3.1.97 \(\int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx\) [97]

Optimal. Leaf size=222 \[ \frac {a^2 (3 A-13 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{256 \sqrt {2} c^{9/2} f}+\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac {a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}}-\frac {a^2 (3 A-13 B) \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (3 A-13 B) \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}} \]

[Out]

1/8*a^2*(A+B)*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(13/2)+1/48*a^2*(3*A-13*B)*cos(f*x+e)^3/f/(c-c*sin(f*x+e))^(
9/2)-1/64*a^2*(3*A-13*B)*cos(f*x+e)/c^2/f/(c-c*sin(f*x+e))^(5/2)+1/256*a^2*(3*A-13*B)*cos(f*x+e)/c^3/f/(c-c*si
n(f*x+e))^(3/2)+1/512*a^2*(3*A-13*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 = 0.36, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3046, 2938, 2759, 2729, 2728, 212} \begin {gather*} \frac {a^2 (3 A-13 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{256 \sqrt {2} c^{9/2} f}+\frac {a^2 (3 A-13 B) \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {a^2 c^2 (A+B) \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}-\frac {a^2 (3 A-13 B) \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(3*A - 13*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(256*Sqrt[2]*c^(9/2)*f)
+ (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(8*f*(c - c*Sin[e + f*x])^(13/2)) + (a^2*(3*A - 13*B)*Cos[e + f*x]^3)/(48*f
*(c - c*Sin[e + f*x])^(9/2)) - (a^2*(3*A - 13*B)*Cos[e + f*x])/(64*c^2*f*(c - c*Sin[e + f*x])^(5/2)) + (a^2*(3
*A - 13*B)*Cos[e + f*x])/(256*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 2729

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

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))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx &=\left (a^2 c^2\right ) \int \frac {\cos ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx\\ &=\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac {1}{16} \left (a^2 (3 A-13 B) c\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{11/2}} \, dx\\ &=\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac {a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}}-\frac {\left (a^2 (3 A-13 B)\right ) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx}{32 c}\\ &=\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac {a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}}-\frac {a^2 (3 A-13 B) \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {\left (a^2 (3 A-13 B)\right ) \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{128 c^3}\\ &=\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac {a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}}-\frac {a^2 (3 A-13 B) \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (3 A-13 B) \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {\left (a^2 (3 A-13 B)\right ) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{512 c^4}\\ &=\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac {a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}}-\frac {a^2 (3 A-13 B) \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (3 A-13 B) \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}-\frac {\left (a^2 (3 A-13 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 )}{256 c^4 f}\\ &=\frac {a^2 (3 A-13 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{256 \sqrt {2} c^{9/2} f}+\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac {a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}}-\frac {a^2 (3 A-13 B) \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 (3 A-13 B) \cos (e+f x)}{256 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 = 1.67, size = 357, normalized size = 1.61 \begin {gather*} \frac {a^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 \left (2013 A \cos \left (\frac {1}{2} (e+f x)\right )+1517 B \cos \left (\frac {1}{2} (e+f x)\right )-999 A \cos \left (\frac {3}{2} (e+f x)\right )-791 B \cos \left (\frac {3}{2} (e+f x)\right )-69 A \cos \left (\frac {5}{2} (e+f x)\right )-725 B \cos \left (\frac {5}{2} (e+f x)\right )-9 A \cos \left (\frac {7}{2} (e+f x)\right )+39 B \cos \left (\frac {7}{2} (e+f x)\right )-(24+24 i) \sqrt [4]{-1} (3 A-13 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+2013 A \sin \left (\frac {1}{2} (e+f x)\right )+1517 B \sin \left (\frac {1}{2} (e+f x)\right )+999 A \sin \left (\frac {3}{2} (e+f x)\right )+791 B \sin \left (\frac {3}{2} (e+f x)\right )-69 A \sin \left (\frac {5}{2} (e+f x)\right )-725 B \sin \left (\frac {5}{2} (e+f x)\right )+9 A \sin \left (\frac {7}{2} (e+f x)\right )-39 B \sin \left (\frac {7}{2} (e+f x)\right )\right )}{6144 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 (c-c \sin (e+f x))^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*(2013*A*Cos[(e + f*x)/2] + 1517*B*Cos[(e + f*x
)/2] - 999*A*Cos[(3*(e + f*x))/2] - 791*B*Cos[(3*(e + f*x))/2] - 69*A*Cos[(5*(e + f*x))/2] - 725*B*Cos[(5*(e +
f*x))/2] - 9*A*Cos[(7*(e + f*x))/2] + 39*B*Cos[(7*(e + f*x))/2] - (24 + 24*I)*(-1)^(1/4)*(3*A - 13*B)*ArcTan[
(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8 + 2013*A*Sin[(e + f*x)/
2] + 1517*B*Sin[(e + f*x)/2] + 999*A*Sin[(3*(e + f*x))/2] + 791*B*Sin[(3*(e + f*x))/2] - 69*A*Sin[(5*(e + f*x)
)/2] - 725*B*Sin[(5*(e + f*x))/2] + 9*A*Sin[(7*(e + f*x))/2] - 39*B*Sin[(7*(e + f*x))/2]))/(6144*f*(Cos[(e + f
*x)/2] + Sin[(e + f*x)/2])^4*(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. \(439\) vs. \(2(195)=390\).
time = 9.96, size = 440, normalized size = 1.98

