\(\int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx\) [165]

Optimal result

Integrand size = 40, antiderivative size = 239 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {8 a^4 (A+B) \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {4 a^3 (A+B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a^2 (A+B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a (A+B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}} \]

[Out]

-a^2*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f/(c-c*sin(f*x+e))^(1/2)-1/3*a*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^
(5/2)/f/(c-c*sin(f*x+e))^(1/2)-1/4*B*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(1/2)-8*a^4*(A+B)*co
s(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)-4*a^3*(A+B)*cos(f*x+e)*(a+a*sin(f*x+
e))^(1/2)/f/(c-c*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3052, 2819, 2816, 2746, 31} \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {8 a^4 (A+B) \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {4 a^3 (A+B) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a^2 (A+B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a (A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}} \]

[In]

Int[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f*x]],x]

[Out]

(-8*a^4*(A + B)*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]]) - (4
*a^3*(A + B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*Sqrt[c - c*Sin[e + f*x]]) - (a^2*(A + B)*Cos[e + f*x]*(
a + a*Sin[e + f*x])^(3/2))/(f*Sqrt[c - c*Sin[e + f*x]]) - (a*(A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/
(3*f*Sqrt[c - c*Sin[e + f*x]]) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*f*Sqrt[c - c*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) (a+a \sin (e+f x))^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}+(A+B) \int \frac {(a+a \sin (e+f x))^{7/2}}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = -\frac {a (A+B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}+(2 a (A+B)) \int \frac {(a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = -\frac {a^2 (A+B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a (A+B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}+\left (4 a^2 (A+B)\right ) \int \frac {(a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = -\frac {4 a^3 (A+B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a^2 (A+B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a (A+B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}+\left (8 a^3 (A+B)\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = -\frac {4 a^3 (A+B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a^2 (A+B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a (A+B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}+\frac {\left (8 a^4 (A+B) c \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {4 a^3 (A+B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a^2 (A+B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a (A+B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}}-\frac {\left (8 a^4 (A+B) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {8 a^4 (A+B) \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 {4 a^3 (A+B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a^2 (A+B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a (A+B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 12.44 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.77 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \sqrt {a (1+\sin (e+f x))} \left (-12 (8 A+15 B) \cos (2 (e+f x))+3 B \cos (4 (e+f x))+1536 (A+B) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+24 (29 A+36 B) \sin (e+f x)-8 (A+4 B) \sin (3 (e+f x))\right )}{96 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \sqrt {c-c \sin (e+f x)}} \]

[In]

Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/Sqrt[c - c*Sin[e + f*x]],x]

[Out]

-1/96*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*(-12*(8*A + 1
5*B)*Cos[2*(e + f*x)] + 3*B*Cos[4*(e + f*x)] + 1536*(A + B)*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 24*(29*
A + 36*B)*Sin[e + f*x] - 8*(A + 4*B)*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*Sqrt[c - c*
Sin[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(548\) vs. \(2(215)=430\).

Time = 3.30 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.30

method	result	size
default	\(\frac {a^{3} \left (-64 A \sin \left (f x +e \right )+64 A +67 B +16 B \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-24 A \cos \left (f x +e \right )-67 B \sin \left (f x +e \right )-192 A \cos \left (f x +e \right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-88 A \sin \left (f x +e \right ) \cos \left (f x +e \right )-112 B \cos \left (f x +e \right ) \sin \left (f x +e \right )+96 A \cos \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-96 A \sin \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+96 B \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-45 \cos \left (f x +e \right ) B -80 B \left (\cos ^{2}\left (f x +e \right )\right )+13 B \left (\cos ^{4}\left (f x +e \right )\right )-192 A \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-192 B \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-32 B \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+4 A \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-20 A \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+192 A \sin \left (f x +e \right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-192 B \cos \left (f x +e \right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )+192 B \sin \left (f x +e \right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )+96 B \cos \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-96 B \sin \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+96 A \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+3 B \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-3 B \left (\cos ^{5}\left (f x +e \right )\right )+4 A \left (\cos ^{4}\left (f x +e \right )\right )+48 B \left (\cos ^{3}\left (f x +e \right )\right )+24 A \left (\cos ^{3}\left (f x +e \right )\right )-68 A \left (\cos ^{2}\left (f x +e \right )\right )\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}{12 f \left (\cos \left (f x +e \right )+\sin \left (f x +e \right )+1\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\)	\(549\)
parts	\(\frac {A \left (\cos ^{4}\left (f x +e \right )+\left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+6 \left (\cos ^{3}\left (f x +e \right )\right )-5 \left (\cos ^{2}\left (f x +e \right )\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 )-48 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \cos \left (f x +e \right )+48 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \sin \left (f x +e \right )-17 \left (\cos ^{2}\left (f x +e \right )\right )-22 \cos \left (f x +e \right ) \sin \left (f x +e \right )+24 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-48 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-6 \cos \left (f x +e \right )-16 \sin \left (f x +e \right )+16\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a^{3}}{3 f \left (\cos \left (f x +e \right )+\sin \left (f x +e \right )+1\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}+\frac {B \left (-3 \left (\cos ^{5}\left (f x +e \right )\right )+3 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+13 \left (\cos ^{4}\left (f x +e \right )\right )+16 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+48 \left (\cos ^{3}\left (f x +e \right )\right )-32 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+96 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \cos \left (f x +e \right )-96 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )-192 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \cos \left (f x +e \right )+192 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \sin \left (f x +e \right )-80 \left (\cos ^{2}\left (f x +e \right )\right )-112 \cos \left (f x +e \right ) \sin \left (f x +e \right )+96 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-192 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-45 \cos \left (f x +e \right )-67 \sin \left (f x +e \right )+67\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a^{3}}{12 f \left (\cos \left (f x +e \right )+\sin \left (f x +e \right )+1\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\)	\(565\)

