Integrand size = 21, antiderivative size = 37 \[ \int \frac {\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {b} \sqrt {a+b} d} \]
Result contains complex when optimal does not.
Time = 0.86 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.62 \[ \int \frac {\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\arctan \left (\frac {\sqrt {b}-i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )+\arctan \left (\frac {\sqrt {b}+i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )}{\sqrt {-a-b} \sqrt {b} d} \]
(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]] + ArcTan[(Sqr t[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]])/(Sqrt[-a - b]*Sqrt[b]*d)
Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3665, 221}
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.
\(\displaystyle \int \frac {\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x)}{a+b \sin (c+d x)^2}dx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle -\frac {\int \frac {1}{-b \cos ^2(c+d x)+a+b}d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {b} d \sqrt {a+b}}\) |
3.1.81.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-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]
Time = 0.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(-\frac {\operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{d \sqrt {\left (a +b \right ) b}}\) | \(29\) |
default | \(-\frac {\operatorname {arctanh}\left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{d \sqrt {\left (a +b \right ) b}}\) | \(29\) |
risch | \(\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{2 \sqrt {-a b -b^{2}}\, d}-\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{2 \sqrt {-a b -b^{2}}\, d}\) | \(116\) |
Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.16 \[ \int \frac {\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx=\left [\frac {\log \left (-\frac {b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a b + b^{2}} \cos \left (d x + c\right ) + a + b}{b \cos \left (d x + c\right )^{2} - a - b}\right )}{2 \, \sqrt {a b + b^{2}} d}, \frac {\sqrt {-a b - b^{2}} \arctan \left (\frac {\sqrt {-a b - b^{2}} \cos \left (d x + c\right )}{a + b}\right )}{{\left (a b + b^{2}\right )} d}\right ] \]
[1/2*log(-(b*cos(d*x + c)^2 - 2*sqrt(a*b + b^2)*cos(d*x + c) + a + b)/(b*c os(d*x + c)^2 - a - b))/(sqrt(a*b + b^2)*d), sqrt(-a*b - b^2)*arctan(sqrt( -a*b - b^2)*cos(d*x + c)/(a + b))/((a*b + b^2)*d)]
Leaf count of result is larger than twice the leaf count of optimal. 367693 vs. \(2 (34) = 68\).
Time = 76.85 (sec) , antiderivative size = 367693, normalized size of antiderivative = 9937.65 \[ \int \frac {\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx=\text {Too large to display} \]
Piecewise((zoo*x/sin(c), Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (log(tan(c/2 + d *x/2))/(b*d), Eq(a, 0)), (2/(b*d*tan(c/2 + d*x/2)**2 - b*d), Eq(a, -b)), ( -cos(c + d*x)/(a*d), Eq(b, 0)), (x*sin(c)/(a + b*sin(c)**2), Eq(d, 0)), (7 4*a**37*b*log(-sqrt(-1 - 2*b/a - 2*sqrt(a*b + b**2)/a) + tan(c/2 + d*x/2)) /(2*a**38*b*d + 5478*a**37*b**2*d - 148*a**37*b*d*sqrt(a*b + b**2) + 25025 32*a**36*b**3*d - 135124*a**36*b**2*d*sqrt(a*b + b**2) + 456961248*a**35*b **4*d - 36983424*a**35*b**3*d*sqrt(a*b + b**2) + 44602414272*a**34*b**5*d - 4809599808*a**34*b**4*d*sqrt(a*b + b**2) + 2698911348224*a**33*b**6*d - 363524561920*a**33*b**5*d*sqrt(a*b + b**2) + 110776036340736*a**32*b**7*d - 17891931206656*a**32*b**6*d*sqrt(a*b + b**2) + 3275718126403584*a**31*b* *8*d - 616808259780608*a**31*b**7*d*sqrt(a*b + b**2) + 72854727629602816*a **30*b**9*d - 15666762815766528*a**30*b**8*d*sqrt(a*b + b**2) + 1258467596 957384704*a**29*b**10*d - 304230303833522176*a**29*b**9*d*sqrt(a*b + b**2) + 17306140891880620032*a**28*b**11*d - 4645206174395269120*a**28*b**10*d* sqrt(a*b + b**2) + 193199008739227598848*a**27*b**12*d - 57001938802859573 248*a**27*b**11*d*sqrt(a*b + b**2) + 1778515685235870400512*a**26*b**13*d - 572029907419376123904*a**26*b**12*d*sqrt(a*b + b**2) + 13673782930644613 988352*a**25*b**14*d - 4761020109769125396480*a**25*b**13*d*sqrt(a*b + b** 2) + 88722183139577965838336*a**24*b**15*d - 33244276082712682430464*a**24 *b**14*d*sqrt(a*b + b**2) + 490030319626953299066880*a**23*b**16*d - 19...
Time = 0.40 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.35 \[ \int \frac {\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\log \left (\frac {b \cos \left (d x + c\right ) - \sqrt {{\left (a + b\right )} b}}{b \cos \left (d x + c\right ) + \sqrt {{\left (a + b\right )} b}}\right )}{2 \, \sqrt {{\left (a + b\right )} b} d} \]
1/2*log((b*cos(d*x + c) - sqrt((a + b)*b))/(b*cos(d*x + c) + sqrt((a + b)* b)))/(sqrt((a + b)*b)*d)
Time = 0.38 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\arctan \left (\frac {b \cos \left (d x + c\right )}{\sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} d} \]
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int \frac {\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,\cos \left (c+d\,x\right )}{\sqrt {a+b}}\right )}{\sqrt {b}\,d\,\sqrt {a+b}} \]