\(\int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx\) [1476]

Optimal result

Integrand size = 35, antiderivative size = 366 \[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\frac {5 a \sec (e+f x) (b+a \sin (e+f x)) \sqrt {a+b \sin (e+f x)}}{6 f \sqrt {d \sin (e+f x)}}+\frac {\sec ^3(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{3 d f}-\frac {5 a (a+b)^{3/2} \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{6 \sqrt {d} f}-\frac {5 a b (a+b) \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} \sqrt {\frac {b+a \csc (e+f x)}{-a+b}} E\left (\arcsin \left (\sqrt {-\frac {b+a \csc (e+f x)}{a-b}}\right )|\frac {-a+b}{a+b}\right ) (1+\sin (e+f x)) \tan (e+f x)}{6 f \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \] Output:

5/6*a*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)+1/3*sec(f*x+e)^3*(d*sin(f*x+e))^(1/2)*(a+b*sin(f*x+e))^(5/2)/d/f-5/6 
*a*(a+b)^(3/2)*(-a*(-1+csc(f*x+e))/(a+b))^(1/2)*(a*(1+csc(f*x+e))/(a-b))^( 
1/2)*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))*tan(f*x+e)/d^(1/2)/f-5/6*a*b*(a+b)*(-a*(-1+csc( 
f*x+e))/(a+b))^(1/2)*((b+a*csc(f*x+e))/(-a+b))^(1/2)*EllipticE((-(b+a*csc( 
f*x+e))/(a-b))^(1/2),((-a+b)/(a+b))^(1/2))*(1+sin(f*x+e))*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)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(4665\) vs. \(2(366)=732\).

Time = 28.61 (sec) , antiderivative size = 4665, normalized size of antiderivative = 12.75 \[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\text {Result too large to show} \] Input:

Integrate[(Sec[e + f*x]^4*(a + b*Sin[e + f*x])^(5/2))/Sqrt[d*Sin[e + f*x]] 
,x]
 

Output:

(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((Sec[e + f*x]^3*(a^2 + b^2 + 2*a*b 
*Sin[e + f*x]))/3 + (Sec[e + f*x]*(5*a^2 - 2*b^2 + 5*a*b*Sin[e + f*x]))/6) 
)/(f*Sqrt[d*Sin[e + f*x]]) + (5*a*Csc[(e + f*x)/2]^4*Sec[(e + f*x)/2]^2*Si 
n[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]]*((5*a^2*Sqrt[a + b*Sin[e + f*x]])/(1 
2*Sqrt[Sin[e + f*x]]) - (5*a*b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]] 
)/6)*(-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*EllipticE[ArcSin[Sqrt[(-b + S 
qrt[-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[ArcS 
in[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])])))/(96*f*Sqrt[d*Sin[e + f*x]]*((5*a*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*Tan[(e + f*x)/2]^2 + (2*Sq 
rt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[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 + Sq...
 

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.

\(\displaystyle \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \sin (e+f x))^{5/2}}{\cos (e+f x)^4 \sqrt {d \sin (e+f x)}}dx\)

\(\Big \downarrow \) 3367

\(\displaystyle \frac {5}{6} a \int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}}dx+\frac {\sec ^3(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{3 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5}{6} a \int \frac {(a+b \sin (e+f x))^{3/2}}{\cos (e+f x)^2 \sqrt {d \sin (e+f x)}}dx+\frac {\sec ^3(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{3 d f}\)

\(\Big \downarrow \) 3404

\(\displaystyle \frac {5}{6} a \int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}}dx+\frac {\sec ^3(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{3 d f}\)

Input:

Int[(Sec[e + f*x]^4*(a + b*Sin[e + f*x])^(5/2))/Sqrt[d*Sin[e + f*x]],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 3042

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

rule 3367

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*( 
x_)])^(m_))/Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*(g*Cos 
[e + f*x])^(p + 1)*Sqrt[d*Sin[e + f*x]]*((a + b*Sin[e + f*x])^m/(d*f*g*(2*m 
 + 1))), x] + Simp[2*a*(m/(g^2*(2*m + 1)))   Int[(g*Cos[e + f*x])^(p + 2)*( 
(a + b*Sin[e + f*x])^(m - 1)/Sqrt[d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, 
e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && EqQ[m + p + 3/2, 0]
 

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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1086\) vs. \(2(324)=648\).

