Integrand size = 15, antiderivative size = 79 \[ \int \frac {1}{(a-a \sin (c+d x))^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a-a \sin (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\cos (c+d x)}{2 d (a-a \sin (c+d x))^{3/2}} \] Output:
1/4*arctanh(1/2*a^(1/2)*cos(d*x+c)*2^(1/2)/(a-a*sin(d*x+c))^(1/2))*2^(1/2) /a^(3/2)/d+1/2*cos(d*x+c)/d/(a-a*sin(d*x+c))^(3/2)
Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.54 \[ \int \frac {1}{(a-a \sin (c+d x))^{3/2}} \, dx=-\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )+(1+i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right ) (-1+\sin (c+d x))\right )}{2 a d (-1+\sin (c+d x)) \sqrt {a-a \sin (c+d x)}} \] Input:
Integrate[(a - a*Sin[c + d*x])^(-3/2),x]
Output:
-1/2*((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d *x)/2] + (1 + I)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(c + d* x)/4])]*(-1 + Sin[c + d*x])))/(a*d*(-1 + Sin[c + d*x])*Sqrt[a - a*Sin[c + d*x]])
Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3129, 3042, 3128, 219}
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.
\(\displaystyle \int \frac {1}{(a-a \sin (c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a-a \sin (c+d x))^{3/2}}dx\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {a-a \sin (c+d x)}}dx}{4 a}+\frac {\cos (c+d x)}{2 d (a-a \sin (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {a-a \sin (c+d x)}}dx}{4 a}+\frac {\cos (c+d x)}{2 d (a-a \sin (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3128 |
\(\displaystyle \frac {\cos (c+d x)}{2 d (a-a \sin (c+d x))^{3/2}}-\frac {\int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{a-a \sin (c+d x)}}d\left (-\frac {a \cos (c+d x)}{\sqrt {a-a \sin (c+d x)}}\right )}{2 a d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a-a \sin (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\cos (c+d x)}{2 d (a-a \sin (c+d x))^{3/2}}\) |
Input:
Int[(a - a*Sin[c + d*x])^(-3/2),x]
Output:
ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sin[c + d*x]])]/(2*Sqrt [2]*a^(3/2)*d) + Cos[c + d*x]/(2*d*(a - a*Sin[c + d*x])^(3/2))
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.56
method | result | size |
default | \(-\frac {\left (\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \sin \left (d x +c \right )-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}-2 \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, a^{\frac {3}{2}}\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}{4 a^{\frac {7}{2}} \cos \left (d x +c \right ) \sqrt {a -a \sin \left (d x +c \right )}\, d}\) | \(123\) |
Input:
int(1/(a-a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/4/a^(7/2)*(2^(1/2)*arctanh(1/2*(a*(1+sin(d*x+c)))^(1/2)*2^(1/2)/a^(1/2) )*a^2*sin(d*x+c)-2^(1/2)*arctanh(1/2*(a*(1+sin(d*x+c)))^(1/2)*2^(1/2)/a^(1 /2))*a^2-2*(a*(1+sin(d*x+c)))^(1/2)*a^(3/2))*(a*(1+sin(d*x+c)))^(1/2)/cos( d*x+c)/(a-a*sin(d*x+c))^(1/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (64) = 128\).
Time = 0.11 (sec) , antiderivative size = 246, normalized size of antiderivative = 3.11 \[ \int \frac {1}{(a-a \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {2} {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} \sqrt {-a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) - 4 \, \sqrt {-a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )}}{8 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - 2 \, a^{2} d + {\left (a^{2} d \cos \left (d x + c\right ) + 2 \, a^{2} d\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(1/(a-a*sin(d*x+c))^(3/2),x, algorithm="fricas")
Output:
1/8*(sqrt(2)*(cos(d*x + c)^2 + (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)*sqrt(a)*log(-(a*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(-a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) + sin(d*x + c) + 1) + 3*a*cos(d*x + c) + (a*cos( d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 + (cos(d*x + c) + 2)*s in(d*x + c) - cos(d*x + c) - 2)) - 4*sqrt(-a*sin(d*x + c) + a)*(cos(d*x + c) + sin(d*x + c) + 1))/(a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - 2*a^2 *d + (a^2*d*cos(d*x + c) + 2*a^2*d)*sin(d*x + c))
\[ \int \frac {1}{(a-a \sin (c+d x))^{3/2}} \, dx=\int \frac {1}{\left (- a \sin {\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(a-a*sin(d*x+c))**(3/2),x)
Output:
Integral((-a*sin(c + d*x) + a)**(-3/2), x)
\[ \int \frac {1}{(a-a \sin (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (-a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a-a*sin(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate((-a*sin(d*x + c) + a)^(-3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (64) = 128\).
Time = 0.18 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.61 \[ \int \frac {1}{(a-a \sin (c+d x))^{3/2}} \, dx=\frac {\frac {2 \, \sqrt {2} \log \left (-\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {\sqrt {2} {\left (\sqrt {a} - \frac {2 \, \sqrt {a} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}\right )} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}{a^{2} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {\sqrt {2} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a^{\frac {3}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{16 \, d} \] Input:
integrate(1/(a-a*sin(d*x+c))^(3/2),x, algorithm="giac")
Output:
1/16*(2*sqrt(2)*log(-(cos(-1/4*pi + 1/2*d*x + 1/2*c) - 1)/(cos(-1/4*pi + 1 /2*d*x + 1/2*c) + 1))/(a^(3/2)*sgn(sin(-1/4*pi + 1/2*d*x + 1/2*c))) + sqrt (2)*(sqrt(a) - 2*sqrt(a)*(cos(-1/4*pi + 1/2*d*x + 1/2*c) - 1)/(cos(-1/4*pi + 1/2*d*x + 1/2*c) + 1))*(cos(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^2*(cos(- 1/4*pi + 1/2*d*x + 1/2*c) - 1)*sgn(sin(-1/4*pi + 1/2*d*x + 1/2*c))) - sqrt (2)*(cos(-1/4*pi + 1/2*d*x + 1/2*c) - 1)/(a^(3/2)*(cos(-1/4*pi + 1/2*d*x + 1/2*c) + 1)*sgn(sin(-1/4*pi + 1/2*d*x + 1/2*c))))/d
Timed out. \[ \int \frac {1}{(a-a \sin (c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a-a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:
int(1/(a - a*sin(c + d*x))^(3/2),x)
Output:
int(1/(a - a*sin(c + d*x))^(3/2), x)
\[ \int \frac {1}{(a-a \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {-\sin \left (d x +c \right )+1}}{\sin \left (d x +c \right )^{2}-2 \sin \left (d x +c \right )+1}d x \right )}{a^{2}} \] Input:
int(1/(a-a*sin(d*x+c))^(3/2),x)
Output:
(sqrt(a)*int(sqrt( - sin(c + d*x) + 1)/(sin(c + d*x)**2 - 2*sin(c + d*x) + 1),x))/a**2