[In]

int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/12*a^3/f*(-64*A*sin(f*x+e)+64*A+67*B-32*B*cos(f*x+e)^2*sin(f*x+e)-24*A*cos(f*x+e)-67*B*sin(f*x+e)-80*B*cos(f
*x+e)^2-68*A*cos(f*x+e)^2-192*A*cos(f*x+e)*ln(csc(f*x+e)-cot(f*x+e)-1)-20*A*sin(f*x+e)*cos(f*x+e)^2-88*A*sin(f
*x+e)*cos(f*x+e)-112*B*cos(f*x+e)*sin(f*x+e)+96*A*cos(f*x+e)*ln(2/(1+cos(f*x+e)))-96*A*sin(f*x+e)*ln(2/(1+cos(
f*x+e)))+13*B*cos(f*x+e)^4+96*B*ln(2/(1+cos(f*x+e)))+3*B*cos(f*x+e)^4*sin(f*x+e)-45*cos(f*x+e)*B+4*A*cos(f*x+e
)^4-192*A*ln(csc(f*x+e)-cot(f*x+e)-1)-192*B*ln(csc(f*x+e)-cot(f*x+e)-1)+24*A*cos(f*x+e)^3+192*A*sin(f*x+e)*ln(
csc(f*x+e)-cot(f*x+e)-1)-192*B*cos(f*x+e)*ln(csc(f*x+e)-cot(f*x+e)-1)+192*B*sin(f*x+e)*ln(csc(f*x+e)-cot(f*x+e
)-1)+96*B*cos(f*x+e)*ln(2/(1+cos(f*x+e)))-96*B*sin(f*x+e)*ln(2/(1+cos(f*x+e)))+48*B*cos(f*x+e)^3+96*A*ln(2/(1+
cos(f*x+e)))-3*B*cos(f*x+e)^5+4*A*cos(f*x+e)^3*sin(f*x+e)+16*B*cos(f*x+e)^3*sin(f*x+e))*(a*(1+sin(f*x+e)))^(1/
2)/(cos(f*x+e)+sin(f*x+e)+1)/(-c*(sin(f*x+e)-1))^(1/2)

Fricas [F]

\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{\sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{\sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(7/2)/sqrt(-c*sin(f*x + e) + c), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (215) = 430\).

Time = 0.78 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.96 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {2 \, \sqrt {2} \sqrt {a} {\left (\frac {6 \, {\left (\sqrt {2} A a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + \sqrt {2} B a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {3 \, \sqrt {2} B a^{3} c^{\frac {7}{2}} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, \sqrt {2} A a^{3} c^{\frac {7}{2}} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, \sqrt {2} B a^{3} c^{\frac {7}{2}} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, \sqrt {2} A a^{3} c^{\frac {7}{2}} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, \sqrt {2} B a^{3} c^{\frac {7}{2}} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 6 \, \sqrt {2} A a^{3} c^{\frac {7}{2}} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 6 \, \sqrt {2} B a^{3} c^{\frac {7}{2}} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{c^{4}}\right )}}{3 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

2/3*sqrt(2)*sqrt(a)*(6*(sqrt(2)*A*a^3*sqrt(c)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + sqrt(2)*B*a^3*sqrt(c)*sgn(
cos(-1/4*pi + 1/2*f*x + 1/2*e)))*log(-cos(-1/4*pi + 1/2*f*x + 1/2*e)^2 + 1)/(c*sgn(sin(-1/4*pi + 1/2*f*x + 1/2
*e))) + (3*sqrt(2)*B*a^3*c^(7/2)*cos(-1/4*pi + 1/2*f*x + 1/2*e)^8*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(
-1/4*pi + 1/2*f*x + 1/2*e)) + 2*sqrt(2)*A*a^3*c^(7/2)*cos(-1/4*pi + 1/2*f*x + 1/2*e)^6*sgn(cos(-1/4*pi + 1/2*f
*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*sqrt(2)*B*a^3*c^(7/2)*cos(-1/4*pi + 1/2*f*x + 1/2*e)^6*sg
n(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*sqrt(2)*A*a^3*c^(7/2)*cos(-1/4*pi +
1/2*f*x + 1/2*e)^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*sqrt(2)*B*a^3*c
^(7/2)*cos(-1/4*pi + 1/2*f*x + 1/2*e)^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)
) + 6*sqrt(2)*A*a^3*c^(7/2)*cos(-1/4*pi + 1/2*f*x + 1/2*e)^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*
pi + 1/2*f*x + 1/2*e)) + 6*sqrt(2)*B*a^3*c^(7/2)*cos(-1/4*pi + 1/2*f*x + 1/2*e)^2*sgn(cos(-1/4*pi + 1/2*f*x +
1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))/c^4)/f

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \]

[In]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(1/2),x)

[Out]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(1/2), x)