3.578 \(\int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx\)
Optimal. Leaf size=229 \[ -\frac{16 a^2 (c+13 d) \cos (e+f x)}{105 d f (c+d)^4 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{8 a^2 (c+13 d) \cos (e+f x)}{105 d f (c+d)^3 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a^2 (c+13 d) \cos (e+f x)}{35 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac{2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}} \]
[Out]
(2*a^2*(c - d)*Cos[e + f*x])/(7*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2)) - (2*a^2*(c +
13*d)*Cos[e + f*x])/(35*d*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (8*a^2*(c + 13*d
)*Cos[e + f*x])/(105*d*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (16*a^2*(c + 13*d)*C
os[e + f*x])/(105*d*(c + d)^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])
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Rubi [A] time = 0.447942, antiderivative size = 229, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used =
{2762, 21, 2772, 2771} \[ -\frac{16 a^2 (c+13 d) \cos (e+f x)}{105 d f (c+d)^4 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{8 a^2 (c+13 d) \cos (e+f x)}{105 d f (c+d)^3 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a^2 (c+13 d) \cos (e+f x)}{35 d f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac{2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(9/2),x]
[Out]
(2*a^2*(c - d)*Cos[e + f*x])/(7*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2)) - (2*a^2*(c +
13*d)*Cos[e + f*x])/(35*d*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (8*a^2*(c + 13*d
)*Cos[e + f*x])/(105*d*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (16*a^2*(c + 13*d)*C
os[e + f*x])/(105*d*(c + d)^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])
Rule 2762
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c
+ a*d)), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*
Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
&& (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))
Rule 21
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])
Rule 2772
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[((b*c - a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]]), x]
+ Dist[((2*n + 3)*(b*c - a*d))/(2*b*(n + 1)*(c^2 - d^2)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n
+ 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &
& LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
Rule 2771
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(3/2), x_Symbol] :> Sim
p[(-2*b^2*Cos[e + f*x])/(f*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b,
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx &=\frac{2 a^2 (c-d) \cos (e+f x)}{7 d (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac{(2 a) \int \frac{-\frac{1}{2} a (c+13 d)-\frac{1}{2} a (c+13 d) \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}} \, dx}{7 d (c+d)}\\ &=\frac{2 a^2 (c-d) \cos (e+f x)}{7 d (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}+\frac{(a (c+13 d)) \int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx}{7 d (c+d)}\\ &=\frac{2 a^2 (c-d) \cos (e+f x)}{7 d (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac{2 a^2 (c+13 d) \cos (e+f x)}{35 d (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}+\frac{(4 a (c+13 d)) \int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx}{35 d (c+d)^2}\\ &=\frac{2 a^2 (c-d) \cos (e+f x)}{7 d (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac{2 a^2 (c+13 d) \cos (e+f x)}{35 d (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac{8 a^2 (c+13 d) \cos (e+f x)}{105 d (c+d)^3 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac{(8 a (c+13 d)) \int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{105 d (c+d)^3}\\ &=\frac{2 a^2 (c-d) \cos (e+f x)}{7 d (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac{2 a^2 (c+13 d) \cos (e+f x)}{35 d (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac{8 a^2 (c+13 d) \cos (e+f x)}{105 d (c+d)^3 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac{16 a^2 (c+13 d) \cos (e+f x)}{105 d (c+d)^4 f \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.53433, size = 193, normalized size = 0.84 \[ -\frac{2 a \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\left (469 c^2 d+35 c^3+191 c d^2+117 d^3\right ) \sin (e+f x)-2 d \left (7 c^2+92 c d+13 d^2\right ) \cos (2 (e+f x))+147 c^2 d+175 c^3-2 c d^2 \sin (3 (e+f x))+253 c d^2-26 d^3 \sin (3 (e+f x))+41 d^3\right )}{105 f (c+d)^4 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(9/2),x]
[Out]
(-2*a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(175*c^3 + 147*c^2*d + 253*c*d^2 + 41*d
^3 - 2*d*(7*c^2 + 92*c*d + 13*d^2)*Cos[2*(e + f*x)] + (35*c^3 + 469*c^2*d + 191*c*d^2 + 117*d^3)*Sin[e + f*x]
- 2*c*d^2*Sin[3*(e + f*x)] - 26*d^3*Sin[3*(e + f*x)]))/(105*(c + d)^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*
(c + d*Sin[e + f*x])^(7/2))
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Maple [B] time = 0.319, size = 979, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(9/2),x)
[Out]
-2/105/f/(c+d)^4*(c+d*sin(f*x+e))^(1/2)*(a*(1+sin(f*x+e)))^(3/2)*(-64*cos(f*x+e)^8*c*d^6+104*sin(f*x+e)*cos(f*
x+e)^8*d^7+8*sin(f*x+e)*cos(f*x+e)^8*c*d^6+29*sin(f*x+e)*cos(f*x+e)^6*c^3*d^4+371*sin(f*x+e)*cos(f*x+e)^6*c^2*
d^5-113*sin(f*x+e)*cos(f*x+e)^6*c*d^6+106*sin(f*x+e)*cos(f*x+e)^4*c^5*d^2+754*sin(f*x+e)*cos(f*x+e)^4*c^4*d^3+
72*sin(f*x+e)*cos(f*x+e)^4*c^3*d^4-944*sin(f*x+e)*cos(f*x+e)^4*c^2*d^5+382*sin(f*x+e)*cos(f*x+e)^4*c*d^6+7*sin
(f*x+e)*cos(f*x+e)^2*c^6*d-887*sin(f*x+e)*cos(f*x+e)^2*c^5*d^2-1797*sin(f*x+e)*cos(f*x+e)^2*c^4*d^3-25*sin(f*x
+e)*cos(f*x+e)^2*c^3*d^4+997*sin(f*x+e)*cos(f*x+e)^2*c^2*d^5-397*sin(f*x+e)*cos(f*x+e)^2*c*d^6+280*c^7+504*c^6
*d+424*c^2*d^5-120*c*d^6-776*c^4*d^3-152*d^7-296*c^5*d^2+136*c^3*d^4-455*sin(f*x+e)*cos(f*x+e)^6*d^7+4*cos(f*x
+e)^6*c^4*d^3-149*cos(f*x+e)^6*c^3*d^4-443*cos(f*x+e)^6*c^2*d^5+345*cos(f*x+e)^6*c*d^6+750*sin(f*x+e)*cos(f*x+
e)^4*d^7-112*cos(f*x+e)^4*c^6*d-670*cos(f*x+e)^4*c^5*d^2-1398*cos(f*x+e)^4*c^4*d^3-4*cos(f*x+e)^8*c^2*d^5+56*c
os(f*x+e)^4*c^3*d^4+1232*cos(f*x+e)^4*c^2*d^5-618*cos(f*x+e)^4*c*d^6-35*sin(f*x+e)*cos(f*x+e)^2*c^7-551*sin(f*
x+e)*cos(f*x+e)^2*d^7-259*cos(f*x+e)^2*c^6*d+1035*cos(f*x+e)^2*c^5*d^2+2185*cos(f*x+e)^2*c^4*d^3-43*cos(f*x+e)
^2*c^3*d^4-1209*cos(f*x+e)^2*c^2*d^5+457*cos(f*x+e)^2*c*d^6-504*sin(f*x+e)*c^6*d+296*sin(f*x+e)*c^5*d^2+776*si
n(f*x+e)*c^4*d^3-136*sin(f*x+e)*c^3*d^4-424*sin(f*x+e)*c^2*d^5+120*sin(f*x+e)*c*d^6-156*cos(f*x+e)^8*d^7+635*c
os(f*x+e)^6*d^7-954*cos(f*x+e)^4*d^7-105*cos(f*x+e)^2*c^7+627*cos(f*x+e)^2*d^7-280*sin(f*x+e)*c^7+152*sin(f*x+
e)*d^7)/cos(f*x+e)^3/(cos(f*x+e)^2*d^2+c^2-d^2)^4
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Maxima [B] time = 2.41802, size = 1013, normalized size = 4.42 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm="maxima")
[Out]
-2/105*((175*c^4 + 133*c^3*d + 69*c^2*d^2 + 15*c*d^3)*a^(3/2) - 3*(35*c^4 - 385*c^3*d - 189*c^2*d^2 - 67*c*d^3