method	result	size
default	\(\frac {a^{2} \left (-12 \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{5} \left (3 A -13 B \right ) \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+24 \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{5} \left (3 A -13 B \right ) \sin \left (f x +e \right )-3 \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{5} \left (3 A -13 B \right ) \left (\cos ^{4}\left (f x +e \right )\right )+24 \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{5} \left (3 A -13 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+18 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {7}{2}} c^{\frac {3}{2}}-132 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {5}{2}}-264 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {7}{2}}+144 A \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {9}{2}}-78 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {7}{2}} c^{\frac {3}{2}}-452 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {5}{2}}+1144 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {7}{2}}-624 B \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {9}{2}}-72 A \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5}+312 B \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{1536 c^{\frac {19}{2}} \left (\sin \left (f x +e \right )-1\right )^{3} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(440\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1536/c^(19/2)*a^2*(-12*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^5*(3*A-13*B)*sin(f*x+e)
*cos(f*x+e)^2+24*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^5*(3*A-13*B)*sin(f*x+e)-3*arcta
nh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^5*(3*A-13*B)*cos(f*x+e)^4+24*arctanh(1/2*(c+c*sin(f*x
+e))^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^5*(3*A-13*B)*cos(f*x+e)^2+18*A*(c+c*sin(f*x+e))^(7/2)*c^(3/2)-132*A*(c+c
*sin(f*x+e))^(5/2)*c^(5/2)-264*A*(c+c*sin(f*x+e))^(3/2)*c^(7/2)+144*A*(c+c*sin(f*x+e))^(1/2)*c^(9/2)-78*B*(c+c
*sin(f*x+e))^(7/2)*c^(3/2)-452*B*(c+c*sin(f*x+e))^(5/2)*c^(5/2)+1144*B*(c+c*sin(f*x+e))^(3/2)*c^(7/2)-624*B*(c
+c*sin(f*x+e))^(1/2)*c^(9/2)-72*A*2^(1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^5+312*B*2^(1/2
)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^5)*(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))^2*(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)^2/(-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. 694 vs. \(2 (205) = 410\).
time = 0.87, size = 694, normalized size = 3.13 \begin {gather*} -\frac {3 \, \sqrt {2} {\left ({\left (3 \, A - 13 \, B\right )} a^{2} \cos \left (f x + e\right )^{5} + 5 \, {\left (3 \, A - 13 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} - 8 \, {\left (3 \, A - 13 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} - 20 \, {\left (3 \, A - 13 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} + 8 \, {\left (3 \, A - 13 \, B\right )} a^{2} \cos \left (f x + e\right ) + 16 \, {\left (3 \, A - 13 \, B\right )} a^{2} - {\left ({\left (3 \, A - 13 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} - 4 \, {\left (3 \, A - 13 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} - 12 \, {\left (3 \, A - 13 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} + 8 \, {\left (3 \, A - 13 \, B\right )} a^{2} \cos \left (f x + e\right ) + 16 \, {\left (3 \, A - 13 \, B\right )} a^{2}\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 (3 \, A - 13 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} + {\left (39 \, A + 343 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} + 2 \, {\left (129 \, A + 209 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 12 \, {\left (13 \, A + 29 \, B\right )} a^{2} \cos \left (f x + e\right ) - 384 \, {\left (A + B\right )} a^{2} - {\left (3 \, {\left (3 \, A - 13 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} - 2 \, {\left (15 \, A + 191 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} + 12 \, {\left (19 \, A + 3 \, B\right )} a^{2} \cos \left (f x + e\right ) + 384 \, {\left (A + B\right )} a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3072 \, {\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))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(9/2),x, algorithm="fricas")

