3.2.6 \(\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx\) [106]

Optimal. Leaf size=217 \[ -\frac {5 a^3 (A-15 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{128 \sqrt {2} c^{9/2} f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 (A-15 B) c \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 (A-15 B) \cos ^3(e+f x)}{192 c f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 (A-15 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}} \]

[Out]

1/8*a^3*(A+B)*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(15/2)+1/48*a^3*(A-15*B)*c*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(
11/2)-5/192*a^3*(A-15*B)*cos(f*x+e)^3/c/f/(c-c*sin(f*x+e))^(7/2)+5/128*a^3*(A-15*B)*cos(f*x+e)/c^3/f/(c-c*sin(
f*x+e))^(3/2)-5/256*a^3*(A-15*B)*arctanh(1/2*cos(f*x+e)*c^(1/2)*2^(1/2)/(c-c*sin(f*x+e))^(1/2))/c^(9/2)/f*2^(1
/2)

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Rubi [A]
time = 0.37, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {3046, 2938, 2759, 2728, 212} \begin {gather*} -\frac {5 a^3 (A-15 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{128 \sqrt {2} c^{9/2} f}+\frac {a^3 c^3 (A+B) \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {5 a^3 (A-15 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {a^3 c (A-15 B) \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 (A-15 B) \cos ^3(e+f x)}{192 c f (c-c \sin (e+f x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(9/2),x]

[Out]

(-5*a^3*(A - 15*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(128*Sqrt[2]*c^(9/2)*f)
+ (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(8*f*(c - c*Sin[e + f*x])^(15/2)) + (a^3*(A - 15*B)*c*Cos[e + f*x]^5)/(48*
f*(c - c*Sin[e + f*x])^(11/2)) - (5*a^3*(A - 15*B)*Cos[e + f*x]^3)/(192*c*f*(c - c*Sin[e + f*x])^(7/2)) + (5*a
^3*(A - 15*B)*Cos[e + f*x])/(128*c^3*f*(c - c*Sin[e + f*x])^(3/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2728

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, b*(C
os[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2759

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[2*g*(g
*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(2*m +
p + 1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 2938

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p
+ 1))), x] + Dist[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^
(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[
m + p], 0]) && NeQ[2*m + p + 1, 0]

Rule 3046

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

Rubi steps

\begin {align*} \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx &=\left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {1}{16} \left (a^3 (A-15 B) c^2\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{13/2}} \, dx\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 (A-15 B) c \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {1}{96} \left (5 a^3 (A-15 B)\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{9/2}} \, dx\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 (A-15 B) c \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 (A-15 B) \cos ^3(e+f x)}{192 c f (c-c \sin (e+f x))^{7/2}}+\frac {\left (5 a^3 (A-15 B)\right ) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx}{128 c^2}\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 (A-15 B) c \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 (A-15 B) \cos ^3(e+f x)}{192 c f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 (A-15 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}-\frac {\left (5 a^3 (A-15 B)\right ) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{256 c^4}\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 (A-15 B) c \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 (A-15 B) \cos ^3(e+f x)}{192 c f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 (A-15 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {\left (5 a^3 (A-15 B)\right ) \text {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{128 c^4 f}\\ &=-\frac {5 a^3 (A-15 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{128 \sqrt {2} c^{9/2} f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{8 f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 (A-15 B) c \cos ^5(e+f x)}{48 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 (A-15 B) \cos ^3(e+f x)}{192 c f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 (A-15 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.95, size = 355, normalized size = 1.64 \begin {gather*} \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 \left (1765 A \cos \left (\frac {1}{2} (e+f x)\right )+405 B \cos \left (\frac {1}{2} (e+f x)\right )-895 A \cos \left (\frac {3}{2} (e+f x)\right )-2703 B \cos \left (\frac {3}{2} (e+f x)\right )-397 A \cos \left (\frac {5}{2} (e+f x)\right )+579 B \cos \left (\frac {5}{2} (e+f x)\right )+15 A \cos \left (\frac {7}{2} (e+f x)\right )+543 B \cos \left (\frac {7}{2} (e+f x)\right )+(120+120 i) \sqrt [4]{-1} (A-15 B) \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8+1765 A \sin \left (\frac {1}{2} (e+f x)\right )+405 B \sin \left (\frac {1}{2} (e+f x)\right )+895 A \sin \left (\frac {3}{2} (e+f x)\right )+2703 B \sin \left (\frac {3}{2} (e+f x)\right )-397 A \sin \left (\frac {5}{2} (e+f x)\right )+579 B \sin \left (\frac {5}{2} (e+f x)\right )-15 A \sin \left (\frac {7}{2} (e+f x)\right )-543 B \sin \left (\frac {7}{2} (e+f x)\right )\right )}{3072 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(9/2),x]

