3.2.68 \(\int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx\) [168]

Optimal. Leaf size=264 \[ \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {a^4 (A+7 B) \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {a^3 (A+7 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 c^3 f \sqrt {c-c \sin (e+f x)}} \]

[Out]

1/6*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(7/2)-1/12*a*(A+7*B)*cos(f*x+e)*(a+a*sin(f*x+e)
)^(5/2)/c/f/(c-c*sin(f*x+e))^(5/2)+1/4*a^2*(A+7*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/c^2/f/(c-c*sin(f*x+e))^(3
/2)+a^4*(A+7*B)*cos(f*x+e)*ln(1-sin(f*x+e))/c^3/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)+1/2*a^3*(A+7*B
)*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/c^3/f/(c-c*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.42, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3051, 2818, 2819, 2816, 2746, 31} \begin {gather*} \frac {a^4 (A+7 B) \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {a^3 (A+7 B) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{2 c^3 f \sqrt {c-c \sin (e+f x)}}+\frac {a^2 (A+7 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac {a (A+7 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(6*f*(c - c*Sin[e + f*x])^(7/2)) - (a*(A + 7*B)*Cos[e + f*x]
*(a + a*Sin[e + f*x])^(5/2))/(12*c*f*(c - c*Sin[e + f*x])^(5/2)) + (a^2*(A + 7*B)*Cos[e + f*x]*(a + a*Sin[e +
f*x])^(3/2))/(4*c^2*f*(c - c*Sin[e + f*x])^(3/2)) + (a^4*(A + 7*B)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^3*f*
Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (a^3*(A + 7*B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(2*
c^3*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 2818

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[-2*b*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^n/(f*(2*n + 1))), x] - Dist[b*((2*m - 1)
/(d*(2*n + 1))), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e
, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] && LtQ[n, -1] &&  !(ILtQ[m + n, 0] && G
tQ[2*m + n + 1, 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 3051

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

Rubi steps

\begin {align*} \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {(A+7 B) \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{5/2}} \, dx}{6 c}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {(a (A+7 B)) \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{3/2}} \, dx}{4 c^2}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac {\left (a^2 (A+7 B)\right ) \int \frac {(a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx}{2 c^3}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {a^3 (A+7 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 c^3 f \sqrt {c-c \sin (e+f x)}}-\frac {\left (a^3 (A+7 B)\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c^3}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {a^3 (A+7 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 c^3 f \sqrt {c-c \sin (e+f x)}}-\frac {\left (a^4 (A+7 B) \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{c^2 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {a^3 (A+7 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 c^3 f \sqrt {c-c \sin (e+f x)}}+\frac {\left (a^4 (A+7 B) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 c f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (A+7 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {a^4 (A+7 B) \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {a^3 (A+7 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 c^3 f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 1.98, size = 244, normalized size = 0.92 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{7/2} \left (8 (A+B)-6 (3 A+5 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+18 (A+3 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4+6 (A+7 B) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6+3 B \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sin (e+f x)\right )}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(7/2)*(8*(A + B) - 6*(3*A + 5*B)*(Cos[(e + f*x)/
2] - Sin[(e + f*x)/2])^2 + 18*(A + 3*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + 6*(A + 7*B)*Log[Cos[(e + f*x
)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6 + 3*B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6
*Sin[e + f*x]))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1454\) vs. \(2(236)=472\).
time = 0.31, size = 1455, normalized size = 5.51

