3.570 \(\int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx\)

Optimal. Leaf size=142 \[ -\frac {16 a \cos (e+f x)}{15 f (c+d)^3 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {8 a \cos (e+f x)}{15 f (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}} \]

[Out]

-2/5*a*cos(f*x+e)/(c+d)/f/(c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2)-8/15*a*cos(f*x+e)/(c+d)^2/f/(c+d*sin(f
*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)-16/15*a*cos(f*x+e)/(c+d)^3/f/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)

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Rubi [A]  time = 0.29, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2772, 2771} \[ -\frac {16 a \cos (e+f x)}{15 f (c+d)^3 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {8 a \cos (e+f x)}{15 f (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(7/2),x]

[Out]

(-2*a*Cos[e + f*x])/(5*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (8*a*Cos[e + f*x])/(15
*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (16*a*Cos[e + f*x])/(15*(c + d)^3*f*Sqrt[a
+ a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])

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]

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx &=-\frac {2 a \cos (e+f x)}{5 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}+\frac {4 \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx}{5 (c+d)}\\ &=-\frac {2 a \cos (e+f x)}{5 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a \cos (e+f x)}{15 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac {8 \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{15 (c+d)^2}\\ &=-\frac {2 a \cos (e+f x)}{5 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a \cos (e+f x)}{15 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {16 a \cos (e+f x)}{15 (c+d)^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 128, normalized size = 0.90 \[ -\frac {2 \sqrt {a (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (15 c^2+4 d (5 c+d) \sin (e+f x)+10 c d+8 d^2 \sin ^2(e+f x)+3 d^2\right )}{15 f (c+d)^3 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(7/2),x]

[Out]

(-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(15*c^2 + 10*c*d + 3*d^2 + 4*d*(5*c + d)*
Sin[e + f*x] + 8*d^2*Sin[e + f*x]^2))/(15*(c + d)^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x
])^(5/2))

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fricas [B]  time = 0.52, size = 556, normalized size = 3.92 \[ \frac {2 \, {\left (8 \, d^{2} \cos \left (f x + e\right )^{3} - 4 \, {\left (5 \, c d - d^{2}\right )} \cos \left (f x + e\right )^{2} - 15 \, c^{2} + 10 \, c d - 7 \, d^{2} - {\left (15 \, c^{2} + 10 \, c d + 11 \, d^{2}\right )} \cos \left (f x + e\right ) - {\left (8 \, d^{2} \cos \left (f x + e\right )^{2} - 15 \, c^{2} + 10 \, c d - 7 \, d^{2} + 4 \, {\left (5 \, c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{15 \, {\left ({\left (c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{4} - 3 \, {\left (c^{4} d^{2} + 3 \, c^{3} d^{3} + 3 \, c^{2} d^{4} + c d^{5}\right )} f \cos \left (f x + e\right )^{3} - {\left (3 \, c^{5} d + 12 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 18 \, c^{2} d^{4} + 9 \, c d^{5} + 2 \, d^{6}\right )} f \cos \left (f x + e\right )^{2} + {\left (c^{6} + 3 \, c^{5} d + 6 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 9 \, c^{2} d^{4} + 3 \, c d^{5}\right )} f \cos \left (f x + e\right ) + {\left (c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f - {\left ({\left (c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{3} + {\left (3 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 12 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{2} - {\left (3 \, c^{5} d + 9 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right ) - {\left (c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f\right )} \sin \left (f x + e\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

