\(\int \frac {\sin (7+3 x)}{\sqrt {3+\sin ^2(7+3 x)}} \, dx\) [115]

Optimal result

Integrand size = 21, antiderivative size = 15 \[ \int \frac {\sin (7+3 x)}{\sqrt {3+\sin ^2(7+3 x)}} \, dx=-\frac {1}{3} \arcsin \left (\frac {1}{2} \cos (7+3 x)\right ) \]

[Out]

-1/3*arcsin(1/2*cos(7+3*x))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3265, 222} \[ \int \frac {\sin (7+3 x)}{\sqrt {3+\sin ^2(7+3 x)}} \, dx=-\frac {1}{3} \arcsin \left (\frac {1}{2} \cos (3 x+7)\right ) \]

[In]

Int[Sin[7 + 3*x]/Sqrt[3 + Sin[7 + 3*x]^2],x]

[Out]

-1/3*ArcSin[Cos[7 + 3*x]/2]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 3265

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {4-x^2}} \, dx,x,\cos (7+3 x)\right )\right ) \\ & = -\frac {1}{3} \arcsin \left (\frac {1}{2} \cos (7+3 x)\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.60 \[ \int \frac {\sin (7+3 x)}{\sqrt {3+\sin ^2(7+3 x)}} \, dx=\frac {1}{3} i \log \left (i \sqrt {2} \cos (7+3 x)+\sqrt {7-\cos (2 (7+3 x))}\right ) \]

[In]

Integrate[Sin[7 + 3*x]/Sqrt[3 + Sin[7 + 3*x]^2],x]

[Out]

(I/3)*Log[I*Sqrt[2]*Cos[7 + 3*x] + Sqrt[7 - Cos[2*(7 + 3*x)]]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(11)=22\).

Time = 0.70 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.80

method	result	size
default	\(-\frac {\sqrt {\left (3+\sin ^{2}\left (7+3 x \right )\right ) \left (\cos ^{2}\left (7+3 x \right )\right )}\, \arcsin \left (-1+\frac {\left (\cos ^{2}\left (7+3 x \right )\right )}{2}\right )}{6 \cos \left (7+3 x \right ) \sqrt {3+\sin ^{2}\left (7+3 x \right )}}\)	\(57\)

[In]

int(sin(7+3*x)/(3+sin(7+3*x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/6*((3+sin(7+3*x)^2)*cos(7+3*x)^2)^(1/2)*arcsin(-1+1/2*cos(7+3*x)^2)/cos(7+3*x)/(3+sin(7+3*x)^2)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (11) = 22\).

Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 6.27 \[ \int \frac {\sin (7+3 x)}{\sqrt {3+\sin ^2(7+3 x)}} \, dx=\frac {1}{6} \, \arctan \left (-\frac {4 \, \cos \left (3 \, x + 7\right ) \sin \left (3 \, x + 7\right ) - {\left (\cos \left (3 \, x + 7\right )^{3} - 2 \, \cos \left (3 \, x + 7\right )\right )} \sqrt {-\cos \left (3 \, x + 7\right )^{2} + 4}}{\cos \left (3 \, x + 7\right )^{4} - 8 \, \cos \left (3 \, x + 7\right )^{2} + 4}\right ) - \frac {1}{6} \, \arctan \left (\frac {\sin \left (3 \, x + 7\right )}{\cos \left (3 \, x + 7\right )}\right ) \]

[In]

integrate(sin(7+3*x)/(3+sin(7+3*x)^2)^(1/2),x, algorithm="fricas")

[Out]

1/6*arctan(-(4*cos(3*x + 7)*sin(3*x + 7) - (cos(3*x + 7)^3 - 2*cos(3*x + 7))*sqrt(-cos(3*x + 7)^2 + 4))/(cos(3
*x + 7)^4 - 8*cos(3*x + 7)^2 + 4)) - 1/6*arctan(sin(3*x + 7)/cos(3*x + 7))

Sympy [F]

\[ \int \frac {\sin (7+3 x)}{\sqrt {3+\sin ^2(7+3 x)}} \, dx=\int \frac {\sin {\left (3 x + 7 \right )}}{\sqrt {\sin ^{2}{\left (3 x + 7 \right )} + 3}}\, dx \]

[In]

integrate(sin(7+3*x)/(3+sin(7+3*x)**2)**(1/2),x)

[Out]

Integral(sin(3*x + 7)/sqrt(sin(3*x + 7)**2 + 3), x)

Maxima [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {\sin (7+3 x)}{\sqrt {3+\sin ^2(7+3 x)}} \, dx=-\frac {1}{3} \, \arcsin \left (\frac {1}{2} \, \cos \left (3 \, x + 7\right )\right ) \]

[In]

integrate(sin(7+3*x)/(3+sin(7+3*x)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/3*arcsin(1/2*cos(3*x + 7))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (11) = 22\).

Time = 0.53 (sec) , antiderivative size = 48, normalized size of antiderivative = 3.20 \[ \int \frac {\sin (7+3 x)}{\sqrt {3+\sin ^2(7+3 x)}} \, dx=\frac {2}{3} \, \arctan \left (-\frac {1}{2} \, \sqrt {3} \tan \left (\frac {3}{2} \, x + \frac {7}{2}\right )^{2} - \frac {1}{2} \, \sqrt {3} + \frac {1}{2} \, \sqrt {3 \, \tan \left (\frac {3}{2} \, x + \frac {7}{2}\right )^{4} + 10 \, \tan \left (\frac {3}{2} \, x + \frac {7}{2}\right )^{2} + 3}\right ) \]

[In]

integrate(sin(7+3*x)/(3+sin(7+3*x)^2)^(1/2),x, algorithm="giac")

[Out]

2/3*arctan(-1/2*sqrt(3)*tan(3/2*x + 7/2)^2 - 1/2*sqrt(3) + 1/2*sqrt(3*tan(3/2*x + 7/2)^4 + 10*tan(3/2*x + 7/2)
^2 + 3))

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin (7+3 x)}{\sqrt {3+\sin ^2(7+3 x)}} \, dx=\int \frac {\sin \left (3\,x+7\right )}{\sqrt {{\sin \left (3\,x+7\right )}^2+3}} \,d x \]

[In]

int(sin(3*x + 7)/(sin(3*x + 7)^2 + 3)^(1/2),x)

[Out]

int(sin(3*x + 7)/(sin(3*x + 7)^2 + 3)^(1/2), x)