[Out]

(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(1765*A*Cos[(e + f*x)/2] + 405*B*Cos[(e + f*x)
/2] - 895*A*Cos[(3*(e + f*x))/2] - 2703*B*Cos[(3*(e + f*x))/2] - 397*A*Cos[(5*(e + f*x))/2] + 579*B*Cos[(5*(e
+ f*x))/2] + 15*A*Cos[(7*(e + f*x))/2] + 543*B*Cos[(7*(e + f*x))/2] + (120 + 120*I)*(-1)^(1/4)*(A - 15*B)*ArcT
an[(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8 + 1765*A*Sin[(e + f*
x)/2] + 405*B*Sin[(e + f*x)/2] + 895*A*Sin[(3*(e + f*x))/2] + 2703*B*Sin[(3*(e + f*x))/2] - 397*A*Sin[(5*(e +
f*x))/2] + 579*B*Sin[(5*(e + f*x))/2] - 15*A*Sin[(7*(e + f*x))/2] - 543*B*Sin[(7*(e + f*x))/2]))/(3072*f*(Cos[
(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(431\) vs. \(2(190)=380\).
time = 9.53, size = 432, normalized size = 1.99

method	result	size
default	\(-\frac {a^{3} \left (-60 \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{4} \left (A -15 B \right ) \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+120 \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{4} \left (A -15 B \right ) \sin \left (f x +e \right )-15 \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{4} \left (A -15 B \right ) \left (\cos ^{4}\left (f x +e \right )\right )+120 \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{4} \left (A -15 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+30 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {7}{2}} \sqrt {c}+292 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {3}{2}}-440 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {5}{2}}+240 A \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {7}{2}}+1086 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {7}{2}} \sqrt {c}-4380 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {3}{2}}+6600 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {5}{2}}-3600 B \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {7}{2}}-120 A \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4}+1800 B \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{768 c^{\frac {17}{2}} \left (\sin \left (f x +e \right )-1\right )^{3} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(432\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(9/2),x,method=_RETURNVERBOSE)

[Out]

-1/768/c^(17/2)*a^3*(-60*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^4*(A-15*B)*sin(f*x+e)*c
os(f*x+e)^2+120*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^4*(A-15*B)*sin(f*x+e)-15*arctanh
(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^4*(A-15*B)*cos(f*x+e)^4+120*arctanh(1/2*(c+c*sin(f*x+e)
)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^4*(A-15*B)*cos(f*x+e)^2+30*A*(c+c*sin(f*x+e))^(7/2)*c^(1/2)+292*A*(c+c*sin(
f*x+e))^(5/2)*c^(3/2)-440*A*(c+c*sin(f*x+e))^(3/2)*c^(5/2)+240*A*(c+c*sin(f*x+e))^(1/2)*c^(7/2)+1086*B*(c+c*si
n(f*x+e))^(7/2)*c^(1/2)-4380*B*(c+c*sin(f*x+e))^(5/2)*c^(3/2)+6600*B*(c+c*sin(f*x+e))^(3/2)*c^(5/2)-3600*B*(c+
c*sin(f*x+e))^(1/2)*c^(7/2)-120*A*2^(1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^4+1800*B*2^(1/
2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^4)*(c*(1+sin(f*x+e)))^(1/2)/(sin(f*x+e)-1)^3/cos(f*x+
e)/(c-c*sin(f*x+e))^(1/2)/f

