3.5 \(\int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx\)

Optimal. Leaf size=45 \[ \frac {\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 a f \sqrt {c-c \sin (e+f x)}} \]

[Out]

1/2*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/a/f/(c-c*sin(f*x+e))^(1/2)

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Rubi [A]  time = 0.28, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2841, 2738} \[ \frac {\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 a f \sqrt {c-c \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/Sqrt[c - c*Sin[e + f*x]],x]

[Out]

(Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*a*f*Sqrt[c - c*Sin[e + f*x]])

Rule 2738

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

Rule 2841

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*
(x_)])^(n_.), x_Symbol] :> Dist[1/(a^(p/2)*c^(p/2)), Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(
n + p/2), x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p
/2]

Rubi steps

\begin {align*} \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx &=\frac {\int (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \, dx}{a c}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 a f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 62, normalized size = 1.38 \[ -\frac {\sec (e+f x) \sqrt {a (\sin (e+f x)+1)} \sqrt {c-c \sin (e+f x)} (\cos (2 (e+f x))-4 \sin (e+f x))}{4 c f} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/Sqrt[c - c*Sin[e + f*x]],x]

[Out]

-1/4*(Sec[e + f*x]*(Cos[2*(e + f*x)] - 4*Sin[e + f*x])*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])/(c
*f)

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fricas [A]  time = 0.42, size = 59, normalized size = 1.31 \[ -\frac {{\left (\cos \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) - 1\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{2 \, c f \cos \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