2/15*(8*d^2*cos(f*x + e)^3 - 4*(5*c*d - d^2)*cos(f*x + e)^2 - 15*c^2 + 10*c*d - 7*d^2 - (15*c^2 + 10*c*d + 11*
d^2)*cos(f*x + e) - (8*d^2*cos(f*x + e)^2 - 15*c^2 + 10*c*d - 7*d^2 + 4*(5*c*d + d^2)*cos(f*x + e))*sin(f*x +
e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^3*d^3 + 3*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e)^4
- 3*(c^4*d^2 + 3*c^3*d^3 + 3*c^2*d^4 + c*d^5)*f*cos(f*x + e)^3 - (3*c^5*d + 12*c^4*d^2 + 20*c^3*d^3 + 18*c^2*d
^4 + 9*c*d^5 + 2*d^6)*f*cos(f*x + e)^2 + (c^6 + 3*c^5*d + 6*c^4*d^2 + 10*c^3*d^3 + 9*c^2*d^4 + 3*c*d^5)*f*cos(
f*x + e) + (c^6 + 6*c^5*d + 15*c^4*d^2 + 20*c^3*d^3 + 15*c^2*d^4 + 6*c*d^5 + d^6)*f - ((c^3*d^3 + 3*c^2*d^4 +
3*c*d^5 + d^6)*f*cos(f*x + e)^3 + (3*c^4*d^2 + 10*c^3*d^3 + 12*c^2*d^4 + 6*c*d^5 + d^6)*f*cos(f*x + e)^2 - (3*
c^5*d + 9*c^4*d^2 + 10*c^3*d^3 + 6*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e) - (c^6 + 6*c^5*d + 15*c^4*d^2 + 20*
c^3*d^3 + 15*c^2*d^4 + 6*c*d^5 + d^6)*f)*sin(f*x + e))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)8*sqrt(2*a)*((((-(54358179840000*d^16-108716359680000*c*d^15-163074539520000*c^2*d^14+434865438720000*c^3*
d^13+108716359680000*c^4*d^12-652298158080000*c^5*d^11+108716359680000*c^6*d^10+434865438720000*c^7*d^9-163074
539520000*c^8*d^8-108716359680000*c^9*d^7+54358179840000*c^10*d^6)*tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi)))^2/(-10
8716359680000*d^17+108716359680000*c*d^16+543581798400000*c^2*d^15-543581798400000*c^3*d^14-1087163596800000*c
^4*d^13+1087163596800000*c^5*d^12+1087163596800000*c^6*d^11-1087163596800000*c^7*d^10-543581798400000*c^8*d^9+
543581798400000*c^9*d^8+108716359680000*c^10*d^7-108716359680000*c^11*d^6)-(-362387865600000*d^16+130459631616
0000*c*d^15-652298158080000*c^2*d^14-2899102924800000*c^3*d^13+3913788948480000*c^4*d^12+869730877440000*c^5*d
^11-4203699240960000*c^6*d^10+1739461754880000*c^7*d^9+1087163596800000*c^8*d^8-1014686023680000*c^9*d^7+21743
2719360000*c^10*d^6)/(-108716359680000*d^17+108716359680000*c*d^16+543581798400000*c^2*d^15-543581798400000*c^
3*d^14-1087163596800000*c^4*d^13+1087163596800000*c^5*d^12+1087163596800000*c^6*d^11-1087163596800000*c^7*d^10
-543581798400000*c^8*d^9+543581798400000*c^9*d^8+108716359680000*c^10*d^7-108716359680000*c^11*d^6))*tan(1/2*(
1/2*f*x+1/4*(2*exp(1)-pi)))^2-(1021933780992000*d^16-4595078135808000*c*d^15+6443256250368000*c^2*d^14+7537667
60448000*c^3*d^13-10944113541120000*c^4*d^12+10465761558528000*c^5*d^11-1203127713792000*c^6*d^10-481251085516
8000*c^7*d^9+4355902144512000*c^8*d^8-1811939328000000*c^9*d^7+326149079040000*c^10*d^6)/(-108716359680000*d^1
7+108716359680000*c*d^16+543581798400000*c^2*d^15-543581798400000*c^3*d^14-1087163596800000*c^4*d^13+108716359
6800000*c^5*d^12+1087163596800000*c^6*d^11-1087163596800000*c^7*d^10-543581798400000*c^8*d^9+543581798400000*c
^9*d^8+108716359680000*c^10*d^7-108716359680000*c^11*d^6))*tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi)))^2-(-3623878656
00000*d^16+1304596316160000*c*d^15-652298158080000*c^2*d^14-2899102924800000*c^3*d^13+3913788948480000*c^4*d^1
2+869730877440000*c^5*d^11-4203699240960000*c^6*d^10+1739461754880000*c^7*d^9+1087163596800000*c^8*d^8-1014686
023680000*c^9*d^7+217432719360000*c^10*d^6)/(-108716359680000*d^17+108716359680000*c*d^16+543581798400000*c^2*
d^15-543581798400000*c^3*d^14-1087163596800000*c^4*d^13+1087163596800000*c^5*d^12+1087163596800000*c^6*d^11-10
87163596800000*c^7*d^10-543581798400000*c^8*d^9+543581798400000*c^9*d^8+108716359680000*c^10*d^7-1087163596800
00*c^11*d^6))*tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi)))^2-(54358179840000*d^16-108716359680000*c*d^15-1630745395200
00*c^2*d^14+434865438720000*c^3*d^13+108716359680000*c^4*d^12-652298158080000*c^5*d^11+108716359680000*c^6*d^1
0+434865438720000*c^7*d^9-163074539520000*c^8*d^8-108716359680000*c^9*d^7+54358179840000*c^10*d^6)/(-108716359
680000*d^17+108716359680000*c*d^16+543581798400000*c^2*d^15-543581798400000*c^3*d^14-1087163596800000*c^4*d^13
+1087163596800000*c^5*d^12+1087163596800000*c^6*d^11-1087163596800000*c^7*d^10-543581798400000*c^8*d^9+5435817
98400000*c^9*d^8+108716359680000*c^10*d^7-108716359680000*c^11*d^6))/sqrt(c*tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi)
))^4+d*tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi)))^4+2*c*tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi)))^2-6*d*tan(1/2*(1/2*f*x+
1/4*(2*exp(1)-pi)))^2+c+d)/(c*tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi)))^4+d*tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi)))^4+
2*c*tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi)))^2-6*d*tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi)))^2+c+d)^2*sign(cos(1/2*(f*x
+exp(1))-1/4*pi))*tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi)))/f

