3.3.97 \(\int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{13/2}} \, dx\) [297]

3.3.97.1 Optimal result

Integrand size = 27, antiderivative size = 150 \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{13/2}} \, dx=-\frac {2 (a+a \sin (c+d x))^{5/2}}{d e (e \cos (c+d x))^{11/2}}+\frac {4 (a+a \sin (c+d x))^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac {16 (a+a \sin (c+d x))^{9/2}}{7 a^2 d e (e \cos (c+d x))^{11/2}}+\frac {32 (a+a \sin (c+d x))^{11/2}}{77 a^3 d e (e \cos (c+d x))^{11/2}} \]

output

-2*(a+a*sin(d*x+c))^(5/2)/d/e/(e*cos(d*x+c))^(11/2)+4*(a+a*sin(d*x+c))^(7/ 
2)/a/d/e/(e*cos(d*x+c))^(11/2)-16/7*(a+a*sin(d*x+c))^(9/2)/a^2/d/e/(e*cos( 
d*x+c))^(11/2)+32/77*(a+a*sin(d*x+c))^(11/2)/a^3/d/e/(e*cos(d*x+c))^(11/2)
 

3.3.97.2 Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.49 \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{13/2}} \, dx=\frac {2 \sqrt {e \cos (c+d x)} \sec ^6(c+d x) (a (1+\sin (c+d x)))^{5/2} \left (5+26 \sin (c+d x)-40 \sin ^2(c+d x)+16 \sin ^3(c+d x)\right )}{77 d e^7} \]

input

Integrate[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(13/2),x]
 

output

(2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^6*(a*(1 + Sin[c + d*x]))^(5/2)*(5 + 2 
6*Sin[c + d*x] - 40*Sin[c + d*x]^2 + 16*Sin[c + d*x]^3))/(77*d*e^7)
 

3.3.97.3 Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3042, 3151, 3042, 3151, 3042, 3151, 3042, 3150}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a \sin (c+d x)+a)^{5/2}}{(e \cos (c+d x))^{13/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (c+d x)+a)^{5/2}}{(e \cos (c+d x))^{13/2}}dx\)

\(\Big \downarrow \) 3151

\(\displaystyle \frac {6 \int \frac {(\sin (c+d x) a+a)^{7/2}}{(e \cos (c+d x))^{13/2}}dx}{a}-\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{11/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \int \frac {(\sin (c+d x) a+a)^{7/2}}{(e \cos (c+d x))^{13/2}}dx}{a}-\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{11/2}}\)

\(\Big \downarrow \) 3151

\(\displaystyle \frac {6 \left (\frac {2 (a \sin (c+d x)+a)^{7/2}}{3 d e (e \cos (c+d x))^{11/2}}-\frac {4 \int \frac {(\sin (c+d x) a+a)^{9/2}}{(e \cos (c+d x))^{13/2}}dx}{3 a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{11/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \left (\frac {2 (a \sin (c+d x)+a)^{7/2}}{3 d e (e \cos (c+d x))^{11/2}}-\frac {4 \int \frac {(\sin (c+d x) a+a)^{9/2}}{(e \cos (c+d x))^{13/2}}dx}{3 a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{11/2}}\)

\(\Big \downarrow \) 3151

\(\displaystyle \frac {6 \left (\frac {2 (a \sin (c+d x)+a)^{7/2}}{3 d e (e \cos (c+d x))^{11/2}}-\frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{9/2}}{7 d e (e \cos (c+d x))^{11/2}}-\frac {2 \int \frac {(\sin (c+d x) a+a)^{11/2}}{(e \cos (c+d x))^{13/2}}dx}{7 a}\right )}{3 a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{11/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {6 \left (\frac {2 (a \sin (c+d x)+a)^{7/2}}{3 d e (e \cos (c+d x))^{11/2}}-\frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{9/2}}{7 d e (e \cos (c+d x))^{11/2}}-\frac {2 \int \frac {(\sin (c+d x) a+a)^{11/2}}{(e \cos (c+d x))^{13/2}}dx}{7 a}\right )}{3 a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{11/2}}\)

\(\Big \downarrow \) 3150

\(\displaystyle \frac {6 \left (\frac {2 (a \sin (c+d x)+a)^{7/2}}{3 d e (e \cos (c+d x))^{11/2}}-\frac {4 \left (\frac {2 (a \sin (c+d x)+a)^{9/2}}{7 d e (e \cos (c+d x))^{11/2}}-\frac {4 (a \sin (c+d x)+a)^{11/2}}{77 a d e (e \cos (c+d x))^{11/2}}\right )}{3 a}\right )}{a}-\frac {2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{11/2}}\)

input

Int[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(13/2),x]
 

output

(-2*(a + a*Sin[c + d*x])^(5/2))/(d*e*(e*Cos[c + d*x])^(11/2)) + (6*((2*(a 
+ a*Sin[c + d*x])^(7/2))/(3*d*e*(e*Cos[c + d*x])^(11/2)) - (4*((2*(a + a*S 
in[c + d*x])^(9/2))/(7*d*e*(e*Cos[c + d*x])^(11/2)) - (4*(a + a*Sin[c + d* 
x])^(11/2))/(77*a*d*e*(e*Cos[c + d*x])^(11/2))))/(3*a)))/a
 

