3.115 \(\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx\)
Optimal. Leaf size=242 \[ -\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac{64 c^2 (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac{512 c^3 (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{105 a^2 f}+\frac{2048 c^4 (7 A-13 B) \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{105 a^2 f}-\frac{(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac{16 c (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f} \]
[Out]
(2048*(7*A - 13*B)*c^4*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(105*a^2*f) - (512*(7*A - 13*B)*c^3*Sec[e + f*x]
*(c - c*Sin[e + f*x])^(3/2))/(105*a^2*f) - (64*(7*A - 13*B)*c^2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(105*
a^2*f) - (16*(7*A - 13*B)*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(105*a^2*f) - ((7*A - 13*B)*Sec[e + f*x]*
(c - c*Sin[e + f*x])^(9/2))/(21*a^2*f) - ((A - B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(13/2))/(3*a^2*c^2*f)
________________________________________________________________________________________
Rubi [A] time = 0.65076, antiderivative size = 242, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used =
{2967, 2855, 2674, 2673} \[ -\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac{64 c^2 (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac{512 c^3 (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{105 a^2 f}+\frac{2048 c^4 (7 A-13 B) \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{105 a^2 f}-\frac{(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac{16 c (7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^2,x]
[Out]
(2048*(7*A - 13*B)*c^4*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(105*a^2*f) - (512*(7*A - 13*B)*c^3*Sec[e + f*x]
*(c - c*Sin[e + f*x])^(3/2))/(105*a^2*f) - (64*(7*A - 13*B)*c^2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(105*
a^2*f) - (16*(7*A - 13*B)*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(105*a^2*f) - ((7*A - 13*B)*Sec[e + f*x]*
(c - c*Sin[e + f*x])^(9/2))/(21*a^2*f) - ((A - B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(13/2))/(3*a^2*c^2*f)
Rule 2967
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]))
Rule 2855
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*(p +
1)), x] + Dist[(b*(a*d*m + b*c*(m + p + 1)))/(a*g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x]
)^(m - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, -1] && LtQ[p, -1]
Rule 2674
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m + p)), x] + Dist[(a*(2*m + p - 1))/(m + p), Int[(
g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0]
&& IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]
Rule 2673
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m - 1)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
Rubi steps
\begin{align*} \int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx &=\frac{\int \sec ^4(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{13/2} \, dx}{a^2 c^2}\\ &=-\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac{(7 A-13 B) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{11/2} \, dx}{6 a^2 c}\\ &=-\frac{(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac{(8 (7 A-13 B)) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{9/2} \, dx}{21 a^2}\\ &=-\frac{16 (7 A-13 B) c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}-\frac{(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac{(32 (7 A-13 B) c) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{7/2} \, dx}{35 a^2}\\ &=-\frac{64 (7 A-13 B) c^2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac{16 (7 A-13 B) c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}-\frac{(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac{\left (256 (7 A-13 B) c^2\right ) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{5/2} \, dx}{105 a^2}\\ &=-\frac{512 (7 A-13 B) c^3 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{105 a^2 f}-\frac{64 (7 A-13 B) c^2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac{16 (7 A-13 B) c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}-\frac{(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}-\frac{\left (1024 (7 A-13 B) c^3\right ) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx}{105 a^2}\\ &=\frac{2048 (7 A-13 B) c^4 \sec (e+f x) \sqrt{c-c \sin (e+f x)}}{105 a^2 f}-\frac{512 (7 A-13 B) c^3 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{105 a^2 f}-\frac{64 (7 A-13 B) c^2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{105 a^2 f}-\frac{16 (7 A-13 B) c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{105 a^2 f}-\frac{(7 A-13 B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{21 a^2 f}-\frac{(A-B) \sec ^3(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^2 c^2 f}\\ \end{align*}
