\(\int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx\) [174]

Optimal result

Integrand size = 40, antiderivative size = 197 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {4 (A-B) c^3 \cos (e+f x) \log (1+\sin (e+f x))}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 (A-B) c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}+\frac {(A-B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a+a \sin (e+f x)}}-\frac {B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt {a+a \sin (e+f x)}} \]

[Out]

1/2*(A-B)*c*cos(f*x+e)*(c-c*sin(f*x+e))^(3/2)/f/(a+a*sin(f*x+e))^(1/2)-1/3*B*cos(f*x+e)*(c-c*sin(f*x+e))^(5/2)
/f/(a+a*sin(f*x+e))^(1/2)+4*(A-B)*c^3*cos(f*x+e)*ln(1+sin(f*x+e))/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1
/2)+2*(A-B)*c^2*cos(f*x+e)*(c-c*sin(f*x+e))^(1/2)/f/(a+a*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3052, 2819, 2816, 2746, 31} \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {4 c^3 (A-B) \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c^2 (A-B) \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}+\frac {c (A-B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}}-\frac {B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt {a \sin (e+f x)+a}} \]

[In]

Int[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/Sqrt[a + a*Sin[e + f*x]],x]

[Out]

(4*(A - B)*c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*
(A - B)*c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]) + ((A - B)*c*Cos[e + f*x]*(c -
c*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + a*Sin[e + f*x]]) - (B*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*f*Sqrt
[a + a*Sin[e + f*x]])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 2816

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

Rule 2819

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

Rule 3052

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

Rubi steps \begin{align*} \text {integral}& = -\frac {B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt {a+a \sin (e+f x)}}+(A-B) \int \frac {(c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx \\ & = \frac {(A-B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a+a \sin (e+f x)}}-\frac {B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt {a+a \sin (e+f x)}}+(2 (A-B) c) \int \frac {(c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx \\ & = \frac {2 (A-B) c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}+\frac {(A-B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a+a \sin (e+f x)}}-\frac {B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt {a+a \sin (e+f x)}}+\left (4 (A-B) c^2\right ) \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx \\ & = \frac {2 (A-B) c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}+\frac {(A-B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a+a \sin (e+f x)}}-\frac {B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt {a+a \sin (e+f x)}}+\frac {\left (4 a (A-B) c^3 \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{a+a \sin (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {2 (A-B) c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}+\frac {(A-B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a+a \sin (e+f x)}}-\frac {B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt {a+a \sin (e+f x)}}+\frac {\left (4 (A-B) c^3 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \sin (e+f x)\right )}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 (A-B) c^3 \cos (e+f x) \log (1+\sin (e+f x))}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 (A-B) c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}+\frac {(A-B) c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a+a \sin (e+f x)}}-\frac {B \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.56 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {c^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-1+\sin (e+f x))^2 \sqrt {c-c \sin (e+f x)} \left (3 (A-3 B) \cos (2 (e+f x))-96 A \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+96 B \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+(36 A-51 B) \sin (e+f x)+B \sin (3 (e+f x))\right )}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \sqrt {a (1+\sin (e+f x))}} \]

[In]

Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/Sqrt[a + a*Sin[e + f*x]],x]

[Out]

-1/12*(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]]*(3*(A - 3*B)*C
os[2*(e + f*x)] - 96*A*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 96*B*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]
] + (36*A - 51*B)*Sin[e + f*x] + B*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*Sqrt[a*(1 + S
in[e + f*x])])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(494\) vs. \(2(177)=354\).

Time = 4.17 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.51

method	result	size
default	\(\frac {c^{2} \left (15 A \sin \left (f x +e \right )+15 A -17 B +2 B \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-3 A \cos \left (f x +e \right )-17 B \sin \left (f x +e \right )+18 A \sin \left (f x +e \right ) \cos \left (f x +e \right )-26 B \cos \left (f x +e \right ) \sin \left (f x +e \right )+24 A \cos \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+24 A \sin \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-24 B \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+9 \cos \left (f x +e \right ) B +19 B \left (\cos ^{2}\left (f x +e \right )\right )-2 B \left (\cos ^{4}\left (f x +e \right )\right )-7 B \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+3 A \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-24 B \cos \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-24 B \sin \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-48 A \cos \left (f x +e \right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )+48 B \cos \left (f x +e \right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-48 A \sin \left (f x +e \right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )+48 B \sin \left (f x +e \right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )+24 A \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-48 A \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )+48 B \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-9 B \left (\cos ^{3}\left (f x +e \right )\right )+3 A \left (\cos ^{3}\left (f x +e \right )\right )-15 A \left (\cos ^{2}\left (f x +e \right )\right )\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}{6 f \left (-\cos \left (f x +e \right )+\sin \left (f x +e \right )-1\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}\)	\(495\)
parts	\(\frac {A \left (\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\cos ^{3}\left (f x +e \right )+6 \cos \left (f x +e \right ) \sin \left (f x +e \right )+8 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )-16 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \sin \left (f x +e \right )-5 \left (\cos ^{2}\left (f x +e \right )\right )+8 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \cos \left (f x +e \right )-16 \cos \left (f x +e \right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )+5 \sin \left (f x +e \right )-\cos \left (f x +e \right )+8 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-16 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )+5\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{2}}{2 f \left (-\cos \left (f x +e \right )+\sin \left (f x +e \right )-1\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}+\frac {B \left (-2 \left (\cos ^{4}\left (f x +e \right )\right )+2 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-9 \left (\cos ^{3}\left (f x +e \right )\right )-7 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+48 \cos \left (f x +e \right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )+48 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \sin \left (f x +e \right )-24 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \cos \left (f x +e \right )-24 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )+19 \left (\cos ^{2}\left (f x +e \right )\right )-26 \cos \left (f x +e \right ) \sin \left (f x +e \right )+48 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-24 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+9 \cos \left (f x +e \right )-17 \sin \left (f x +e \right )-17\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{2}}{6 f \left (-\cos \left (f x +e \right )+\sin \left (f x +e \right )-1\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}\)	\(517\)