-1/2*(cos(f*x + e)^2 - 2*sin(f*x + e) - 1)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c*f*cos(f*x + e
))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (8*pi/x/2)>(-8*pi/
x/2)4*sqrt(2*a)*((sqrt(c*tan(1/2*exp(1))^2+c)*(-324259173170675712*tan(1/2*exp(1))^7+1188950301625810944*tan(1
/2*exp(1))^5-684547143360315392*tan(1/2*exp(1))^3+108086391056891904*tan(1/2*exp(1)))+sqrt(c*tan(1/2*exp(1))^2
+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))^2*(40532396646334464*tan(1/2*exp(1))^8-8106479
3292668928*tan(1/2*exp(1))^7-378302368699121664*tan(1/2*exp(1))^6+297237575406452736*tan(1/2*exp(1))^5+5674535
53048682496*tan(1/2*exp(1))^4-171136785840078848*tan(1/2*exp(1))^3-162129586585337856*tan(1/2*exp(1))^2+270215
97764222976*tan(1/2*exp(1))+4503599627370496)+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1
/2*(1/2*f*x+2*exp(1))))^3*(1125899906842624*tan(1/2*exp(1))^9-6755399441055744*tan(1/2*exp(1))^8-4053239664633
4464*tan(1/2*exp(1))^7+103582791429521408*tan(1/2*exp(1))^6+141863388262170624*tan(1/2*exp(1))^5-1351079888211
14880*tan(1/2*exp(1))^4-94575592174780416*tan(1/2*exp(1))^3+40532396646334464*tan(1/2*exp(1))^2+10133099161583
616*tan(1/2*exp(1))-2251799813685248)+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*
f*x+2*exp(1))))*(-4503599627370496*tan(1/2*exp(1))^9-27021597764222976*tan(1/2*exp(1))^8+162129586585337856*ta
n(1/2*exp(1))^7+414331165718085632*tan(1/2*exp(1))^6-567453553048682496*tan(1/2*exp(1))^5-540431955284459520*t
an(1/2*exp(1))^4+378302368699121664*tan(1/2*exp(1))^3+162129586585337856*tan(1/2*exp(1))^2-40532396646334464*t
an(1/2*exp(1))-9007199254740992)+sqrt(c*tan(1/2*exp(1))^2+c)*(324259173170675712*tan(1/2*exp(1))^7-11889503016
25810944*tan(1/2*exp(1))^5+684547143360315392*tan(1/2*exp(1))^3-108086391056891904*tan(1/2*exp(1)))*tan(1/4*ex
p(1))^6+sqrt(c*tan(1/2*exp(1))^2+c)*(-648518346341351424*tan(1/2*exp(1))^8+4107282860161892352*tan(1/2*exp(1))
^6-7133701809754865664*tan(1/2*exp(1))^4+1945555039024054272*tan(1/2*exp(1))^2)*tan(1/4*exp(1))^5+sqrt(c*tan(1
/2*exp(1))^2+c)*(2161727821137838080*tan(1/2*exp(1))^8-13690942867206307840*tan(1/2*exp(1))^6+2377900603251621
8880*tan(1/2*exp(1))^4-6485183463413514240*tan(1/2*exp(1))^2)*tan(1/4*exp(1))^3+sqrt(c*tan(1/2*exp(1))^2+c)*(4
863887597560135680*tan(1/2*exp(1))^7-17834254524387164160*tan(1/2*exp(1))^5+10268207150404730880*tan(1/2*exp(1
))^3-1621295865853378560*tan(1/2*exp(1)))*tan(1/4*exp(1))^2+sqrt(c*tan(1/2*exp(1))^2+c)*(-648518346341351424*t
an(1/2*exp(1))^8+4107282860161892352*tan(1/2*exp(1))^6-7133701809754865664*tan(1/2*exp(1))^4+19455550390240542
72*tan(1/2*exp(1))^2)*tan(1/4*exp(1))+sqrt(c*tan(1/2*exp(1))^2+c)*(-4863887597560135680*tan(1/2*exp(1))^7+1783
4254524387164160*tan(1/2*exp(1))^5-10268207150404730880*tan(1/2*exp(1))^3+1621295865853378560*tan(1/2*exp(1)))
*tan(1/4*exp(1))^4+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))^2*(
90071992547409920*tan(1/2*exp(1))^9+540431955284459520*tan(1/2*exp(1))^8-3242591731706757120*tan(1/2*exp(1))^7
-3422735716801576960*tan(1/2*exp(1))^6+11349071060973649920*tan(1/2*exp(1))^5+5944751508129054720*tan(1/2*exp(
1))^4-7566047373982433280*tan(1/2*exp(1))^3-1621295865853378560*tan(1/2*exp(1))^2+810647932926689280*tan(1/2*e
xp(1)))*tan(1/4*exp(1))^3+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1)