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maple [B]  time = 0.36, size = 430, normalized size = 3.03 \[ \frac {2 \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \left (19 c^{3} \left (\cos ^{2}\left (f x +e \right )\right ) d^{2}-11 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) d^{5}+25 \left (\cos ^{2}\left (f x +e \right )\right ) c^{4} d +21 \left (\cos ^{4}\left (f x +e \right )\right ) c^{2} d^{3}-2 c \left (\cos ^{4}\left (f x +e \right )\right ) d^{4}+13 c \left (\cos ^{2}\left (f x +e \right )\right ) d^{4}-11 c \,d^{4}-6 c^{2} d^{3}-22 c^{3} d^{2}-35 c^{4} d +4 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right ) c \,d^{4}+7 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{3} d^{2}+3 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d^{3}-15 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{4}+7 d^{5} \sin \left (f x +e \right )+6 c^{2} d^{3} \sin \left (f x +e \right )+11 c \,d^{4} \sin \left (f x +e \right )+35 \sin \left (f x +e \right ) c^{4} d +22 \sin \left (f x +e \right ) c^{3} d^{2}-7 d^{5}+4 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right ) d^{5}-15 \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d^{3}+8 \left (\cos ^{6}\left (f x +e \right )\right ) d^{5}+22 \left (\cos ^{2}\left (f x +e \right )\right ) d^{5}-23 \left (\cos ^{4}\left (f x +e \right )\right ) d^{5}-15 c^{5}+15 c^{5} \sin \left (f x +e \right )\right )}{15 f \cos \left (f x +e \right ) \left (\left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+c^{2}-d^{2}\right )^{3} \left (c +d \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(7/2),x)