Mathematica [B] time = 6.8705, size = 953, normalized size = 3.94 \[ -\frac{(26 A-83 B) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^4 \sin \left (\frac{3}{2} (e+f x)\right ) (c-c \sin (e+f x))^{9/2}}{12 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}-\frac{(2 A-13 B) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^4 \sin \left (\frac{5}{2} (e+f x)\right ) (c-c \sin (e+f x))^{9/2}}{20 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}-\frac{B \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^4 \sin \left (\frac{7}{2} (e+f x)\right ) (c-c \sin (e+f x))^{9/2}}{28 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}+\frac{(164 A-351 B) \sin \left (\frac{1}{2} (e+f x)\right ) \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}}{4 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}+\frac{(164 A-351 B) \cos \left (\frac{1}{2} (e+f x)\right ) \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}}{4 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}+\frac{(26 A-83 B) \cos \left (\frac{3}{2} (e+f x)\right ) \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}}{12 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}-\frac{(2 A-13 B) \cos \left (\frac{5}{2} (e+f x)\right ) \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}}{20 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}+\frac{B \cos \left (\frac{7}{2} (e+f x)\right ) \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}}{28 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}+\frac{32 (2 A-3 B) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^3 (c-c \sin (e+f x))^{9/2}}{f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2}-\frac{32 (A-B) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^{9/2}}{3 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 (\sin (e+f x) a+a)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^2,x]
[Out]
(-32*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^(9/2))/(3*f*(Cos[(e + f*x)/2] - Sin[(e
+ f*x)/2])^9*(a + a*Sin[e + f*x])^2) + (32*(2*A - 3*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e +
f*x])^(9/2))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) + ((164*A - 351*B)*Cos[(e + f
*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2))/(4*f*(Cos[(e + f*x)/2] - Sin[(e + f
*x)/2])^9*(a + a*Sin[e + f*x])^2) + ((26*A - 83*B)*Cos[(3*(e + f*x))/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^
4*(c - c*Sin[e + f*x])^(9/2))/(12*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) - ((2*A -
13*B)*Cos[(5*(e + f*x))/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2))/(20*f*(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) + (B*Cos[(7*(e + f*x))/2]*(Cos[(e + f*x)/2] + Sin[(e +
f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2))/(28*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) +
((164*A - 351*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2))/(4*f*(C
os[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) - ((26*A - 83*B)*(Cos[(e + f*x)/2] + Sin[(e + f*
x)/2])^4*(c - c*Sin[e + f*x])^(9/2)*Sin[(3*(e + f*x))/2])/(12*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a
*Sin[e + f*x])^2) - ((2*A - 13*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2)*Sin[(5*(e
+ f*x))/2])/(20*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a + a*Sin[e + f*x])^2) - (B*(Cos[(e + f*x)/2] + Si
n[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2)*Sin[(7*(e + f*x))/2])/(28*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])
^9*(a + a*Sin[e + f*x])^2)
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Maple [A] time = 1.054, size = 143, normalized size = 0.6 \begin{align*} -{\frac{2\,{c}^{5} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( 15\,B\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( 196\,A-544\,B \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) + \left ( 7448\,A-13592\,B \right ) \sin \left ( fx+e \right ) + \left ( 21\,A-114\,B \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( -1848\,A+3732\,B \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+6888\,A-13032\,B \right ) }{105\,{a}^{2} \left ( 1+\sin \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) f}{\frac{1}{\sqrt{c-c\sin \left ( fx+e \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^2,x)
[Out]
-2/105*c^5/a^2*(-1+sin(f*x+e))/(1+sin(f*x+e))*(15*B*sin(f*x+e)*cos(f*x+e)^4+(196*A-544*B)*cos(f*x+e)^2*sin(f*x
+e)+(7448*A-13592*B)*sin(f*x+e)+(21*A-114*B)*cos(f*x+e)^4+(-1848*A+3732*B)*cos(f*x+e)^2+6888*A-13032*B)/cos(f*
x+e)/(c-c*sin(f*x+e))^(1/2)/f
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Maxima [B] time = 1.60448, size = 1029, normalized size = 4.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^2,x, algorithm="maxima")
[Out]
-2/105*(7*(723*c^(9/2) + 2184*c^(9/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 5370*c^(9/2)*sin(f*x + e)^2/(cos(f*x +
e) + 1)^2 + 10696*c^(9/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15021*c^(9/2)*sin(f*x + e)^4/(cos(f*x + e) +
1)^4 + 21168*c^(9/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 20748*c^(9/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 +
21168*c^(9/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 15021*c^(9/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 10696
*c^(9/2)*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 5370*c^(9/2)*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 2184*c^(9/