)))^2*(607985949695016960*tan(1/2*exp(1))^8-1215971899390033920*tan(1/2*exp(1))^7-5674535530486824960*tan(1/2*
exp(1))^6+4458563631096791040*tan(1/2*exp(1))^5+8511803295730237440*tan(1/2*exp(1))^4-2567051787601182720*tan(
1/2*exp(1))^3-2431943798780067840*tan(1/2*exp(1))^2+405323966463344640*tan(1/2*exp(1))+67553994410557440)*tan(
1/4*exp(1))^4+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))^2*(-2702
1597764222976*tan(1/2*exp(1))^9-162129586585337856*tan(1/2*exp(1))^8+972777519512027136*tan(1/2*exp(1))^7+1026
820715040473088*tan(1/2*exp(1))^6-3404721318292094976*tan(1/2*exp(1))^5-1783425452438716416*tan(1/2*exp(1))^4+
2269814212194729984*tan(1/2*exp(1))^3+486388759756013568*tan(1/2*exp(1))^2-243194379878006784*tan(1/2*exp(1)))
*tan(1/4*exp(1))^5+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))^2*(
-27021597764222976*tan(1/2*exp(1))^9-162129586585337856*tan(1/2*exp(1))^8+972777519512027136*tan(1/2*exp(1))^7
+1026820715040473088*tan(1/2*exp(1))^6-3404721318292094976*tan(1/2*exp(1))^5-1783425452438716416*tan(1/2*exp(1
))^4+2269814212194729984*tan(1/2*exp(1))^3+486388759756013568*tan(1/2*exp(1))^2-243194379878006784*tan(1/2*exp
(1)))*tan(1/4*exp(1))+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))^
2*(-40532396646334464*tan(1/2*exp(1))^8+81064793292668928*tan(1/2*exp(1))^7+378302368699121664*tan(1/2*exp(1))
^6-297237575406452736*tan(1/2*exp(1))^5-567453553048682496*tan(1/2*exp(1))^4+171136785840078848*tan(1/2*exp(1)
)^3+162129586585337856*tan(1/2*exp(1))^2-27021597764222976*tan(1/2*exp(1))-4503599627370496)*tan(1/4*exp(1))^6
+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))^2*(-60798594969501696
0*tan(1/2*exp(1))^8+1215971899390033920*tan(1/2*exp(1))^7+5674535530486824960*tan(1/2*exp(1))^6-44585636310967
91040*tan(1/2*exp(1))^5-8511803295730237440*tan(1/2*exp(1))^4+2567051787601182720*tan(1/2*exp(1))^3+2431943798
780067840*tan(1/2*exp(1))^2-405323966463344640*tan(1/2*exp(1))-67553994410557440)*tan(1/4*exp(1))^2+sqrt(c*tan
(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))^3*(16888498602639360*tan(1/2*exp
(1))^9-101330991615836160*tan(1/2*exp(1))^8-607985949695016960*tan(1/2*exp(1))^7+1553741871442821120*tan(1/2*e
xp(1))^6+2127950823932559360*tan(1/2*exp(1))^5-2026619832316723200*tan(1/2*exp(1))^4-1418633882621706240*tan(1
/2*exp(1))^3+607985949695016960*tan(1/2*exp(1))^2+151996487423754240*tan(1/2*exp(1))-33776997205278720)*tan(1/
4*exp(1))^4+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))^3*(4503599
6273704960*tan(1/2*exp(1))^9-202661983231672320*tan(1/2*exp(1))^8-810647932926689280*tan(1/2*exp(1))^7+1891511
843495608320*tan(1/2*exp(1))^6+2702159776422297600*tan(1/2*exp(1))^5-2837267765243412480*tan(1/2*exp(1))^4-207
1655828590428160*tan(1/2*exp(1))^3+810647932926689280*tan(1/2*exp(1))^2+135107988821114880*tan(1/2*exp(1))-225
17998136852480)*tan(1/4*exp(1))^3+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+
2*exp(1))))^3*(-1125899906842624*tan(1/2*exp(1))^9+6755399441055744*tan(1/2*exp(1))^8+40532396646334464*tan(1/
2*exp(1))^7-103582791429521408*tan(1/2*exp(1))^6-141863388262170624*tan(1/2*exp(1))^5+135107988821114880*tan(1
/2*exp(1))^4+94575592174780416*tan(1/2*exp(1))^3-40532396646334464*tan(1/2*exp(1))^2-10133099161583616*tan(1/2