[Out]

2/15/f*(a*(1+sin(f*x+e)))^(1/2)*(c+d*sin(f*x+e))^(1/2)*(4*sin(f*x+e)*cos(f*x+e)^4*d^5+4*sin(f*x+e)*cos(f*x+e)^
4*c*d^4+7*sin(f*x+e)*cos(f*x+e)^2*c^3*d^2+3*sin(f*x+e)*cos(f*x+e)^2*c^2*d^3-15*sin(f*x+e)*cos(f*x+e)^2*c*d^4-1
1*c*d^4-6*c^2*d^3-22*c^3*d^2-35*c^4*d-15*cos(f*x+e)^2*c^2*d^3-23*cos(f*x+e)^4*d^5+22*cos(f*x+e)^2*d^5+8*cos(f*
x+e)^6*d^5+7*d^5*sin(f*x+e)-2*c*cos(f*x+e)^4*d^4+19*c^3*cos(f*x+e)^2*d^2+13*c*cos(f*x+e)^2*d^4+6*c^2*d^3*sin(f
*x+e)+11*c*d^4*sin(f*x+e)+21*cos(f*x+e)^4*c^2*d^3-11*sin(f*x+e)*cos(f*x+e)^2*d^5+25*cos(f*x+e)^2*c^4*d+35*sin(
f*x+e)*c^4*d+22*sin(f*x+e)*c^3*d^2-7*d^5-15*c^5+15*c^5*sin(f*x+e))/cos(f*x+e)/(cos(f*x+e)^2*d^2+c^2-d^2)^3/(c+
d)^3

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maxima [B]  time = 1.34, size = 544, normalized size = 3.83 \[ -\frac {2 \, {\left ({\left (15 \, c^{3} + 10 \, c^{2} d + 3 \, c d^{2}\right )} \sqrt {a} - \frac {{\left (15 \, c^{3} - 60 \, c^{2} d - 25 \, c d^{2} - 6 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {{\left (45 \, c^{3} - 40 \, c^{2} d + 93 \, c d^{2} + 10 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {5 \, {\left (9 \, c^{3} - 22 \, c^{2} d + 13 \, c d^{2} - 12 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, {\left (9 \, c^{3} - 22 \, c^{2} d + 13 \, c d^{2} - 12 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {{\left (45 \, c^{3} - 40 \, c^{2} d + 93 \, c d^{2} + 10 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {{\left (15 \, c^{3} - 60 \, c^{2} d - 25 \, c d^{2} - 6 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {{\left (15 \, c^{3} + 10 \, c^{2} d + 3 \, c d^{2}\right )} \sqrt {a} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{3}}{15 \, {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3} + \frac {3 \, {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {{\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )} {\left (c + \frac {2 \, d \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{\frac {7}{2}} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

-2/15*((15*c^3 + 10*c^2*d + 3*c*d^2)*sqrt(a) - (15*c^3 - 60*c^2*d - 25*c*d^2 - 6*d^3)*sqrt(a)*sin(f*x + e)/(co
s(f*x + e) + 1) + (45*c^3 - 40*c^2*d + 93*c*d^2 + 10*d^3)*sqrt(a)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 5*(9*c
^3 - 22*c^2*d + 13*c*d^2 - 12*d^3)*sqrt(a)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*(9*c^3 - 22*c^2*d + 13*c*d^
2 - 12*d^3)*sqrt(a)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - (45*c^3 - 40*c^2*d + 93*c*d^2 + 10*d^3)*sqrt(a)*sin(
f*x + e)^5/(cos(f*x + e) + 1)^5 + (15*c^3 - 60*c^2*d - 25*c*d^2 - 6*d^3)*sqrt(a)*sin(f*x + e)^6/(cos(f*x + e)
+ 1)^6 - (15*c^3 + 10*c^2*d + 3*c*d^2)*sqrt(a)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7)*(sin(f*x + e)^2/(cos(f*x +
e) + 1)^2 + 1)^3/((c^3 + 3*c^2*d + 3*c*d^2 + d^3 + 3*(c^3 + 3*c^2*d + 3*c*d^2 + d^3)*sin(f*x + e)^2/(cos(f*x
+ e) + 1)^2 + 3*(c^3 + 3*c^2*d + 3*c*d^2 + d^3)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + (c^3 + 3*c^2*d + 3*c*d^2
+ d^3)*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)^(7/2)*f)