2)*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 723*c^(9/2)*sin(f*x + e)^12/(cos(f*x + e) + 1)^12)*A/((a^2 + 3*a^2*
sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(9/2)) - 2*(4707*c^(9/2) + 14121*c^(9/2)*sin(f*x + e)/(cos(
f*x + e) + 1) + 35250*c^(9/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 68549*c^(9/2)*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3 + 99549*c^(9/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 134802*c^(9/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)
^5 + 138012*c^(9/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 134802*c^(9/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 +
99549*c^(9/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 68549*c^(9/2)*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 35250
*c^(9/2)*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 14121*c^(9/2)*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 4707*c^
(9/2)*sin(f*x + e)^12/(cos(f*x + e) + 1)^12)*B/((a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x +
e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1
)^(9/2)))/f
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Fricas [A] time = 1.57781, size = 396, normalized size = 1.64 \begin{align*} \frac{2 \,{\left (3 \,{\left (7 \, A - 38 \, B\right )} c^{4} \cos \left (f x + e\right )^{4} - 12 \,{\left (154 \, A - 311 \, B\right )} c^{4} \cos \left (f x + e\right )^{2} + 24 \,{\left (287 \, A - 543 \, B\right )} c^{4} +{\left (15 \, B c^{4} \cos \left (f x + e\right )^{4} + 4 \,{\left (49 \, A - 136 \, B\right )} c^{4} \cos \left (f x + e\right )^{2} + 8 \,{\left (931 \, A - 1699 \, B\right )} c^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{105 \,{\left (a^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{2} f \cos \left (f x + e\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^2,x, algorithm="fricas")
[Out]
2/105*(3*(7*A - 38*B)*c^4*cos(f*x + e)^4 - 12*(154*A - 311*B)*c^4*cos(f*x + e)^2 + 24*(287*A - 543*B)*c^4 + (1
5*B*c^4*cos(f*x + e)^4 + 4*(49*A - 136*B)*c^4*cos(f*x + e)^2 + 8*(931*A - 1699*B)*c^4)*sin(f*x + e))*sqrt(-c*s
in(f*x + e) + c)/(a^2*f*cos(f*x + e)*sin(f*x + e) + a^2*f*cos(f*x + e))
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(9/2)/(a+a*sin(f*x+e))**2,x)
[Out]
Timed out
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Giac [B] time = 2.53848, size = 1754, normalized size = 7.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^2,x, algorithm="giac")
[Out]
-1/105*(4*(3577*sqrt(2)*A*a^16*sqrt(c) - 7483*sqrt(2)*B*a^16*sqrt(c) - 5110*A*a^16*sqrt(c) + 10690*B*a^16*sqrt
(c) - 1610*sqrt(2)*A*c^(25/2) + 2870*sqrt(2)*B*c^(25/2) + 2100*A*c^(25/2) - 3780*B*c^(25/2))*sgn(tan(1/2*f*x +
1/2*e) - 1)/(5*sqrt(2)*a^2*c^8 - 7*a^2*c^8) + ((((((((2261*A*a^14*c^8*sgn(tan(1/2*f*x + 1/2*e) - 1) - 4934*B*
a^14*c^8*sgn(tan(1/2*f*x + 1/2*e) - 1))*tan(1/2*f*x + 1/2*e)/c^12 + 105*(17*A*a^14*c^8*sgn(tan(1/2*f*x + 1/2*e
) - 1) - 32*B*a^14*c^8*sgn(tan(1/2*f*x + 1/2*e) - 1))/c^12)*tan(1/2*f*x + 1/2*e) + 7*(913*A*a^14*c^8*sgn(tan(1
/2*f*x + 1/2*e) - 1) - 1972*B*a^14*c^8*sgn(tan(1/2*f*x + 1/2*e) - 1))/c^12)*tan(1/2*f*x + 1/2*e) + 35*(169*A*a
^14*c^8*sgn(tan(1/2*f*x + 1/2*e) - 1) - 346*B*a^14*c^8*sgn(tan(1/2*f*x + 1/2*e) - 1))/c^12)*tan(1/2*f*x + 1/2*
e) + 35*(169*A*a^14*c^8*sgn(tan(1/2*f*x + 1/2*e) - 1) - 346*B*a^14*c^8*sgn(tan(1/2*f*x + 1/2*e) - 1))/c^12)*ta
n(1/2*f*x + 1/2*e) + 7*(913*A*a^14*c^8*sgn(tan(1/2*f*x + 1/2*e) - 1) - 1972*B*a^14*c^8*sgn(tan(1/2*f*x + 1/2*e
) - 1))/c^12)*tan(1/2*f*x + 1/2*e) + 105*(17*A*a^14*c^8*sgn(tan(1/2*f*x + 1/2*e) - 1) - 32*B*a^14*c^8*sgn(tan(
1/2*f*x + 1/2*e) - 1))/c^12)*tan(1/2*f*x + 1/2*e) + (2261*A*a^14*c^8*sgn(tan(1/2*f*x + 1/2*e) - 1) - 4934*B*a^
14*c^8*sgn(tan(1/2*f*x + 1/2*e) - 1))/c^12)/(c*tan(1/2*f*x + 1/2*e)^2 + c)^(7/2) + 2240*(3*(sqrt(c)*tan(1/2*f*
x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^5*A*c^5*sgn(tan(1/2*f*x + 1/2*e) - 1) - 6*(sqrt(c)*tan(1/2*f*
x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^5*B*c^5*sgn(tan(1/2*f*x + 1/2*e) - 1) + 15*(sqrt(c)*tan(1/2*f
*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^4*A*c^(11/2)*sgn(tan(1/2*f*x + 1/2*e) - 1) - 24*(sqrt(c)*tan
(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^4*B*c^(11/2)*sgn(tan(1/2*f*x + 1/2*e) - 1) - 10*(sqrt(
c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^3*A*c^6*sgn(tan(1/2*f*x + 1/2*e) - 1) + 16*(sqrt
(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^3*B*c^6*sgn(tan(1/2*f*x + 1/2*e) - 1) - 30*(sqr
t(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^2*A*c^(13/2)*sgn(tan(1/2*f*x + 1/2*e) - 1) + 4
8*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^2*B*c^(13/2)*sgn(tan(1/2*f*x + 1/2*e) -
1) + 27*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))*A*c^7*sgn(tan(1/2*f*x + 1/2*e) - 1
) - 42*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))*B*c^7*sgn(tan(1/2*f*x + 1/2*e) - 1)
- 5*A*c^(15/2)*sgn(tan(1/2*f*x + 1/2*e) - 1) + 8*B*c^(15/2)*sgn(tan(1/2*f*x + 1/2*e) - 1))/(((sqrt(c)*tan(1/2
*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^2 + 2*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x +
1/2*e)^2 + c))*sqrt(c) - c)^3*a^2))/f