\(\int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx\) [541]

Optimal result

Integrand size = 23, antiderivative size = 91 \[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\frac {2 b}{d e \sqrt {e \cos (c+d x)}}-\frac {2 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {\cos (c+d x)}}+\frac {2 a \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}} \] Output:

2*b/d/e/(e*cos(d*x+c))^(1/2)-2*a*(e*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d* 
x+1/2*c),2^(1/2))/d/e^2/cos(d*x+c)^(1/2)+2*a*sin(d*x+c)/d/e/(e*cos(d*x+c)) 
^(1/2)
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.59 \[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\frac {2 \left (b-a \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+a \sin (c+d x)\right )}{d e \sqrt {e \cos (c+d x)}} \] Input:

Integrate[(a + b*Sin[c + d*x])/(e*Cos[c + d*x])^(3/2),x]
 

Output:

(2*(b - a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + a*Sin[c + d*x]))/ 
(d*e*Sqrt[e*Cos[c + d*x]])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 3148, 3042, 3116, 3042, 3121, 3042, 3119}

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.

\(\displaystyle \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}}dx\)

\(\Big \downarrow \) 3148

\(\displaystyle a \int \frac {1}{(e \cos (c+d x))^{3/2}}dx+\frac {2 b}{d e \sqrt {e \cos (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle a \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx+\frac {2 b}{d e \sqrt {e \cos (c+d x)}}\)

\(\Big \downarrow \) 3116

\(\displaystyle a \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \cos (c+d x)}dx}{e^2}\right )+\frac {2 b}{d e \sqrt {e \cos (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle a \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2}\right )+\frac {2 b}{d e \sqrt {e \cos (c+d x)}}\)

\(\Big \downarrow \) 3121

\(\displaystyle a \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{e^2 \sqrt {\cos (c+d x)}}\right )+\frac {2 b}{d e \sqrt {e \cos (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle a \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \sqrt {\cos (c+d x)}}\right )+\frac {2 b}{d e \sqrt {e \cos (c+d x)}}\)

\(\Big \downarrow \) 3119

\(\displaystyle a \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}}\right )+\frac {2 b}{d e \sqrt {e \cos (c+d x)}}\)

Input:

Int[(a + b*Sin[c + d*x])/(e*Cos[c + d*x])^(3/2),x]
 

Output:

(2*b)/(d*e*Sqrt[e*Cos[c + d*x]]) + a*((-2*Sqrt[e*Cos[c + d*x]]*EllipticE[( 
c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (2*Sin[c + d*x])/(d*e*Sqrt[e* 
Cos[c + d*x]]))
 

Defintions of rubi rules used

rule 3042

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

rule 3116

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1))   I 
nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && 
 IntegerQ[2*n]
 

rule 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* 
(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3121

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) 
^n/Sin[c + d*x]^n   Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt 
Q[-1, n, 1] && IntegerQ[2*n]
 

rule 3148

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + 
 Simp[a   Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && 
 (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
 

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.31

method	result	size
default	\(\frac {4 a \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a +2 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )}{e \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d}\)	\(119\)
parts	\(-\frac {2 a \left (-2 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} e +\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} e +\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{e \sqrt {-e \left (2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\, d}+\frac {2 b}{d e \sqrt {e \cos \left (d x +c \right )}}\)	\(219\)

Input:

int((a+b*sin(d*x+c))/(e*cos(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2/e/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/sin(1/2*d*x+1/2*c)*(2*a*sin(1/2*d* 
x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+ 
1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a+b*sin(1/2*d*x+1/ 
2*c))/d
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.22 \[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left (i \, \sqrt {\frac {1}{2}} a \sqrt {e} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - i \, \sqrt {\frac {1}{2}} a \sqrt {e} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + b\right )}\right )}}{d e^{2} \cos \left (d x + c\right )} \] Input:

integrate((a+b*sin(d*x+c))/(e*cos(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

-2*(I*sqrt(1/2)*a*sqrt(e)*cos(d*x + c)*weierstrassZeta(-4, 0, weierstrassP 
Inverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - I*sqrt(1/2)*a*sqrt(e)*cos 
(d*x + c)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - 
 I*sin(d*x + c))) - sqrt(e*cos(d*x + c))*(a*sin(d*x + c) + b))/(d*e^2*cos( 
d*x + c))
 

Sympy [F]

\[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\int \frac {a + b \sin {\left (c + d x \right )}}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((a+b*sin(d*x+c))/(e*cos(d*x+c))**(3/2),x)
 

Output:

Integral((a + b*sin(c + d*x))/(e*cos(c + d*x))**(3/2), x)
 

Maxima [F]

\[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\int { \frac {b \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b*sin(d*x+c))/(e*cos(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*sin(d*x + c) + a)/(e*cos(d*x + c))^(3/2), x)
 

Giac [F]

\[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\int { \frac {b \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b*sin(d*x+c))/(e*cos(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

integrate((b*sin(d*x + c) + a)/(e*cos(d*x + c))^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\int \frac {a+b\,\sin \left (c+d\,x\right )}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:

int((a + b*sin(c + d*x))/(e*cos(c + d*x))^(3/2),x)
 

Output:

int((a + b*sin(c + d*x))/(e*cos(c + d*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\frac {\sqrt {e}\, \left (\cos \left (d x +c \right ) \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a d +2 \sqrt {\cos \left (d x +c \right )}\, b \right )}{\cos \left (d x +c \right ) d \,e^{2}} \] Input:

int((a+b*sin(d*x+c))/(e*cos(d*x+c))^(3/2),x)
 

Output:

(sqrt(e)*(cos(c + d*x)*int(sqrt(cos(c + d*x))/cos(c + d*x)**2,x)*a*d + 2*s 
qrt(cos(c + d*x))*b))/(cos(c + d*x)*d*e**2)