[Out]

-1/3072*(3*sqrt(2)*((3*A - 13*B)*a^2*cos(f*x + e)^5 + 5*(3*A - 13*B)*a^2*cos(f*x + e)^4 - 8*(3*A - 13*B)*a^2*c
os(f*x + e)^3 - 20*(3*A - 13*B)*a^2*cos(f*x + e)^2 + 8*(3*A - 13*B)*a^2*cos(f*x + e) + 16*(3*A - 13*B)*a^2 - (
(3*A - 13*B)*a^2*cos(f*x + e)^4 - 4*(3*A - 13*B)*a^2*cos(f*x + e)^3 - 12*(3*A - 13*B)*a^2*cos(f*x + e)^2 + 8*(
3*A - 13*B)*a^2*cos(f*x + e) + 16*(3*A - 13*B)*a^2)*sin(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 - 2*sqrt(2)*s
qrt(-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*(3*A - 13*B
)*a^2*cos(f*x + e)^4 + (39*A + 343*B)*a^2*cos(f*x + e)^3 + 2*(129*A + 209*B)*a^2*cos(f*x + e)^2 - 12*(13*A + 2
9*B)*a^2*cos(f*x + e) - 384*(A + B)*a^2 - (3*(3*A - 13*B)*a^2*cos(f*x + e)^3 - 2*(15*A + 191*B)*a^2*cos(f*x +
e)^2 + 12*(19*A + 3*B)*a^2*cos(f*x + e) + 384*(A + B)*a^2)*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))**2*(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. 781 vs. \(2 (205) = 410\).
time = 0.64, size = 781, normalized size = 3.52 \begin {gather*} \frac {\frac {24 \, \sqrt {2} {\left (3 \, A a^{2} \sqrt {c} - 13 \, B a^{2} \sqrt {c}\right )} \log \left (-\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}\right )}{c^{5} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} {\left (3 \, A a^{2} \sqrt {c} + 3 \, B a^{2} \sqrt {c} + \frac {32 \, B a^{2} \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - \frac {24 \, A a^{2} \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} + \frac {72 \, B a^{2} \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} - \frac {96 \, B a^{2} \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {150 \, A a^{2} \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}} - \frac {650 \, B a^{2} \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}}\right )} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}}{c^{5} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\frac {96 \, \sqrt {2} B a^{2} c^{\frac {31}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {24 \, \sqrt {2} A a^{2} c^{\frac {31}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} - \frac {72 \, \sqrt {2} B a^{2} c^{\frac {31}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} - \frac {32 \, \sqrt {2} B a^{2} c^{\frac {31}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} - \frac {3 \, \sqrt {2} A a^{2} c^{\frac {31}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}} - \frac {3 \, \sqrt {2} B a^{2} c^{\frac {31}{2}} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}}}{c^{20}}}{24576 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24576*(24*sqrt(2)*(3*A*a^2*sqrt(c) - 13*B*a^2*sqrt(c))*log(-(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*p
i + 1/2*f*x + 1/2*e) + 1))/(c^5*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) - sqrt(2)*(3*A*a^2*sqrt(c) + 3*B*a^2*sqrt
(c) + 32*B*a^2*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^2*sq
rt(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 + 72*B*a^2*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 - 96*B*a^2*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 + 150*A*a^2*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 - 650*B*a^2*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)*(cos(-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))) - (96*sqrt(2)*B*a^2*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^2*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 - 72
*sqrt(2)*B*a^2*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*p
i + 1/2*f*x + 1/2*e) + 1)^2 - 32*sqrt(2)*B*a^2*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^2*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^2*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)/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 )}^2}{{\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))^2)/(c - c*sin(e + f*x))^(9/2),x)

[Out]

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

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