method	result	size
default	\(\text {Expression too large to display}\)	\(1455\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/f*(63*B*ln(2/(1+cos(f*x+e)))*cos(f*x+e)^3-126*B*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))*cos(f*x+e)^3+6*
A*cos(f*x+e)^3*sin(f*x+e)*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))-3*A*cos(f*x+e)^3*sin(f*x+e)*ln(2/(1+cos(f
*x+e)))+14*A*cos(f*x+e)^2*sin(f*x+e)+98*B*cos(f*x+e)^2*sin(f*x+e)+20*A+116*B-168*B*ln(-(-1+cos(f*x+e)+sin(f*x+
e))/sin(f*x+e))*sin(f*x+e)*cos(f*x+e)+84*B*ln(2/(1+cos(f*x+e)))*sin(f*x+e)*cos(f*x+e)-18*A*cos(f*x+e)^3*ln(-(-
1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))+9*A*cos(f*x+e)^3*ln(2/(1+cos(f*x+e)))-6*A*cos(f*x+e)-21*B*ln(2/(1+cos(f*x
+e)))*cos(f*x+e)^3*sin(f*x+e)-116*B*sin(f*x+e)-20*A*sin(f*x+e)+6*A*cos(f*x+e)^3+57*B*cos(f*x+e)^3-54*B*cos(f*x
+e)-41*B*cos(f*x+e)^3*sin(f*x+e)+84*B*ln(2/(1+cos(f*x+e)))*cos(f*x+e)^2*sin(f*x+e)-168*B*ln(-(-1+cos(f*x+e)+si
n(f*x+e))/sin(f*x+e))*cos(f*x+e)^2*sin(f*x+e)+44*B*cos(f*x+e)^4-28*A*cos(f*x+e)^2-160*B*cos(f*x+e)^2-8*A*cos(f
*x+e)^3*sin(f*x+e)-24*A*cos(f*x+e)^2*ln(2/(1+cos(f*x+e)))-12*A*cos(f*x+e)*ln(2/(1+cos(f*x+e)))+42*B*ln(-(-1+co
s(f*x+e)+sin(f*x+e))/sin(f*x+e))*cos(f*x+e)^3*sin(f*x+e)+336*B*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))*cos(
f*x+e)^2-3*B*cos(f*x+e)^5+3*A*cos(f*x+e)^4*ln(2/(1+cos(f*x+e)))-6*A*cos(f*x+e)^4*ln(-(-1+cos(f*x+e)+sin(f*x+e)
)/sin(f*x+e))+21*B*ln(2/(1+cos(f*x+e)))*cos(f*x+e)^4-42*B*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))*cos(f*x+e
)^4-168*B*ln(2/(1+cos(f*x+e)))*sin(f*x+e)+62*B*sin(f*x+e)*cos(f*x+e)+168*B*cos(f*x+e)*ln(-(-1+cos(f*x+e)+sin(f
*x+e))/sin(f*x+e))+336*B*sin(f*x+e)*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))-24*A*cos(f*x+e)^2*sin(f*x+e)*ln
(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))+12*A*cos(f*x+e)^2*sin(f*x+e)*ln(2/(1+cos(f*x+e)))-24*A*ln(2/(1+cos(f*
x+e)))*sin(f*x+e)+24*A*cos(f*x+e)*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))+48*A*sin(f*x+e)*ln(-(-1+cos(f*x+e
)+sin(f*x+e))/sin(f*x+e))-84*B*cos(f*x+e)*ln(2/(1+cos(f*x+e)))+8*A*cos(f*x+e)^4-24*A*cos(f*x+e)*sin(f*x+e)*ln(
-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))+12*A*cos(f*x+e)*sin(f*x+e)*ln(2/(1+cos(f*x+e)))+24*A*ln(2/(1+cos(f*x+e
)))-48*A*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))+168*B*ln(2/(1+cos(f*x+e)))-336*B*ln(-(-1+cos(f*x+e)+sin(f*
x+e))/sin(f*x+e))+48*A*cos(f*x+e)^2*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))-168*B*ln(2/(1+cos(f*x+e)))*cos(
f*x+e)^2+14*A*sin(f*x+e)*cos(f*x+e)-3*B*sin(f*x+e)*cos(f*x+e)^4)*(a*(1+sin(f*x+e)))^(7/2)/(cos(f*x+e)^4+cos(f*
x+e)^3*sin(f*x+e)+3*cos(f*x+e)^3-4*cos(f*x+e)^2*sin(f*x+e)-8*cos(f*x+e)^2-4*cos(f*x+e)*sin(f*x+e)-4*cos(f*x+e)
+8*sin(f*x+e)+8)/(-c*(sin(f*x+e)-1))^(7/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 809 vs. \(2 (252) = 504\).
time = 0.55, size = 809, normalized size = 3.06 \begin {gather*} -\frac {B {\left (\frac {42 \, a^{\frac {7}{2}} \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c^{\frac {7}{2}}} - \frac {21 \, a^{\frac {7}{2}} \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}{c^{\frac {7}{2}}} + \frac {2 \, {\left (\frac {21 \, a^{\frac {7}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {102 \, a^{\frac {7}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {227 \, a^{\frac {7}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {228 \, a^{\frac {7}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {227 \, a^{\frac {7}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {102 \, a^{\frac {7}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {21 \, a^{\frac {7}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}\right )}}{c^{\frac {7}{2}} - \frac {6 \, c^{\frac {7}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {16 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {26 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {30 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {26 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {16 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {6 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {c^{\frac {7}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}}\right )} + A {\left (\frac {6 \, a^{\frac {7}{2}} \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c^{\frac {7}{2}}} - \frac {3 \, a^{\frac {7}{2}} \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}{c^{\frac {7}{2}}} + \frac {4 \, {\left (\frac {3 \, a^{\frac {7}{2}} \sqrt {c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {6 \, a^{\frac {7}{2}} \sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {22 \, a^{\frac {7}{2}} \sqrt {c} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {6 \, a^{\frac {7}{2}} \sqrt {c} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {3 \, a^{\frac {7}{2}} \sqrt {c} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{c^{4} - \frac {6 \, c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {15 \, c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {20 \, c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {6 \, c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}}\right )}}{3 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(B*(42*a^(7/2)*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/c^(7/2) - 21*a^(7/2)*log(sin(f*x + e)^2/(cos(f*x
+ e) + 1)^2 + 1)/c^(7/2) + 2*(21*a^(7/2)*sin(f*x + e)/(cos(f*x + e) + 1) - 102*a^(7/2)*sin(f*x + e)^2/(cos(f*x
+ e) + 1)^2 + 227*a^(7/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 228*a^(7/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)
^4 + 227*a^(7/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 102*a^(7/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 21*a^
(7/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7)/(c^(7/2) - 6*c^(7/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 16*c^(7/2)*s
in(f*x + e)^2/(cos(f*x + e) + 1)^2 - 26*c^(7/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 30*c^(7/2)*sin(f*x + e)^
4/(cos(f*x + e) + 1)^4 - 26*c^(7/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 16*c^(7/2)*sin(f*x + e)^6/(cos(f*x +
e) + 1)^6 - 6*c^(7/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + c^(7/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8)) + A
*(6*a^(7/2)*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/c^(7/2) - 3*a^(7/2)*log(sin(f*x + e)^2/(cos(f*x + e) + 1)
^2 + 1)/c^(7/2) + 4*(3*a^(7/2)*sqrt(c)*sin(f*x + e)/(cos(f*x + e) + 1) - 6*a^(7/2)*sqrt(c)*sin(f*x + e)^2/(cos
(f*x + e) + 1)^2 + 22*a^(7/2)*sqrt(c)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 6*a^(7/2)*sqrt(c)*sin(f*x + e)^4/(
cos(f*x + e) + 1)^4 + 3*a^(7/2)*sqrt(c)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/(c^4 - 6*c^4*sin(f*x + e)/(cos(f*
x + e) + 1) + 15*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 20*c^4*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15*c^4
*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 6*c^4*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + c^4*sin(f*x + e)^6/(cos(f*x
+ e) + 1)^6)))/f