Time = 6.00 (sec) , antiderivative size = 1087, normalized size of antiderivative = 2.97

method	result	size
default	\(\text {Expression too large to display}\)	\(1087\)

Input:

int(sec(f*x+e)^4*(a+b*sin(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x,method=_RET 
URNVERBOSE)
 

Output:

1/12/f/(a+b*sin(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)*(-2^(1/2)*(5*cos(f*x+e) 
+5-4*sec(f*x+e)-6*sec(f*x+e)^3)*a^2*b+2^(1/2)*a*b^2*(-5*sin(f*x+e)-tan(f*x 
+e)+6*tan(f*x+e)*sec(f*x+e)^2)+2^(1/2)*a^3*(5*tan(f*x+e)+2*tan(f*x+e)*sec( 
f*x+e)^2)+2*sin(f*x+e)*tan(f*x+e)^3*2^(1/2)*b^3+(-10*cos(f*x+e)-10)*(-a^2+ 
b^2)^(1/2)*(-(-a*cot(f*x+e)-(-a^2+b^2)^(1/2)+b+a*csc(f*x+e))/(-a^2+b^2)^(1 
/2))^(1/2)*(-(csc(f*x+e)-cot(f*x+e))/(b+(-a^2+b^2)^(1/2))*a)^(1/2)*((-a*co 
t(f*x+e)+(-a^2+b^2)^(1/2)+b+a*csc(f*x+e))/(b+(-a^2+b^2)^(1/2)))^(1/2)*b^2* 
EllipticE(((-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b+a*csc(f*x+e))/(b+(-a^2+b^2)^( 
1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))+(1 
0*cos(f*x+e)+10)*(-(-a*cot(f*x+e)-(-a^2+b^2)^(1/2)+b+a*csc(f*x+e))/(-a^2+b 
^2)^(1/2))^(1/2)*(-(csc(f*x+e)-cot(f*x+e))/(b+(-a^2+b^2)^(1/2))*a)^(1/2)*( 
(-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b+a*csc(f*x+e))/(b+(-a^2+b^2)^(1/2)))^(1/2 
)*a^2*b*EllipticE(((-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b+a*csc(f*x+e))/(b+(-a^ 
2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^( 
1/2))+(-10*cos(f*x+e)-10)*(-(-a*cot(f*x+e)-(-a^2+b^2)^(1/2)+b+a*csc(f*x+e) 
)/(-a^2+b^2)^(1/2))^(1/2)*(-(csc(f*x+e)-cot(f*x+e))/(b+(-a^2+b^2)^(1/2))*a 
)^(1/2)*((-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b+a*csc(f*x+e))/(b+(-a^2+b^2)^(1/ 
2)))^(1/2)*b^3*EllipticE(((-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b+a*csc(f*x+e))/ 
(b+(-a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^( 
1/2))^(1/2))+(5*cos(f*x+e)+5)*(-a^2+b^2)^(1/2)*(-(-a*cot(f*x+e)-(-a^2+b...
 

Fricas [F]

\[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sec \left (f x + e\right )^{4}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \] Input:

integrate(sec(f*x+e)^4*(a+b*sin(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algo 
rithm="fricas")
 

Output:

integral((2*a*b*sec(f*x + e)^4*sin(f*x + e) - (b^2*cos(f*x + e)^2 - a^2 - 
b^2)*sec(f*x + e)^4)*sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e))/(d*sin( 
f*x + e)), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \] Input:

integrate(sec(f*x+e)**4*(a+b*sin(f*x+e))**(5/2)/(d*sin(f*x+e))**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sec \left (f x + e\right )^{4}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \] Input:

integrate(sec(f*x+e)^4*(a+b*sin(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algo 
rithm="maxima")
 

Output:

integrate((b*sin(f*x + e) + a)^(5/2)*sec(f*x + e)^4/sqrt(d*sin(f*x + e)), 
x)
 

Giac [F]

\[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sec \left (f x + e\right )^{4}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \] Input:

integrate(sec(f*x+e)^4*(a+b*sin(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algo 
rithm="giac")
 

Output:

integrate((b*sin(f*x + e) + a)^(5/2)*sec(f*x + e)^4/sqrt(d*sin(f*x + e)), 
x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\cos \left (e+f\,x\right )}^4\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \] Input:

int((a + b*sin(e + f*x))^(5/2)/(cos(e + f*x)^4*(d*sin(e + f*x))^(1/2)),x)
 

Output:

int((a + b*sin(e + f*x))^(5/2)/(cos(e + f*x)^4*(d*sin(e + f*x))^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt {d \sin (e+f x)}} \, dx=\frac {\sqrt {d}\, \left (\left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\sin \left (f x +e \right ) b +a}\, \sec \left (f x +e \right )^{4}}{\sin \left (f x +e \right )}d x \right ) a^{2}+\left (\int \sqrt {\sin \left (f x +e \right )}\, \sqrt {\sin \left (f x +e \right ) b +a}\, \sec \left (f x +e \right )^{4} \sin \left (f x +e \right )d x \right ) b^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )}\, \sqrt {\sin \left (f x +e \right ) b +a}\, \sec \left (f x +e \right )^{4}d x \right ) a b \right )}{d} \] Input:

int(sec(f*x+e)^4*(a+b*sin(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x)
 

Output:

(sqrt(d)*(int((sqrt(sin(e + f*x))*sqrt(sin(e + f*x)*b + a)*sec(e + f*x)**4 
)/sin(e + f*x),x)*a**2 + int(sqrt(sin(e + f*x))*sqrt(sin(e + f*x)*b + a)*s 
ec(e + f*x)**4*sin(e + f*x),x)*b**2 + 2*int(sqrt(sin(e + f*x))*sqrt(sin(e 
+ f*x)*b + a)*sec(e + f*x)**4,x)*a*b))/d