*exp(1))+2251799813685248)*tan(1/4*exp(1))^6+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/
2*(1/2*f*x+2*exp(1))))^3*(-13510798882111488*tan(1/2*exp(1))^9+60798594969501696*tan(1/2*exp(1))^8+24319437987
8006784*tan(1/2*exp(1))^7-567453553048682496*tan(1/2*exp(1))^6-810647932926689280*tan(1/2*exp(1))^5+8511803295
73023744*tan(1/2*exp(1))^4+621496748577128448*tan(1/2*exp(1))^3-243194379878006784*tan(1/2*exp(1))^2-405323966
46334464*tan(1/2*exp(1))+6755399441055744)*tan(1/4*exp(1))^5+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*e
xp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))^3*(-13510798882111488*tan(1/2*exp(1))^9+60798594969501696*tan(1/2*exp(1
))^8+243194379878006784*tan(1/2*exp(1))^7-567453553048682496*tan(1/2*exp(1))^6-810647932926689280*tan(1/2*exp(
1))^5+851180329573023744*tan(1/2*exp(1))^4+621496748577128448*tan(1/2*exp(1))^3-243194379878006784*tan(1/2*exp
(1))^2-40532396646334464*tan(1/2*exp(1))+6755399441055744)*tan(1/4*exp(1))+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/
2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))^3*(-16888498602639360*tan(1/2*exp(1))^9+10133099161583616
0*tan(1/2*exp(1))^8+607985949695016960*tan(1/2*exp(1))^7-1553741871442821120*tan(1/2*exp(1))^6-212795082393255
9360*tan(1/2*exp(1))^5+2026619832316723200*tan(1/2*exp(1))^4+1418633882621706240*tan(1/2*exp(1))^3-60798594969
5016960*tan(1/2*exp(1))^2-151996487423754240*tan(1/2*exp(1))+33776997205278720)*tan(1/4*exp(1))^2+sqrt(c*tan(1
/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))*(4503599627370496*tan(1/2*exp(1))^
9+27021597764222976*tan(1/2*exp(1))^8-162129586585337856*tan(1/2*exp(1))^7-414331165718085632*tan(1/2*exp(1))^
6+567453553048682496*tan(1/2*exp(1))^5+540431955284459520*tan(1/2*exp(1))^4-378302368699121664*tan(1/2*exp(1))
^3-162129586585337856*tan(1/2*exp(1))^2+40532396646334464*tan(1/2*exp(1))+9007199254740992)*tan(1/4*exp(1))^6+
sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))*(67553994410557440*tan
(1/2*exp(1))^9+405323966463344640*tan(1/2*exp(1))^8-2431943798780067840*tan(1/2*exp(1))^7-6214967485771284480*
tan(1/2*exp(1))^6+8511803295730237440*tan(1/2*exp(1))^5+8106479329266892800*tan(1/2*exp(1))^4-5674535530486824
960*tan(1/2*exp(1))^3-2431943798780067840*tan(1/2*exp(1))^2+607985949695016960*tan(1/2*exp(1))[phone]
880)*tan(1/4*exp(1))^2+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))
*(180143985094819840*tan(1/2*exp(1))^9+810647932926689280*tan(1/2*exp(1))^8-3242591731706757120*tan(1/2*exp(1)
)^7-7566047373982433280*tan(1/2*exp(1))^6+10808639105689190400*tan(1/2*exp(1))^5+11349071060973649920*tan(1/2*
exp(1))^4-8286623314361712640*tan(1/2*exp(1))^3-3242591731706757120*tan(1/2*exp(1))^2+540431955284459520*tan(1
/2*exp(1))+90071992547409920)*tan(1/4*exp(1))^3+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan
(1/2*(1/2*f*x+2*exp(1))))*(-54043195528445952*tan(1/2*exp(1))^9-243194379878006784*tan(1/2*exp(1))^8+972777519
512027136*tan(1/2*exp(1))^7+2269814212194729984*tan(1/2*exp(1))^6-3242591731706757120*tan(1/2*exp(1))^5-340472
1318292094976*tan(1/2*exp(1))^4+2485986994308513792*tan(1/2*exp(1))^3+972777519512027136*tan(1/2*exp(1))^2-162
129586585337856*tan(1/2*exp(1))-27021597764222976)*tan(1/4*exp(1))^5+sqrt(c*tan(1/2*exp(1))^2+c)*(tan(1/2*(1/2
*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))*(-54043195528445952*tan(1/2*exp(1))^9-243194379878006784*tan(1/
