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)*(-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-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)
Leaf count is larger than twice the leaf count of optimal. \(5569\) vs. \(2(312)=624\).
Time = 29.72 (sec) , antiderivative size = 5569, normalized size of antiderivative = 17.85 \[ \int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx=\text {Result too large to show} \] Input:
Integrate[(Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]] ,x]
Output:
Result too large to show
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
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1018\) vs. \(2(282)=564\).
Time = 4.05 (sec) , antiderivative size = 1019, normalized size of antiderivative = 3.27
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)*((2*cos(f*x+e)+2)*(-a^2 +b^2)^(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*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)*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))+( -2*cos(f*x+e)-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*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^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))+(2*cos(f*x+e)+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*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)*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))+(-1-cos(f*x+e))*(-a^2+b^2)^(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*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^2*EllipticF(((-a*cot(f*x+e)+(-a^2+b^2)^(1...
\[ \int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{2}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \] 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)
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
\[ \int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{2}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \] 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)
\[ \int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{2}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \] 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)
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)
\[ \int \frac {\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)}} \, dx=\frac {\sqrt {d}\, \left (\left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\sin \left (f x +e \right ) b +a}\, \sec \left (f x +e \right )^{2}}{\sin \left (f x +e \right )}d x \right ) a +\left (\int \sqrt {\sin \left (f x +e \right )}\, \sqrt {\sin \left (f x +e \right ) b +a}\, \sec \left (f x +e \right )^{2}d x \right ) b \right )}{d} \] Input:
int(sec(f*x+e)^2*(a+b*sin(f*x+e))^(3/2)/(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)**2 )/sin(e + f*x),x)*a + int(sqrt(sin(e + f*x))*sqrt(sin(e + f*x)*b + a)*sec( e + f*x)**2,x)*b))/d