3.15.79 \(\int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx\) [1479]

3.15.79.1 Optimal result

Integrand size = 35, antiderivative size = 312 \[ \int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx=\frac {\sec (e+f x) (b+a \sin (e+f x)) \sqrt {a+b \sin (e+f x)}}{f \sqrt {d \sin (e+f x)}}-\frac {(a+b)^{3/2} \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{\sqrt {d} f}-\frac {b (a+b) \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} \sqrt {\frac {b+a \csc (e+f x)}{-a+b}} E\left (\arcsin \left (\sqrt {-\frac {b+a \csc (e+f x)}{a-b}}\right )|\frac {-a+b}{a+b}\right ) (1+\sin (e+f x)) \tan (e+f x)}{f \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \]

output

sec(f*x+e)*(b+a*sin(f*x+e))*(a+b*sin(f*x+e))^(1/2)/f/(d*sin(f*x+e))^(1/2)- 
(a+b)^(3/2)*EllipticF(d^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(d*sin(f* 
x+e))^(1/2),((-a-b)/(a-b))^(1/2))*(-a*(-1+csc(f*x+e))/(a+b))^(1/2)*(a*(1+c 
sc(f*x+e))/(a-b))^(1/2)*tan(f*x+e)/f/d^(1/2)-b*(a+b)*EllipticE(((-b-a*csc( 
f*x+e))/(a-b))^(1/2),((-a+b)/(a+b))^(1/2))*(1+sin(f*x+e))*(-a*(-1+csc(f*x+ 
e))/(a+b))^(1/2)*((b+a*csc(f*x+e))/(-a+b))^(1/2)*tan(f*x+e)/f/(a*(1+csc(f* 
x+e))/(a-b))^(1/2)/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2)
 

3.15.79.2 Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(4593\) vs. \(2(312)=624\).

Time = 28.39 (sec) , antiderivative size = 4593, normalized size of antiderivative = 14.72 \[ \int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx=\text {Result too large to show} \]

input

Integrate[(Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]] 
,x]
 

output

(Csc[(e + f*x)/2]^4*Sec[(e + f*x)/2]^2*Sin[e + f*x]^4*Sqrt[a + b*Sin[e + f 
*x]]*((a*Sqrt[a + b*Sin[e + f*x]])/(2*Sqrt[Sin[e + f*x]]) - b*Sqrt[Sin[e + 
 f*x]]*Sqrt[a + b*Sin[e + f*x]])*(-2*b*Tan[(e + f*x)/2]^2 + (2*Sqrt[-a^2 + 
 b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*(-(b*E 
llipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^ 
2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + 
 f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x 
)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^ 
2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + 
 f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/((a + b*Sin[e + f*x])*Sqrt[(a*Tan[(e 
+ f*x)/2])/(-b + Sqrt[-a^2 + b^2])])))/(16*f*Sqrt[d*Sin[e + f*x]]*((b*Cos[ 
e + f*x]*Csc[(e + f*x)/2]^4*Sec[(e + f*x)/2]^2*Sin[e + f*x]^(7/2)*(-2*b*Ta 
n[(e + f*x)/2]^2 + (2*Sqrt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*S 
in[e + f*x]))/(a^2 - b^2)]*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^ 
2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/ 
(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + 
Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt 
[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt 
[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/((a 
+ b*Sin[e + f*x])*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])))...
 

3.15.79.3 Rubi [F]

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3404

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

input

Int[(Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]],x]
 

output

$Aborted
 

3.15.79.3.1 Defintions of rubi rules used

rule 3042

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

rule 3404

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Unin 
tegrable[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, 
x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[a^2 - b^2, 0]
 

3.15.79.4 Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1911\) vs. \(2(287)=574\).

Time = 2.29 (sec) , antiderivative size = 1912, normalized size of antiderivative = 6.13

method	result	size
default	\(\text {Expression too large to display}\)	\(1912\)

input

int(sec(f*x+e)^2*(a+b*sin(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2),x,method=_RET 
URNVERBOSE)
 

output

1/2/f/(a+b*sin(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)*((-a^2+b^2)^(1/2)*(1/(b+ 
(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)*(- 
(a*csc(f*x+e)-a*cot(f*x+e)-(-a^2+b^2)^(1/2)+b)/(-a^2+b^2)^(1/2))^(1/2)*(-a 
/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2 
+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^( 
1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2*cos(f*x+e)-2*(-a^2 
+b^2)^(1/2)*(1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^ 
(1/2)+b))^(1/2)*(-(a*csc(f*x+e)-a*cot(f*x+e)-(-a^2+b^2)^(1/2)+b)/(-a^2+b^2 
)^(1/2))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*Ell 
ipticE((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2) 
+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^2* 
cos(f*x+e)+2*(1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2) 
^(1/2)+b))^(1/2)*(-(a*csc(f*x+e)-a*cot(f*x+e)-(-a^2+b^2)^(1/2)+b)/(-a^2+b^ 
2)^(1/2))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*El 
lipticE((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2 
)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2 
*b*cos(f*x+e)-2*(1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b 
^2)^(1/2)+b))^(1/2)*(-(a*csc(f*x+e)-a*cot(f*x+e)-(-a^2+b^2)^(1/2)+b)/(-a^2 
+b^2)^(1/2))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2) 
*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2...
 

3.15.79.5 Fricas [F]

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

input

integrate(sec(f*x+e)^2*(a+b*sin(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2),x, algo 
rithm="fricas")
 

output

integral((b*sec(f*x + e)^2*sin(f*x + e) + a*sec(f*x + e)^2)*sqrt(b*sin(f*x 
 + e) + a)*sqrt(d*sin(f*x + e))/(d*sin(f*x + e)), x)
 

3.15.79.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \]

input

integrate(sec(f*x+e)**2*(a+b*sin(f*x+e))**(3/2)/(d*sin(f*x+e))**(1/2),x)
 

output

Timed out
 

3.15.79.7 Maxima [F]

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

input

integrate(sec(f*x+e)^2*(a+b*sin(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2),x, algo 
rithm="maxima")
 

output

integrate((b*sin(f*x + e) + a)^(3/2)*sec(f*x + e)^2/sqrt(d*sin(f*x + e)), 
x)
 

3.15.79.8 Giac [F]

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

input

integrate(sec(f*x+e)^2*(a+b*sin(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2),x, algo 
rithm="giac")
 

output

integrate((b*sin(f*x + e) + a)^(3/2)*sec(f*x + e)^2/sqrt(d*sin(f*x + e)), 
x)
 

3.15.79.9 Mupad [F(-1)]

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

input

int((a + b*sin(e + f*x))^(3/2)/(cos(e + f*x)^2*(d*sin(e + f*x))^(1/2)),x)
 

output

int((a + b*sin(e + f*x))^(3/2)/(cos(e + f*x)^2*(d*sin(e + f*x))^(1/2)), x)