2*exp(1))^8+972777519512027136*tan(1/2*exp(1))^7+2269814212194729984*tan(1/2*exp(1))^6-3242591731706757120*tan
(1/2*exp(1))^5-3404721318292094976*tan(1/2*exp(1))^4+2485986994308513792*tan(1/2*exp(1))^3+972777519512027136*
tan(1/2*exp(1))^2-162129586585337856*tan(1/2*exp(1))-27021597764222976)*tan(1/4*exp(1))+sqrt(c*tan(1/2*exp(1))
^2+c)*(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))*(-67553994410557440*tan(1/2*exp(1))^9-405323
966463344640*tan(1/2*exp(1))^8+2431943798780067840*tan(1/2*exp(1))^7+6214967485771284480*tan(1/2*exp(1))^6-851
1803295730237440*tan(1/2*exp(1))^5-8106479329266892800*tan(1/2*exp(1))^4+5674535530486824960*tan(1/2*exp(1))^3
+2431943798780067840*tan(1/2*exp(1))^2-607985949695016960*tan(1/2*exp(1))-135107988821114880)*tan(1/4*exp(1))^
4)/(-(tan(1/2*(1/2*f*x+2*exp(1)))-1/tan(1/2*(1/2*f*x+2*exp(1))))^2-4)^2/((2251799813685248*sqrt(2)*c*tan(1/2*e
xp(1))^10+11258999068426240*sqrt(2)*c*tan(1/2*exp(1))^8+22517998136852480*sqrt(2)*c*tan(1/2*exp(1))^6+22517998
136852480*sqrt(2)*c*tan(1/2*exp(1))^4+11258999068426240*sqrt(2)*c*tan(1/2*exp(1))^2+2251799813685248*sqrt(2)*c
)*tan(1/4*exp(1))^6+(6755399441055744*sqrt(2)*c*tan(1/2*exp(1))^10+33776997205278720*sqrt(2)*c*tan(1/2*exp(1))
^8+67553994410557440*sqrt(2)*c*tan(1/2*exp(1))^6+67553994410557440*sqrt(2)*c*tan(1/2*exp(1))^4+337769972052787
20*sqrt(2)*c*tan(1/2*exp(1))^2+6755399441055744*sqrt(2)*c)*tan(1/4*exp(1))^4+(6755399441055744*sqrt(2)*c*tan(1
/2*exp(1))^10+33776997205278720*sqrt(2)*c*tan(1/2*exp(1))^8+67553994410557440*sqrt(2)*c*tan(1/2*exp(1))^6+6755
3994410557440*sqrt(2)*c*tan(1/2*exp(1))^4+33776997205278720*sqrt(2)*c*tan(1/2*exp(1))^2+6755399441055744*sqrt(
2)*c)*tan(1/4*exp(1))^2+2251799813685248*sqrt(2)*c*tan(1/2*exp(1))^10+11258999068426240*sqrt(2)*c*tan(1/2*exp(
1))^8+22517998136852480*sqrt(2)*c*tan(1/2*exp(1))^6+22517998136852480*sqrt(2)*c*tan(1/2*exp(1))^4+112589990684
26240*sqrt(2)*c*tan(1/2*exp(1))^2+2251799813685248*sqrt(2)*c)+1/4*(1/2*pi*sign(tan(1/2*(1/2*f*x+2*exp(1))))+at
an(1/2*(tan(1/2*(1/2*f*x+2*exp(1)))^2-1)/tan(1/2*(1/2*f*x+2*exp(1)))))*(sqrt(c*tan(1/2*exp(1))^2+c)*(107374182
4*sqrt(2)*tan(1/2*exp(1))^3-3221225472*sqrt(2)*tan(1/2*exp(1)))+sqrt(c*tan(1/2*exp(1))^2+c)*(16106127360*sqrt(
2)*tan(1/2*exp(1))^3-48318382080*sqrt(2)*tan(1/2*exp(1)))*tan(1/4*exp(1))^4+sqrt(c*tan(1/2*exp(1))^2+c)*(-1073
741824*sqrt(2)*tan(1/2*exp(1))^3+3221225472*sqrt(2)*tan(1/2*exp(1)))*tan(1/4*exp(1))^6+sqrt(c*tan(1/2*exp(1))^
2+c)*(-19327352832*sqrt(2)*tan(1/2*exp(1))^2+6442450944*sqrt(2))*tan(1/4*exp(1))^5+sqrt(c*tan(1/2*exp(1))^2+c)
*(64424509440*sqrt(2)*tan(1/2*exp(1))^2-21474836480*sqrt(2))*tan(1/4*exp(1))^3+sqrt(c*tan(1/2*exp(1))^2+c)*(-1
6106127360*sqrt(2)*tan(1/2*exp(1))^3+48318382080*sqrt(2)*tan(1/2*exp(1)))*tan(1/4*exp(1))^2+sqrt(c*tan(1/2*exp
(1))^2+c)*(-19327352832*sqrt(2)*tan(1/2*exp(1))^2+6442450944*sqrt(2))*tan(1/4*exp(1)))/(-(2147483648*c*tan(1/2
*exp(1))^4+4294967296*c*tan(1/2*exp(1))^2+2147483648*c)*tan(1/4*exp(1))^6-(6442450944*c*tan(1/2*exp(1))^4+1288
4901888*c*tan(1/2*exp(1))^2+6442450944*c)*tan(1/4*exp(1))^4-(6442450944*c*tan(1/2*exp(1))^4+12884901888*c*tan(
1/2*exp(1))^2+6442450944*c)*tan(1/4*exp(1))^2-2147483648*c*tan(1/2*exp(1))^4-4294967296*c*tan(1/2*exp(1))^2-21
47483648*c))/f

