\(\int \frac {\sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\) [63]

Optimal result

Integrand size = 21, antiderivative size = 72 \[ \int \frac {\sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{\sqrt {a} d}-\frac {2 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}} \]

[Out]

arctanh(1/2*cos(d*x+c)*a^(1/2)*2^(1/2)/(a+a*sin(d*x+c))^(1/2))*2^(1/2)/d/a^(1/2)-2*cos(d*x+c)/d/(a+a*sin(d*x+c
))^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2830, 2728, 212} \[ \int \frac {\sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {2 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}} \]

[In]

Int[Sin[c + d*x]/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d) - (2*Cos[c + d*x])/(d
*Sqrt[a + a*Sin[c + d*x]])

Rule 212

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

Rule 2728

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, b*(C
os[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2830

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d
)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*S
in[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m
, -2^(-1)]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}-\int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {2 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {2 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d} \\ & = \frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{\sqrt {a} d}-\frac {2 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.36 \[ \int \frac {\sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {2 \left ((1+i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right )+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d \sqrt {a (1+\sin (c+d x))}} \]

[In]

Integrate[Sin[c + d*x]/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-2*((1 + I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + Cos[(c + d*x)/2] - Sin[(c +
d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Sin[c + d*x])])

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.33

method	result	size
default	\(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (-\sqrt {a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )+2 \sqrt {a -a \sin \left (d x +c \right )}\right )}{a \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(96\)
risch	\(-\frac {\left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2} \sqrt {2}\, {\mathrm e}^{-i \left (d x +c \right )}}{d \sqrt {-a \left (i {\mathrm e}^{2 i \left (d x +c \right )}-i-2 \,{\mathrm e}^{i \left (d x +c \right )}\right ) {\mathrm e}^{-i \left (d x +c \right )}}}+\frac {2 i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) \left (\arctan \left (\frac {\sqrt {-i a \,{\mathrm e}^{i \left (d x +c \right )}}}{\sqrt {a}}\right ) a \sqrt {-i a \,{\mathrm e}^{i \left (d x +c \right )}}+a^{\frac {3}{2}}\right ) \sqrt {2}\, {\mathrm e}^{-i \left (d x +c \right )}}{d \,a^{\frac {3}{2}} \sqrt {-a \left (i {\mathrm e}^{2 i \left (d x +c \right )}-i-2 \,{\mathrm e}^{i \left (d x +c \right )}\right ) {\mathrm e}^{-i \left (d x +c \right )}}}\)	\(199\)

[In]

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

[Out]

-(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(-a^(1/2)*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2)
)+2*(a-a*sin(d*x+c))^(1/2))/a/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (61) = 122\).

Time = 0.30 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.65 \[ \int \frac {\sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\frac {\sqrt {2} {\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + \frac {2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt {a}} - 4 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{2 \, {\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\right )}} \]

[In]

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

[Out]

1/2*(sqrt(2)*(a*cos(d*x + c) + a*sin(d*x + c) + a)*log(-(cos(d*x + c)^2 - (cos(d*x + c) - 2)*sin(d*x + c) + 2*
sqrt(2)*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/sqrt(a) + 3*cos(d*x + c) + 2)/(cos(d*x + c)
^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2))/sqrt(a) - 4*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) -
sin(d*x + c) + 1))/(a*d*cos(d*x + c) + a*d*sin(d*x + c) + a*d)

Sympy [F]

\[ \int \frac {\sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {\sin {\left (c + d x \right )}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \]

[In]

integrate(sin(d*x+c)/(a+a*sin(d*x+c))**(1/2),x)

[Out]

Integral(sin(c + d*x)/sqrt(a*(sin(c + d*x) + 1)), x)

Maxima [F]

\[ \int \frac {\sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\sin \left (d x + c\right )}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \]

[In]

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

[Out]

integrate(sin(d*x + c)/sqrt(a*sin(d*x + c) + a), x)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.64 \[ \int \frac {\sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {\frac {\sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {\sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {4 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{2 \, d} \]

[In]

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

[Out]

-1/2*(sqrt(2)*log(sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(sqrt(a)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - sqrt(2)*
log(-sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(sqrt(a)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 4*sqrt(2)*sin(-1/4*pi
+ 1/2*d*x + 1/2*c)/(sqrt(a)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))))/d

Mupad [B] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.38 \[ \int \frac {\sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {\left (4\,\mathrm {E}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (c+d\,x\right )}}{2}\right )\middle |1\right )-2\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (c+d\,x\right )}}{2}\right )\middle |1\right )\right )\,\sqrt {{\cos \left (c+d\,x\right )}^2}\,\sqrt {\frac {a+a\,\sin \left (c+d\,x\right )}{2\,a}}}{d\,\cos \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \]

[In]

int(sin(c + d*x)/(a + a*sin(c + d*x))^(1/2),x)

[Out]

-((4*ellipticE(asin((2^(1/2)*(1 - sin(c + d*x))^(1/2))/2), 1) - 2*ellipticF(asin((2^(1/2)*(1 - sin(c + d*x))^(
1/2))/2), 1))*(cos(c + d*x)^2)^(1/2)*((a + a*sin(c + d*x))/(2*a))^(1/2))/(d*cos(c + d*x)*(a + a*sin(c + d*x))^
(1/2))