3.3.97.3.1 Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3150

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/(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 3151

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/(a*f*g*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl 
ify[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[Simpli 
fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]
 

3.3.97.4 Maple [A] (verified)

Time = 2.65 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.65

method	result	size
default	\(-\frac {2 \left (\sec ^{5}\left (d x +c \right )\right ) \left (16 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-40 \left (\cos ^{2}\left (d x +c \right )\right )-42 \sin \left (d x +c \right )+35\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, a^{2} \left (2 \sin \left (d x +c \right )-\left (\cos ^{2}\left (d x +c \right )\right )+2\right )}{77 d \sqrt {e \cos \left (d x +c \right )}\, e^{6}}\)	\(98\)

input

int((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(13/2),x,method=_RETURNVERBOSE)
 

output

-2/77/d*sec(d*x+c)^5*(16*cos(d*x+c)^2*sin(d*x+c)-40*cos(d*x+c)^2-42*sin(d* 
x+c)+35)*(a*(1+sin(d*x+c)))^(1/2)*a^2*(2*sin(d*x+c)-cos(d*x+c)^2+2)/(e*cos 
(d*x+c))^(1/2)/e^6
 

3.3.97.5 Fricas [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.80 \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{13/2}} \, dx=-\frac {2 \, {\left (40 \, a^{2} \cos \left (d x + c\right )^{2} - 35 \, a^{2} - 2 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{2} - 21 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{77 \, {\left (d e^{7} \cos \left (d x + c\right )^{4} + 2 \, d e^{7} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, d e^{7} \cos \left (d x + c\right )^{2}\right )}} \]

input

integrate((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(13/2),x, algorithm="frica 
s")
 

output

-2/77*(40*a^2*cos(d*x + c)^2 - 35*a^2 - 2*(8*a^2*cos(d*x + c)^2 - 21*a^2)* 
sin(d*x + c))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)/(d*e^7*cos(d*x 
 + c)^4 + 2*d*e^7*cos(d*x + c)^2*sin(d*x + c) - 2*d*e^7*cos(d*x + c)^2)
 

3.3.97.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{13/2}} \, dx=\text {Timed out} \]

input

integrate((a+a*sin(d*x+c))**(5/2)/(e*cos(d*x+c))**(13/2),x)
 

output

Timed out
 

3.3.97.7 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 = 2.38 \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{13/2}} \, dx=\frac {2 \, {\left (5 \, a^{\frac {5}{2}} \sqrt {e} + \frac {52 \, a^{\frac {5}{2}} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {150 \, a^{\frac {5}{2}} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {180 \, a^{\frac {5}{2}} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {180 \, a^{\frac {5}{2}} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {150 \, a^{\frac {5}{2}} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {52 \, a^{\frac {5}{2}} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {5 \, a^{\frac {5}{2}} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{77 \, {\left (e^{7} + \frac {4 \, e^{7} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, e^{7} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, e^{7} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {e^{7} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {13}{2}}} \]

input

integrate((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(13/2),x, algorithm="maxim 
a")
 

output

2/77*(5*a^(5/2)*sqrt(e) + 52*a^(5/2)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 
1) - 150*a^(5/2)*sqrt(e)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 180*a^(5/2) 
*sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 180*a^(5/2)*sqrt(e)*sin(d*x 
 + c)^5/(cos(d*x + c) + 1)^5 + 150*a^(5/2)*sqrt(e)*sin(d*x + c)^6/(cos(d*x 
 + c) + 1)^6 - 52*a^(5/2)*sqrt(e)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 5* 
a^(5/2)*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^7 + 4*e^7*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 
 6*e^7*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*e^7*sin(d*x + c)^6/(cos(d*x 
 + c) + 1)^6 + e^7*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)*d*(sin(d*x + c)/(c 
os(d*x + c) + 1) + 1)^(3/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2))
 

3.3.97.8 Giac [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{13/2}} \, dx=\text {Timed out} \]

input

integrate((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(13/2),x, algorithm="giac" 
)
 

output

Timed out
 

3.3.97.9 Mupad [B] (verification not implemented)

Time = 12.36 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.55 \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{13/2}} \, dx=\frac {30\,a^2\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-40\,a^2\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+8\,a^2\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-76\,a^2\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\frac {77\,d\,e^6\,\cos \left (3\,c+3\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}+77\,d\,e^6\,\sin \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}-\frac {385\,d\,e^6\,\cos \left (c+d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}} \]

input

int((a + a*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(13/2),x)
 

output

(30*a^2*(a + a*sin(c + d*x))^(1/2) - 40*a^2*cos(2*c + 2*d*x)*(a + a*sin(c 
+ d*x))^(1/2) + 8*a^2*sin(3*c + 3*d*x)*(a + a*sin(c + d*x))^(1/2) - 76*a^2 
*sin(c + d*x)*(a + a*sin(c + d*x))^(1/2))/((77*d*e^6*cos(3*c + 3*d*x)*((e* 
exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*1i))/2)^(1/2))/4 + 77*d*e^6*si 
n(2*c + 2*d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*1i))/2)^(1/ 
2) - (385*d*e^6*cos(c + d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d 
*x*1i))/2)^(1/2))/4)