sin \left (4\,c+4\,d\,x\right )-2\,\sin \left (6\,c+6\,d\,x\right )\right )}{35\,d\,e^4\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (15\,\cos \left (2\,c+2\,d\,x\right )+6\,\cos \left (4\,c+4\,d\,x\right )+\cos \left (6\,c+6\,d\,x\right )+10\right )} \]

[In]

int((a + a*sin(c + d*x))^(1/2)/(e*cos(c + d*x))^(9/2),x)

[Out]

-(16*(a*(sin(c + d*x) + 1))^(1/2)*(23*cos(c + d*x) + 11*cos(3*c + 3*d*x) + 2*cos(5*c + 5*d*x) - 16*sin(2*c + 2
*d*x) - 11*sin(4*c + 4*d*x) - 2*sin(6*c + 6*d*x)))/(35*d*e^4*(e*cos(c + d*x))^(1/2)*(15*cos(2*c + 2*d*x) + 6*c
os(4*c + 4*d*x) + cos(6*c + 6*d*x) + 10))