- 10*d^4)*a^(3/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 18*(35*c^4 - 28*c^3*d + 166*c^2*d^2 + 44*c*d^3 + 7*d^4)*a
^(3/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 14*(35*c^4 - 220*c^3*d + 102*c^2*d^2 - 244*c*d^3 - 25*d^4)*a^(3/2
)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 42*(20*c^4 - 61*c^3*d + 117*c^2*d^2 - 55*c*d^3 + 35*d^4)*a^(3/2)*sin(f
*x + e)^4/(cos(f*x + e) + 1)^4 - 42*(20*c^4 - 61*c^3*d + 117*c^2*d^2 - 55*c*d^3 + 35*d^4)*a^(3/2)*sin(f*x + e)
^5/(cos(f*x + e) + 1)^5 + 14*(35*c^4 - 220*c^3*d + 102*c^2*d^2 - 244*c*d^3 - 25*d^4)*a^(3/2)*sin(f*x + e)^6/(c
os(f*x + e) + 1)^6 - 18*(35*c^4 - 28*c^3*d + 166*c^2*d^2 + 44*c*d^3 + 7*d^4)*a^(3/2)*sin(f*x + e)^7/(cos(f*x +
e) + 1)^7 + 3*(35*c^4 - 385*c^3*d - 189*c^2*d^2 - 67*c*d^3 - 10*d^4)*a^(3/2)*sin(f*x + e)^8/(cos(f*x + e) + 1
)^8 - (175*c^4 + 133*c^3*d + 69*c^2*d^2 + 15*c*d^3)*a^(3/2)*sin(f*x + e)^9/(cos(f*x + e) + 1)^9)*(sin(f*x + e)
^2/(cos(f*x + e) + 1)^2 + 1)^3/((c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 + 3*(c^4 + 4*c^3*d + 6*c^2*d^2 + 4*
c*d^3 + d^4)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*(c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*sin(f*x + e)^
4/(cos(f*x + e) + 1)^4 + (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6)*(c +
2*d*sin(f*x + e)/(cos(f*x + e) + 1) + c*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)^(9/2)*f)
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Fricas [B] time = 3.65345, size = 2163, normalized size = 9.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm="fricas")
[Out]
2/105*(8*(a*c*d^2 + 13*a*d^3)*cos(f*x + e)^4 - 140*a*c^3 + 308*a*c^2*d - 244*a*c*d^2 + 76*a*d^3 + 4*(7*a*c^2*d
+ 92*a*c*d^2 + 13*a*d^3)*cos(f*x + e)^3 - (35*a*c^3 + 441*a*c^2*d - 167*a*c*d^2 + 195*a*d^3)*cos(f*x + e)^2 -
(175*a*c^3 + 161*a*c^2*d + 437*a*c*d^2 + 67*a*d^3)*cos(f*x + e) + (140*a*c^3 - 308*a*c^2*d + 244*a*c*d^2 - 76
*a*d^3 + 8*(a*c*d^2 + 13*a*d^3)*cos(f*x + e)^3 - 4*(7*a*c^2*d + 90*a*c*d^2 - 13*a*d^3)*cos(f*x + e)^2 - (35*a*
c^3 + 469*a*c^2*d + 193*a*c*d^2 + 143*a*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f
*x + e) + c)/((c^4*d^4 + 4*c^3*d^5 + 6*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e)^5 + (4*c^5*d^3 + 17*c^4*d^4 + 2
8*c^3*d^5 + 22*c^2*d^6 + 8*c*d^7 + d^8)*f*cos(f*x + e)^4 - 2*(3*c^6*d^2 + 12*c^5*d^3 + 19*c^4*d^4 + 16*c^3*d^5
+ 9*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e)^3 - 2*(2*c^7*d + 11*c^6*d^2 + 28*c^5*d^3 + 43*c^4*d^4 + 42*c^3*d^
5 + 25*c^2*d^6 + 8*c*d^7 + d^8)*f*cos(f*x + e)^2 + (c^8 + 4*c^7*d + 12*c^6*d^2 + 28*c^5*d^3 + 38*c^4*d^4 + 28*
c^3*d^5 + 12*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e) + (c^8 + 8*c^7*d + 28*c^6*d^2 + 56*c^5*d^3 + 70*c^4*d^4 +
56*c^3*d^5 + 28*c^2*d^6 + 8*c*d^7 + d^8)*f + ((c^4*d^4 + 4*c^3*d^5 + 6*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e
)^4 - 4*(c^5*d^3 + 4*c^4*d^4 + 6*c^3*d^5 + 4*c^2*d^6 + c*d^7)*f*cos(f*x + e)^3 - 2*(3*c^6*d^2 + 14*c^5*d^3 + 2
7*c^4*d^4 + 28*c^3*d^5 + 17*c^2*d^6 + 6*c*d^7 + d^8)*f*cos(f*x + e)^2 + 4*(c^7*d + 4*c^6*d^2 + 7*c^5*d^3 + 8*c
^4*d^4 + 7*c^3*d^5 + 4*c^2*d^6 + c*d^7)*f*cos(f*x + e) + (c^8 + 8*c^7*d + 28*c^6*d^2 + 56*c^5*d^3 + 70*c^4*d^4
+ 56*c^3*d^5 + 28*c^2*d^6 + 8*c*d^7 + d^8)*f)*sin(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+a*sin(f*x+e))**(3/2)/(c+d*sin(f*x+e))**(9/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm="giac")
[Out]
integrate((a*sin(f*x + e) + a)^(3/2)/(d*sin(f*x + e) + c)^(9/2), x)