3.6.9 \(\int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx\) [509]

Optimal. Leaf size=333 \[ -\frac {d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (3 c^2+20 c d+9 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 a (c-d)^3 (c+d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(3 c+5 d) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 a (c-d)^2 (c+d) f \sqrt {c+d \sin (e+f x)}} \]

[Out]

-1/3*d*(3*c+5*d)*cos(f*x+e)/a/(c-d)^2/(c+d)/f/(c+d*sin(f*x+e))^(3/2)-cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))/(c+d*
sin(f*x+e))^(3/2)-1/3*d*(3*c^2+20*c*d+9*d^2)*cos(f*x+e)/a/(c-d)^3/(c+d)^2/f/(c+d*sin(f*x+e))^(1/2)+1/3*(3*c^2+
20*c*d+9*d^2)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x
),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/a/(c-d)^3/(c+d)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-1/3*(3*c+
5*d)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)
*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/a/(c-d)^2/(c+d)/f/(c+d*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.33, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2847, 2833, 2831, 2742, 2740, 2734, 2732} \begin {gather*} -\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a f (c-d)^3 (c+d)^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 a f (c-d)^3 (c+d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {d (3 c+5 d) \cos (e+f x)}{3 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}+\frac {(3 c+5 d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 a f (c-d)^2 (c+d) \sqrt {c+d \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2)),x]

[Out]

-1/3*(d*(3*c + 5*d)*Cos[e + f*x])/(a*(c - d)^2*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - Cos[e + f*x]/((c - d)*f
*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) - (d*(3*c^2 + 20*c*d + 9*d^2)*Cos[e + f*x])/(3*a*(c - d)^3*(
c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((3*c^2 + 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*
Sqrt[c + d*Sin[e + f*x]])/(3*a*(c - d)^3*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((3*c + 5*d)*Ellipt
icF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a*(c - d)^2*(c + d)*f*Sqrt[c + d
*Sin[e + f*x]])

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

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

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

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

Rule 2831

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

Rule 2833

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

Rule 2847

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

Rubi steps

\begin {align*} \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx &=-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}+\frac {d \int \frac {-\frac {5 a}{2}+\frac {3}{2} a \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{a^2 (c-d)}\\ &=-\frac {d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {(2 d) \int \frac {\frac {3}{4} a (5 c+3 d)-\frac {1}{4} a (3 c+5 d) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 a^2 (c-d)^2 (c+d)}\\ &=-\frac {d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(4 d) \int \frac {-\frac {1}{8} a \left (15 c^2+12 c d+5 d^2\right )-\frac {1}{8} a \left (3 c^2+20 c d+9 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 a^2 (c-d)^3 (c+d)^2}\\ &=-\frac {d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(3 c+5 d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{6 a (c-d)^2 (c+d)}-\frac {\left (3 c^2+20 c d+9 d^2\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{6 a (c-d)^3 (c+d)^2}\\ &=-\frac {d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (\left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{6 a (c-d)^3 (c+d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left ((3 c+5 d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{6 a (c-d)^2 (c+d) \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (3 c^2+20 c d+9 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 a (c-d)^3 (c+d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(3 c+5 d) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 a (c-d)^2 (c+d) f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 2.95, size = 367, normalized size = 1.10 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (\frac {\left (3 c^2+20 c d+9 d^2\right ) (c+d \sin (e+f x))+d \left (15 c^2+12 c d+5 d^2\right ) F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+\left (3 c^2+20 c d+9 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(c+d)^2}+2 (c+d \sin (e+f x)) \left (\frac {3 \sin \left (\frac {1}{2} (e+f x)\right )}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}-\frac {3 c^2+13 c d+6 d^2+\frac {d^2 \cos (e+f x) \left (8 c^2+3 c d-d^2+d (7 c+3 d) \sin (e+f x)\right )}{(c+d \sin (e+f x))^2}}{(c+d)^2}\right )\right )}{3 a (c-d)^3 f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2)),x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(((3*c^2 + 20*c*d + 9*d^2)*(c + d*Sin[e + f*x]) + d*(15*c^2 + 12*c*d
+ 5*d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (3*c^2 + 20*c*d
+ 9*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(
c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(c + d)^2 + 2*(c + d*Sin[e + f*x])*((3*Sin[(e + f*x)/2])/(Cos[(e
+ f*x)/2] + Sin[(e + f*x)/2]) - (3*c^2 + 13*c*d + 6*d^2 + (d^2*Cos[e + f*x]*(8*c^2 + 3*c*d - d^2 + d*(7*c + 3*
d)*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2)/(c + d)^2)))/(3*a*(c - d)^3*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e +
f*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1290\) vs. \(2(377)=754\).
time = 24.00, size = 1291, normalized size = 3.88