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),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^4*cos(f*x + e)^4 - 8*c^
4*cos(f*x + e)^2 + 8*c^4 + 4*(c^4*cos(f*x + e)^2 - 2*c^4)*sin(f*x + e)), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.58, size = 376, normalized size = 1.42 \begin {gather*} -\frac {\sqrt {2} {\left (\frac {12 \, \sqrt {2} B a^{3} \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 )}{c^{\frac {7}{2}} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {6 \, \sqrt {2} {\left (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 ) + 7 \, 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 (-192 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 192\right )}{c^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} {\left (11 \, 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 ) + 41 \, 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 ) + 18 \, {\left (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 ) + 3 \, 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 )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3 \, {\left (9 \, 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 ) + 31 \, 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 )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} c^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )} \sqrt {a}}{12 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(2)*(12*sqrt(2)*B*a^3*cos(-1/4*pi + 1/2*f*x + 1/2*e)^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))/(c^(7/2)*
sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) + 6*sqrt(2)*(A*a^3*sqrt(c)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 7*B*a^3*
sqrt(c)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*log(-192*cos(-1/4*pi + 1/2*f*x + 1/2*e)^2 + 192)/(c^4*sgn(sin(-1/
4*pi + 1/2*f*x + 1/2*e))) - sqrt(2)*(11*A*a^3*sqrt(c)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 41*B*a^3*sqrt(c)*s
gn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 18*(A*a^3*sqrt(c)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*B*a^3*sqrt(c)*s
gn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-1/4*pi + 1/2*f*x + 1/2*e)^4 - 3*(9*A*a^3*sqrt(c)*sgn(cos(-1/4*pi + 1/
2*f*x + 1/2*e)) + 31*B*a^3*sqrt(c)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-1/4*pi + 1/2*f*x + 1/2*e)^2)/((co
s(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 1)^3*c^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))))*sqrt(a)/f

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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