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(9/2),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^3/(-c*sin(f*x + e) + c)^(9/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (200) = 400\).
time = 0.39, size = 673, normalized size = 3.10 \begin {gather*} -\frac {15 \, \sqrt {2} {\left ({\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} + 5 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 8 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 20 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 8 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right ) + 16 \, {\left (A - 15 \, B\right )} a^{3} - {\left ({\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 4 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 12 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 8 \, {\left (A - 15 \, B\right )} a^{3} \cos \left (f x + e\right ) + 16 \, {\left (A - 15 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (3 \, {\left (5 \, A + 181 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - {\left (191 \, A - 561 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 2 \, {\left (169 \, A + 537 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 12 \, {\left (21 \, A - 59 \, B\right )} a^{3} \cos \left (f x + e\right ) + 384 \, {\left (A + B\right )} a^{3} - {\left (3 \, {\left (5 \, A + 181 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 2 \, {\left (103 \, A - 9 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 12 \, {\left (11 \, A + 91 \, B\right )} a^{3} \cos \left (f x + e\right ) - 384 \, {\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{1536 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(9/2),x, algorithm="fricas")

[Out]

-1/1536*(15*sqrt(2)*((A - 15*B)*a^3*cos(f*x + e)^5 + 5*(A - 15*B)*a^3*cos(f*x + e)^4 - 8*(A - 15*B)*a^3*cos(f*
x + e)^3 - 20*(A - 15*B)*a^3*cos(f*x + e)^2 + 8*(A - 15*B)*a^3*cos(f*x + e) + 16*(A - 15*B)*a^3 - ((A - 15*B)*
a^3*cos(f*x + e)^4 - 4*(A - 15*B)*a^3*cos(f*x + e)^3 - 12*(A - 15*B)*a^3*cos(f*x + e)^2 + 8*(A - 15*B)*a^3*cos
(f*x + e) + 16*(A - 15*B)*a^3)*sin(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) +
c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/
(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(3*(5*A + 181*B)*a^3*cos(f*x + e)^4
- (191*A - 561*B)*a^3*cos(f*x + e)^3 - 2*(169*A + 537*B)*a^3*cos(f*x + e)^2 + 12*(21*A - 59*B)*a^3*cos(f*x +
e) + 384*(A + B)*a^3 - (3*(5*A + 181*B)*a^3*cos(f*x + e)^3 + 2*(103*A - 9*B)*a^3*cos(f*x + e)^2 - 12*(11*A + 9
1*B)*a^3*cos(f*x + e) - 384*(A + B)*a^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^5*f*cos(f*x + e)^5 + 5*c^
5*f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^3 - 20*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f - (c^5
*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 - 12*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f)*sin(f*
x + e))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(9/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3877 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 974 vs. \(2 (200) = 400\).
time = 0.69, size = 974, normalized size = 4.49 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(9/2),x, algorithm="giac")

[Out]

-1/12288*(120*sqrt(2)*(A*a^3 - 15*B*a^3)*log(-(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/
2*e) + 1))/(c^(9/2)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) + sqrt(2)*(3*A*a^3*sqrt(c) + 3*B*a^3*sqrt(c) + 16*A*a
^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 48*B*a^3*sqrt(c)*(cos(-
1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 24*A*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x
+ 1/2*e) - 1)^2/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 + 312*B*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1
)^2/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 - 48*A*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^3/(cos(-1/4
*pi + 1/2*f*x + 1/2*e) + 1)^3 + 1392*B*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^3/(cos(-1/4*pi + 1/2*f
*x + 1/2*e) + 1)^3 - 250*A*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4/(cos(-1/4*pi + 1/2*f*x + 1/2*e)
+ 1)^4 + 3750*B*a^3*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^4)*(co
s(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^4/(c^5*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/
2*e))) + (48*sqrt(2)*A*a^3*c^(31/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(
cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) - 1392*sqrt(2)*B*a^3*c^(31/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)*sgn(sin
(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) - 24*sqrt(2)*A*a^3*c^(31/2)*(cos(-1/4*pi + 1
/2*f*x + 1/2*e) - 1)^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 - 312*sqrt(2
)*B*a^3*c^(31/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2
*f*x + 1/2*e) + 1)^2 - 16*sqrt(2)*A*a^3*c^(31/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^3*sgn(sin(-1/4*pi + 1/2*
f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3 - 48*sqrt(2)*B*a^3*c^(31/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*
e) - 1)^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3 - 3*sqrt(2)*A*a^3*c^(31/2
)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) +
1)^4 - 3*sqrt(2)*B*a^3*c^(31/2)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(c
os(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^4)/c^20)/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 )}^3}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,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))^3)/(c - c*sin(e + f*x))^(9/2),x)

[Out]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^(9/2), x)

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