method	result	size
default	\(\text {Expression too large to display}\)	\(1291\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/a*(-d/(c-d)*(2/3/(c^2-d^2)/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(
sin(f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^2*c/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^
4-6*c^2*d^2+3*d^4)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-1-sin(f*x+e))*d/(c
-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2
))+8/3*c*d/(c^2-d^2)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-1-sin(f*x+e))*
d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-
d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+1/(c-d)^2*(-(-sin(f*x+e)^2*d-
c*sin(f*x+e)+d*sin(f*x+e)+c)/(c-d)/((-d*sin(f*x+e)-c)*(sin(f*x+e)-1)*(1+sin(f*x+e)))^(1/2)-2*d/(2*c-2*d)*(c/d-
1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-1-sin(f*x+e))*d/(c-d))^(1/2)/(-(-d*sin(f*x
+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-d/(c-d)*(c/d-1)*((c+d
*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-1-sin(f*x+e))*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*c
os(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin
(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))-d/(c-d)^2*(2*d*cos(f*x+e)^2/(c^2-d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x
+e)^2)^(1/2)+2*c/(c^2-d^2)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-1-sin(f*x+
e))*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+
d))^(1/2))+2/(c^2-d^2)*d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*((-1-sin(f*x+e)
)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((
c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))))/cos(f*x+e)/(c+d*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(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(5/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.22, size = 2243, normalized size = 6.74 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