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mupad [B]  time = 16.35, size = 501, normalized size = 3.53 \[ -\frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}\,\left (\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,32{}\mathrm {i}}{15\,d\,f\,{\left (c+d\right )}^3}-\frac {32\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d\,f\,{\left (c+d\right )}^3}+\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (240\,c^2+80\,d^2\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^3\,f\,{\left (c+d\right )}^3}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (c^2\,240{}\mathrm {i}+d^2\,80{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^3\,f\,{\left (c+d\right )}^3}+\frac {32\,c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,d^2\,f\,{\left (c+d\right )}^3}-\frac {c\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,32{}\mathrm {i}}{3\,d^2\,f\,{\left (c+d\right )}^3}\right )}{{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}+\frac {{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3}{{\left (c+d\right )}^3}-\frac {3\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (4\,c^2+2\,c\,d+d^2\right )}{d^2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (6\,c+d\right )}{d}+\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{d^3}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (c\,6{}\mathrm {i}+d\,1{}\mathrm {i}\right )}{d}-\frac {3\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3\,\left (4\,c^2+2\,c\,d+d^2\right )}{d^2\,{\left (c+d\right )}^3}+\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{d^3\,{\left (c+d\right )}^3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(7/2),x)

[Out]

-((c + d*sin(e + f*x))^(1/2)*((exp(e*1i + f*x*1i)*(a + a*sin(e + f*x))^(1/2)*32i)/(15*d*f*(c + d)^3) - (32*exp
(e*6i + f*x*6i)*(a + a*sin(e + f*x))^(1/2))/(15*d*f*(c + d)^3) + (exp(e*4i + f*x*4i)*(240*c^2 + 80*d^2)*(a + a
*sin(e + f*x))^(1/2))/(15*d^3*f*(c + d)^3) - (exp(e*3i + f*x*3i)*(c^2*240i + d^2*80i)*(a + a*sin(e + f*x))^(1/
2))/(15*d^3*f*(c + d)^3) + (32*c*exp(e*2i + f*x*2i)*(a + a*sin(e + f*x))^(1/2))/(3*d^2*f*(c + d)^3) - (c*exp(e
*5i + f*x*5i)*(a + a*sin(e + f*x))^(1/2)*32i)/(3*d^2*f*(c + d)^3)))/(exp(e*7i + f*x*7i) + (c*1i + d*1i)^3/(c +
d)^3 - (3*exp(e*5i + f*x*5i)*(2*c*d + 4*c^2 + d^2))/d^2 - (exp(e*1i + f*x*1i)*(6*c + d))/d + (exp(e*3i + f*x*
3i)*(12*c*d^2 + 12*c^2*d + 8*c^3 + 3*d^3))/d^3 + (exp(e*6i + f*x*6i)*(c*6i + d*1i))/d - (3*exp(e*2i + f*x*2i)*
(c*1i + d*1i)^3*(2*c*d + 4*c^2 + d^2))/(d^2*(c + d)^3) + (exp(e*4i + f*x*4i)*(c*1i + d*1i)^3*(12*c*d^2 + 12*c^
2*d + 8*c^3 + 3*d^3))/(d^3*(c + d)^3))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**(1/2)/(c+d*sin(f*x+e))**(7/2),x)

[Out]

Timed out

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