Integrand size = 16, antiderivative size = 101 \[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {b \cos (e+f x) \sin (e+f x)}{a (a+b) f \sqrt {a+b \sin ^2(e+f x)}}+\frac {E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{a (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}} \]
b*cos(f*x+e)*sin(f*x+e)/a/(a+b)/f/(a+b*sin(f*x+e)^2)^(1/2)+(cos(f*x+e)^2)^ (1/2)/cos(f*x+e)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*(a+b*sin(f*x+e)^2)^(1/ 2)/a/(a+b)/f/(1+b*sin(f*x+e)^2/a)^(1/2)
Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {2 a \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )+\sqrt {2} b \sin (2 (e+f x))}{2 a (a+b) f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
(2*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + S qrt[2]*b*Sin[2*(e + f*x)])/(2*a*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x) ]])
Time = 0.42 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3042, 3663, 25, 3042, 3657, 3042, 3656}
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.
\(\displaystyle \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a+b \sin (e+f x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3663 |
\(\displaystyle \frac {b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\int -\sqrt {b \sin ^2(e+f x)+a}dx}{a (a+b)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \sqrt {b \sin ^2(e+f x)+a}dx}{a (a+b)}+\frac {b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt {a+b \sin ^2(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {b \sin (e+f x)^2+a}dx}{a (a+b)}+\frac {b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt {a+b \sin ^2(e+f x)}}\) |
\(\Big \downarrow \) 3657 |
\(\displaystyle \frac {\sqrt {a+b \sin ^2(e+f x)} \int \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}dx}{a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt {a+b \sin ^2(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a+b \sin ^2(e+f x)} \int \sqrt {\frac {b \sin (e+f x)^2}{a}+1}dx}{a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt {a+b \sin ^2(e+f x)}}\) |
\(\Big \downarrow \) 3656 |
\(\displaystyle \frac {b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt {a+b \sin ^2(e+f x)}}+\frac {\sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{a f (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}\) |
(b*Cos[e + f*x]*Sin[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + ( EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(a*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])
3.2.60.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a ]/f)*EllipticE[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + b*(Sin[e + f*x]^2/a)] Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(p + 1)*(a + b))), x] + Simp[1/(2*a*(p + 1)*(a + b)) Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && LtQ[p, -1]
Time = 1.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, a E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )+\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) b}{a \left (a +b \right ) \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(103\) |
((cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*a*EllipticE(sin(f* x+e),(-1/a*b)^(1/2))+cos(f*x+e)^2*sin(f*x+e)*b)/a/(a+b)/cos(f*x+e)/(a+b*si n(f*x+e)^2)^(1/2)/f
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 938, normalized size of antiderivative = 9.29 \[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} b^{3} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (2 \, {\left (i \, b^{3} \cos \left (f x + e\right )^{2} - i \, a b^{2} - i \, b^{3}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} - {\left (2 i \, a^{2} b + 3 i \, a b^{2} + i \, b^{3} + {\left (-2 i \, a b^{2} - i \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) - {\left (2 \, {\left (-i \, b^{3} \cos \left (f x + e\right )^{2} + i \, a b^{2} + i \, b^{3}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} - {\left (-2 i \, a^{2} b - 3 i \, a b^{2} - i \, b^{3} + {\left (2 i \, a b^{2} + i \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + 2 \, {\left (2 \, {\left (-i \, a^{2} b - 2 i \, a b^{2} - i \, b^{3} + {\left (i \, a b^{2} + i \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} + {\left (2 i \, a^{3} + 3 i \, a^{2} b + i \, a b^{2} + {\left (-2 i \, a^{2} b - i \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + 2 \, {\left (2 \, {\left (i \, a^{2} b + 2 i \, a b^{2} + i \, b^{3} + {\left (-i \, a b^{2} - i \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} + {\left (-2 i \, a^{3} - 3 i \, a^{2} b - i \, a b^{2} + {\left (2 i \, a^{2} b + i \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}})}{2 \, {\left ({\left (a^{2} b^{3} + a b^{4}\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4}\right )} f\right )}} \]
-1/2*(2*sqrt(-b*cos(f*x + e)^2 + a + b)*b^3*cos(f*x + e)*sin(f*x + e) - (2 *(I*b^3*cos(f*x + e)^2 - I*a*b^2 - I*b^3)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - (2*I*a^2*b + 3*I*a*b^2 + I*b^3 + (-2*I*a*b^2 - I*b^3)*cos(f*x + e)^2)*sqr t(-b))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqr t((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + e)) ), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) - (2 *(-I*b^3*cos(f*x + e)^2 + I*a*b^2 + I*b^3)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - (-2*I*a^2*b - 3*I*a*b^2 - I*b^3 + (2*I*a*b^2 + I*b^3)*cos(f*x + e)^2)*sq rt(-b))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sq rt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e) )), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + 2 *(2*(-I*a^2*b - 2*I*a*b^2 - I*b^3 + (I*a*b^2 + I*b^3)*cos(f*x + e)^2)*sqrt (-b)*sqrt((a^2 + a*b)/b^2) + (2*I*a^3 + 3*I*a^2*b + I*a*b^2 + (-2*I*a^2*b - I*a*b^2)*cos(f*x + e)^2)*sqrt(-b))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*( cos(f*x + e) + I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sq rt((a^2 + a*b)/b^2))/b^2) + 2*(2*(I*a^2*b + 2*I*a*b^2 + I*b^3 + (-I*a*b^2 - I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) + (-2*I*a^3 - 3*I* a^2*b - I*a*b^2 + (2*I*a^2*b + I*a*b^2)*cos(f*x + e)^2)*sqrt(-b))*sqrt((2* b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt(...
\[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]