1/18*((sqrt(2)*(6*c^3*d^2 - 5*c^2*d^3 - 18*c*d^4 - 15*d^5)*cos(f*x + e)^3 + sqrt(2)*(12*c^4*d - 4*c^3*d^2 - 41
*c^2*d^3 - 48*c*d^4 - 15*d^5)*cos(f*x + e)^2 - sqrt(2)*(6*c^5 - 5*c^4*d - 12*c^3*d^2 - 20*c^2*d^3 - 18*c*d^4 -
15*d^5)*cos(f*x + e) + (sqrt(2)*(6*c^3*d^2 - 5*c^2*d^3 - 18*c*d^4 - 15*d^5)*cos(f*x + e)^2 - 2*sqrt(2)*(6*c^4
*d - 5*c^3*d^2 - 18*c^2*d^3 - 15*c*d^4)*cos(f*x + e) - sqrt(2)*(6*c^5 + 7*c^4*d - 22*c^3*d^2 - 56*c^2*d^3 - 48
*c*d^4 - 15*d^5))*sin(f*x + e) - sqrt(2)*(6*c^5 + 7*c^4*d - 22*c^3*d^2 - 56*c^2*d^3 - 48*c*d^4 - 15*d^5))*sqrt
(I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3
*I*d*sin(f*x + e) - 2*I*c)/d) + (sqrt(2)*(6*c^3*d^2 - 5*c^2*d^3 - 18*c*d^4 - 15*d^5)*cos(f*x + e)^3 + sqrt(2)*
(12*c^4*d - 4*c^3*d^2 - 41*c^2*d^3 - 48*c*d^4 - 15*d^5)*cos(f*x + e)^2 - sqrt(2)*(6*c^5 - 5*c^4*d - 12*c^3*d^2
- 20*c^2*d^3 - 18*c*d^4 - 15*d^5)*cos(f*x + e) + (sqrt(2)*(6*c^3*d^2 - 5*c^2*d^3 - 18*c*d^4 - 15*d^5)*cos(f*x
+ e)^2 - 2*sqrt(2)*(6*c^4*d - 5*c^3*d^2 - 18*c^2*d^3 - 15*c*d^4)*cos(f*x + e) - sqrt(2)*(6*c^5 + 7*c^4*d - 22
*c^3*d^2 - 56*c^2*d^3 - 48*c*d^4 - 15*d^5))*sin(f*x + e) - sqrt(2)*(6*c^5 + 7*c^4*d - 22*c^3*d^2 - 56*c^2*d^3
- 48*c*d^4 - 15*d^5))*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^
3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*(sqrt(2)*(3*I*c^2*d^3 + 20*I*c*d^4 + 9*I*d^5)*co
s(f*x + e)^3 + sqrt(2)*(6*I*c^3*d^2 + 43*I*c^2*d^3 + 38*I*c*d^4 + 9*I*d^5)*cos(f*x + e)^2 + sqrt(2)*(-3*I*c^4*
d - 20*I*c^3*d^2 - 12*I*c^2*d^3 - 20*I*c*d^4 - 9*I*d^5)*cos(f*x + e) + (sqrt(2)*(3*I*c^2*d^3 + 20*I*c*d^4 + 9*
I*d^5)*cos(f*x + e)^2 + 2*sqrt(2)*(-3*I*c^3*d^2 - 20*I*c^2*d^3 - 9*I*c*d^4)*cos(f*x + e) + sqrt(2)*(-3*I*c^4*d
- 26*I*c^3*d^2 - 52*I*c^2*d^3 - 38*I*c*d^4 - 9*I*d^5))*sin(f*x + e) + sqrt(2)*(-3*I*c^4*d - 26*I*c^3*d^2 - 52
*I*c^2*d^3 - 38*I*c*d^4 - 9*I*d^5))*sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c
*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e
) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 3*(sqrt(2)*(-3*I*c^2*d^3 - 20*I*c*d^4 - 9*I*d^5)*cos(f*x + e)^3 + sqrt(2
)*(-6*I*c^3*d^2 - 43*I*c^2*d^3 - 38*I*c*d^4 - 9*I*d^5)*cos(f*x + e)^2 + sqrt(2)*(3*I*c^4*d + 20*I*c^3*d^2 + 12
*I*c^2*d^3 + 20*I*c*d^4 + 9*I*d^5)*cos(f*x + e) + (sqrt(2)*(-3*I*c^2*d^3 - 20*I*c*d^4 - 9*I*d^5)*cos(f*x + e)^
2 + 2*sqrt(2)*(3*I*c^3*d^2 + 20*I*c^2*d^3 + 9*I*c*d^4)*cos(f*x + e) + sqrt(2)*(3*I*c^4*d + 26*I*c^3*d^2 + 52*I
*c^2*d^3 + 38*I*c*d^4 + 9*I*d^5))*sin(f*x + e) + sqrt(2)*(3*I*c^4*d + 26*I*c^3*d^2 + 52*I*c^2*d^3 + 38*I*c*d^4
+ 9*I*d^5))*sqrt(-I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstras
sPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x +
e) + 2*I*c)/d)) + 6*(3*c^4*d - 6*c^2*d^3 + 3*d^5 - (3*c^2*d^3 + 20*c*d^4 + 9*d^5)*cos(f*x + e)^3 + (6*c^3*d^2
+ 25*c^2*d^3 + 6*c*d^4 - 5*d^5)*cos(f*x + e)^2 + (3*c^4*d + 6*c^3*d^2 + 22*c^2*d^3 + 26*c*d^4 + 7*d^5)*cos(f*x
+ e) - (3*c^4*d - 6*c^2*d^3 + 3*d^5 - (3*c^2*d^3 + 20*c*d^4 + 9*d^5)*cos(f*x + e)^2 - 2*(3*c^3*d^2 + 14*c^2*d
^3 + 13*c*d^4 + 2*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(d*sin(f*x + e) + c))/((a*c^5*d^3 - a*c^4*d^4 - 2*a*c^3
*d^5 + 2*a*c^2*d^6 + a*c*d^7 - a*d^8)*f*cos(f*x + e)^3 + (2*a*c^6*d^2 - a*c^5*d^3 - 5*a*c^4*d^4 + 2*a*c^3*d^5
+ 4*a*c^2*d^6 - a*c*d^7 - a*d^8)*f*cos(f*x + e)^2 - (a*c^7*d - a*c^6*d^2 - a*c^5*d^3 + a*c^4*d^4 - a*c^3*d^5 +
a*c^2*d^6 + a*c*d^7 - a*d^8)*f*cos(f*x + e) - (a*c^7*d + a*c^6*d^2 - 3*a*c^5*d^3 - 3*a*c^4*d^4 + 3*a*c^3*d^5
+ 3*a*c^2*d^6 - a*c*d^7 - a*d^8)*f + ((a*c^5*d^3 - a*c^4*d^4 - 2*a*c^3*d^5 + 2*a*c^2*d^6 + a*c*d^7 - a*d^8)*f*
cos(f*x + e)^2 - 2*(a*c^6*d^2 - a*c^5*d^3 - 2*a*c^4*d^4 + 2*a*c^3*d^5 + a*c^2*d^6 - a*c*d^7)*f*cos(f*x + e) -
(a*c^7*d + a*c^6*d^2 - 3*a*c^5*d^3 - 3*a*c^4*d^4 + 3*a*c^3*d^5 + 3*a*c^2*d^6 - a*c*d^7 - a*d^8)*f)*sin(f*x + e
))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} + 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))**(5/2),x)

[Out]

Integral(1/(c**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x) + c**2*sqrt(c + d*sin(e + f*x)) + 2*c*d*sqrt(c + d*sin(
e + f*x))*sin(e + f*x)**2 + 2*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x) + d**2*sqrt(c + d*sin(e + f*x))*sin(e
+ f*x)**3 + d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2), x)/a

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(5/2)), x)

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2)),x)

[Out]

\text{Hanged}

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