[In]

int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/6*c^2/f*(15*A*sin(f*x+e)+15*A-17*B-7*B*cos(f*x+e)^2*sin(f*x+e)-3*A*cos(f*x+e)-17*B*sin(f*x+e)+19*B*cos(f*x+e
)^2-15*A*cos(f*x+e)^2+3*A*sin(f*x+e)*cos(f*x+e)^2+18*A*sin(f*x+e)*cos(f*x+e)-26*B*cos(f*x+e)*sin(f*x+e)+24*A*c
os(f*x+e)*ln(2/(1+cos(f*x+e)))+24*A*sin(f*x+e)*ln(2/(1+cos(f*x+e)))-2*B*cos(f*x+e)^4-24*B*ln(2/(1+cos(f*x+e)))
+9*cos(f*x+e)*B+3*A*cos(f*x+e)^3-24*B*cos(f*x+e)*ln(2/(1+cos(f*x+e)))-24*B*sin(f*x+e)*ln(2/(1+cos(f*x+e)))-48*
A*cos(f*x+e)*ln(-cot(f*x+e)+csc(f*x+e)+1)+48*B*cos(f*x+e)*ln(-cot(f*x+e)+csc(f*x+e)+1)-48*A*sin(f*x+e)*ln(-cot
(f*x+e)+csc(f*x+e)+1)+48*B*sin(f*x+e)*ln(-cot(f*x+e)+csc(f*x+e)+1)-9*B*cos(f*x+e)^3+24*A*ln(2/(1+cos(f*x+e)))+
2*B*cos(f*x+e)^3*sin(f*x+e)-48*A*ln(-cot(f*x+e)+csc(f*x+e)+1)+48*B*ln(-cot(f*x+e)+csc(f*x+e)+1))*(-c*(sin(f*x+
e)-1))^(1/2)/(-cos(f*x+e)+sin(f*x+e)-1)/(a*(1+sin(f*x+e)))^(1/2)

Fricas [F]

\[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]

[In]

integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(-((A - 2*B)*c^2*cos(f*x + e)^2 - 2*(A - B)*c^2 + (B*c^2*cos(f*x + e)^2 + 2*(A - B)*c^2)*sin(f*x + e))
*sqrt(-c*sin(f*x + e) + c)/sqrt(a*sin(f*x + e) + a), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(5/2)/(a+a*sin(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]

[In]

integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(-c*sin(f*x + e) + c)^(5/2)/sqrt(a*sin(f*x + e) + a), x)

Giac [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.53 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\sqrt {2} \sqrt {c} {\left (\frac {6 \, \sqrt {2} {\left (A \sqrt {a} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B \sqrt {a} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \log \left (-2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} {\left (4 \, B a^{\frac {5}{2}} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 3 \, A a^{\frac {5}{2}} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 3 \, B a^{\frac {5}{2}} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 6 \, A a^{\frac {5}{2}} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 6 \, B a^{\frac {5}{2}} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )}}{3 \, f} \]

[In]

integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

-1/3*sqrt(2)*sqrt(c)*(6*sqrt(2)*(A*sqrt(a)*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - B*sqrt(a)*c^2*sgn(sin(-1/
4*pi + 1/2*f*x + 1/2*e)))*log(-2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 + 2)/(a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))
- sqrt(2)*(4*B*a^(5/2)*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^6 - 3*A*a^(5/2)
*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^4 + 3*B*a^(5/2)*c^2*sgn(sin(-1/4*pi +
1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^4 - 6*A*a^(5/2)*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-
1/4*pi + 1/2*f*x + 1/2*e)^2 + 6*B*a^(5/2)*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*
e)^2)/(a^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))))/f

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}}{\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \]

[In]

int(((A + B*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2))/(a + a*sin(e + f*x))^(1/2),x)

[Out]

int(((A + B*sin(e + f*x))*(c - c*sin(e + f*x))^(5/2))/(a + a*sin(e + f*x))^(1/2), x)