\(\int \frac {1}{\sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx\) [787]

Optimal result

Integrand size = 29, antiderivative size = 399 \[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 (3-b) \sqrt {3+b} d E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(b c-3 d)^2 (c-d) \sqrt {c+d} f}+\frac {2 \sqrt {3+b} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(b c-3 d) (c-d) \sqrt {c+d} f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2880, 2897, 3075} \[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}+\frac {2 d (a-b) \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)^2} \]

[In]

Int[1/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)),x]

[Out]

(2*(a - b)*Sqrt[a + b]*d*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e
+ f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(
c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e +
f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)^2*f) + (2*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e +
f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c
- a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c
+ d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)*f)

Rule 2880

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :
> Dist[1/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] - Dist[b/(a - b), Int[(1 +
Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &
& NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2897

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Si
mp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e +
f*x])/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/((a - b)*(c + d*Sin[e + f*x]))
)]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c -
d)/((a - b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c
^2 - d^2, 0] && PosQ[(c + d)/(a + b)]

Rule 3075

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin
[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c +
d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d)
)*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin
[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; FreeQ[{a, b, c, d, e, f, A,
B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{c-d}-\frac {d \int \frac {1+\sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx}{c-d} \\ & = \frac {2 (a-b) \sqrt {a+b} d E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(c-d) \sqrt {c+d} (b c-a d)^2 f}+\frac {2 \sqrt {a+b} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(c-d) \sqrt {c+d} (b c-a d) f} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(77809\) vs. \(2(399)=798\).

Time = 38.58 (sec) , antiderivative size = 77809, normalized size of antiderivative = 195.01 \[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\text {Result too large to show} \]

[In]

Integrate[1/(Sqrt[3 + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)),x]

[Out]

Result too large to show

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(36776\) vs. \(2(375)=750\).

Time = 11.44 (sec) , antiderivative size = 36777, normalized size of antiderivative = 92.17

method	result	size
default	\(\text {Expression too large to display}\)	\(36777\)

[In]

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

[Out]

result too large to display

Fricas [F]

\[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{\sqrt {a + b \sin {\left (e + f x \right )}} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral(1/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**(3/2)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{\sqrt {a+b\,\sin \left (e+f\,x\right )}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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