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maple [B]  time = 0.38, size = 94, normalized size = 2.09 \[ \frac {\sin \left (f x +e \right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \left (\sin \left (f x +e \right ) \cos \left (f x +e \right )-\left (\cos ^{2}\left (f x +e \right )\right )+\sin \left (f x +e \right )+2 \cos \left (f x +e \right )-1\right )}{2 f \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \left (1-\cos \left (f x +e \right )+\sin \left (f x +e \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2),x)

[Out]

1/2/f*sin(f*x+e)*(a*(1+sin(f*x+e)))^(1/2)*(sin(f*x+e)*cos(f*x+e)-cos(f*x+e)^2+sin(f*x+e)+2*cos(f*x+e)-1)/(-c*(
sin(f*x+e)-1))^(1/2)/(1-cos(f*x+e)+sin(f*x+e))

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maxima [B]  time = 0.50, size = 387, normalized size = 8.60 \[ -\frac {\frac {2 \, \sqrt {a} \sqrt {c} + \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sqrt {a} \sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{c + \frac {2 \, c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {c \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} - \frac {2 \, \sqrt {a} \sqrt {c} - \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{c + \frac {2 \, c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {c \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac {2 \, {\left (\frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{c + \frac {2 \, c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {c \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}}}{2 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 8.96, size = 73, normalized size = 1.62 \[ -\frac {\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (\cos \left (e+f\,x\right )+\cos \left (3\,e+3\,f\,x\right )-4\,\sin \left (2\,e+2\,f\,x\right )\right )}{4\,c\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(e + f*x)^2*(a + a*sin(e + f*x))^(1/2))/(c - c*sin(e + f*x))^(1/2),x)

[Out]

-((a*(sin(e + f*x) + 1))^(1/2)*(-c*(sin(e + f*x) - 1))^(1/2)*(cos(e + f*x) + cos(3*e + 3*f*x) - 4*sin(2*e + 2*
f*x)))/(4*c*f*(cos(2*e + 2*f*x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \cos ^{2}{\left (e + f x \right )}}{\sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)**2*(a+a*sin(f*x+e))**(1/2)/(c-c*sin(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))*cos(e + f*x)**2/sqrt(-c*(sin(e + f*x) - 1)), x)

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