3.1.81 \(\int \frac {\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx\) [81]
Optimal. Leaf size=37 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {b} \sqrt {a+b} d} \]
[Out]
-arctanh(cos(d*x+c)*b^(1/2)/(a+b)^(1/2))/d/b^(1/2)/(a+b)^(1/2)
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Rubi [A]
time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3265, 214}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {b} d \sqrt {a+b}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sin[c + d*x]/(a + b*Sin[c + d*x]^2),x]
[Out]
-(ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b]*d))
Rule 214
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]
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*} \int \frac {\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {b} \sqrt {a+b} d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.09, size = 97, normalized size = 2.62 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b}-i \sqrt {a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a-b}}\right )+\tan ^{-1}\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} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sin[c + d*x]/(a + b*Sin[c + d*x]^2),x]
[Out]
(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/S
qrt[-a - b]])/(Sqrt[-a - b]*Sqrt[b]*d)
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Maple [A]
time = 0.17, size = 29, normalized size = 0.78
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(-\frac {\arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{d \sqrt {\left (a +b \right ) b}}\) |
\(29\) |
| default |
\(-\frac {\arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (a +b \right ) b}}\right )}{d \sqrt {\left (a +b \right ) b}}\) |
\(29\) |
| risch |
\(\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{2 \sqrt {-a b -b^{2}}\, d}-\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (d x +c \right )}}{\sqrt {-a b -b^{2}}}+1\right )}{2 \sqrt {-a b -b^{2}}\, d}\) |
\(116\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(d*x+c)/(a+sin(d*x+c)^2*b),x,method=_RETURNVERBOSE)
[Out]
-1/d/((a+b)*b)^(1/2)*arctanh(b*cos(d*x+c)/((a+b)*b)^(1/2))
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Maxima [A]
time = 0.54, size = 50, normalized size = 1.35 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)/(a+b*sin(d*x+c)^2),x, algorithm="maxima")
[Out]
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)
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Fricas [A]
time = 0.41, size = 117, normalized size = 3.16 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)/(a+b*sin(d*x+c)^2),x, algorithm="fricas")
[Out]
[1/2*log(-(b*cos(d*x + c)^2 - 2*sqrt(a*b + b^2)*cos(d*x + c) + a + b)/(b*cos(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)]
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 367693 vs.
\(2 (34) = 68\).
time = 86.31, size = 367693, normalized size = 9937.65 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)/(a+b*sin(d*x+c)**2),x)
[Out]
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)),
(74*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) + 2502532*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) + 12
58467596957384704*a**29*b**10*d - 304230303833522176*a**29*b**9*d*sqrt(a*b + b**2) + 17306140891880620032*a**2
8*b**11*d - 4645206174395269120*a**28*b**10*d*sqrt(a*b + b**2) + 193199008739227598848*a**27*b**12*d - 5700193
8802859573248*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) + 13673782930644613988352*a**25*b**14*d - 4761020109769125396480*a**25*b**13*d*sq
rt(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 - 196589323247525507891200*a**23*b**15*d*sqrt(a*b + b**2) + 2320264
659880999460536320*a**22*b**17*d - 992185245208510642257920*a**22*b**16*d*sqrt(a*b + b**2) + 94732374230503145
65550080*a**21*b**18*d - 4301031135604201236725760*a**21*b**17*d*sqrt(a*b + b**2) + 33508135815008970573086720
*a**20*b**19*d - 16096759227611109665013760*a**20*b**18*d*sqrt(a*b + b**2) + 103066001007281297455841280*a**19
*b**20*d - 52224655042483407940485120*a**19*b**19*d*sqrt(a*b + b**2) + 276460659949410743463444480*a**18*b**21
*d - 147353756231010598099353600*a**18*b**20*d*sqrt(a*b + b**2) + 648017072918162395858206720*a**17*b**22*d -
362406002426292494841937920*a**17*b**21*d*sqrt(a*b + b**2) + 1328967367029204488741191680*a**16*b**23*d - 7780
70309276565523251855360*a**16*b**22*d*sqrt(a*b + b**2) + 2385704631316968523085905920*a**15*b**24*d - 14592140
48234341782006005760*a**15*b**23*d*sqrt(a*b + b**2) + 3747529655246565869805895680*a**14*b**25*d - 23901430764
92438985108357120*a**14*b**24*d*sqrt(a*b + b**2) + 5144960127422757831513735168*a**13*b**26*d - 34157267952978
30225523507200*a**13*b**25*d*sqrt(a*b + b**2) + 6160343368926179873261617152*a**12*b**27*d - 42504345736272201
70723295232*a**12*b**26*d*sqrt(a*b + b**2) + 6412553052386662194178162688*a**11*b**28*d - 45913902003618170204
75375616*a**11*b**27*d*sqrt(a*b + b**2) + 5777443964131114181336236032*a**10*b**29*d - 42868479174145184964448
09216*a**10*b**28*d*sqrt(a*b + b**2) + 4478521132381256779508482048*a**9*b**30*d - 343931694395295527387476787
2*a**9*b**29*d*sqrt(a*b + b**2) + 2963524367539447964941418496*a**8*b**31*d - 2352691778556737395705774080*a**
8*b**30*d*sqrt(a*b + b**2) + 1656691644402709111026745344*a**7*b**32*d - 1358109508879153777946394624*a**7*b**
31*d*sqrt(a*b + b**2) + 771639221199942160467623936*a**6*b**33*d - 652517637350019802209452032*a**6*b**32*d*sq
rt(a*b + b**2) + 293857629218373137558667264*a**5*b**34*d - 256080136201453725194125312*a**5*b**33*d*sqrt(a*b
+ b**2) + 89102865177381478141526016*a**4*b**35*d - 79945311123381698013691904*a**4*b**34*d*sqrt(a*b + b**2) +
20683374899158687330336768*a**3*b**36*d - 19090166507000540776366080*a**3*b**35*d*sqrt(a*b + b**2) + 34508693
07356993239908352*a**2*b**37*d - 3273780564249381544394752*a**2*b**36*d*sqrt(a*b + b**2) + 3683445856638323266
68288*a*b**38*d - 358899852698093036240896*a*b**37*d*sqrt(a*b + b**2) + 18889465931478580854784*b**39*d - 1888
9465931478580854784*b**38*d*sqrt(a*b + b**2)) + 74*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) + 2502532*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) + 327571812
6403584*a**31*b**8*d - 616808259780608*a**31*b**7*d*sqrt(a*b + b**2) + 72854727629602816*a**30*b**9*d - 156667
62815766528*a**30*b**8*d*sqrt(a*b + b**2) + 1258467596957384704*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 - 57001938802859573248*a**27*b**11*d*sqrt(a*b + b**2) + 17785156852358704
00512*a**26*b**13*d - 572029907419376123904*a**26*b**12*d*sqrt(a*b + b**2) + 13673782930644613988352*a**25*b**
14*d - 4761020109769125396480*a**25*b**13*d*sqr...
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Giac [A]
time = 0.44, size = 37, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (\frac {b \cos \left (d x + c\right )}{\sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)/(a+b*sin(d*x+c)^2),x, algorithm="giac")
[Out]
arctan(b*cos(d*x + c)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*d)
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Mupad [B]
time = 0.09, size = 29, normalized size = 0.78 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,\cos \left (c+d\,x\right )}{\sqrt {a+b}}\right )}{\sqrt {b}\,d\,\sqrt {a+b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(c + d*x)/(a + b*sin(c + d*x)^2),x)
[Out]
-atanh((b^(1/2)*cos(c + d*x))/(a + b)^(1/2))/(b^(1/2)*d*(a + b)^(1/2))
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