3.2.67 \(\int \frac {\sin ^4(e+f x)}{(a+b \sin ^2(e+f x))^{5/2}} \, dx\) [167]

Optimal. Leaf size=269 \[ \frac {a \cos (e+f x) \sin (e+f x)}{3 b (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (a+2 b) \cos (e+f x) \sin (e+f x)}{3 b (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a+2 b) \sqrt {\cos ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 b^2 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {(2 a+3 b) \sqrt {\cos ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 b^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}} \]

[Out]

1/3*a*cos(f*x+e)*sin(f*x+e)/b/(a+b)/f/(a+b*sin(f*x+e)^2)^(3/2)-2/3*(a+2*b)*cos(f*x+e)*sin(f*x+e)/b/(a+b)^2/f/(
a+b*sin(f*x+e)^2)^(1/2)-2/3*(a+2*b)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(cos(f*x+e)^2)^(1/2)*(a+b*si
n(f*x+e)^2)^(1/2)/b^2/(a+b)^2/f/(1+b*sin(f*x+e)^2/a)^(1/2)+1/3*(2*a+3*b)*EllipticF(sin(f*x+e),(-b/a)^(1/2))*se
c(f*x+e)*(cos(f*x+e)^2)^(1/2)*(1+b*sin(f*x+e)^2/a)^(1/2)/b^2/(a+b)/f/(a+b*sin(f*x+e)^2)^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3267, 481, 541, 538, 437, 435, 432, 430} \begin {gather*} \frac {(2 a+3 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (\text {ArcSin}(\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 b^2 f (a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a+2 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\text {ArcSin}(\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 b^2 f (a+b)^2 \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {2 (a+2 b) \sin (e+f x) \cos (e+f x)}{3 b f (a+b)^2 \sqrt {a+b \sin ^2(e+f x)}}+\frac {a \sin (e+f x) \cos (e+f x)}{3 b f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),x]

[Out]

(a*Cos[e + f*x]*Sin[e + f*x])/(3*b*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (2*(a + 2*b)*Cos[e + f*x]*Sin[e +
f*x])/(3*b*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e
+ f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^2*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])
+ ((2*a + 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f
*x]^2)/a])/(3*b^2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 432

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

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 437

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

Rule 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 538

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c]))))))

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 3267

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Sin[e + f*x], x]}, Dist[ff^(m + 1)*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])), Subst[Int[x^m*((a + b*ff^2*
x^2)^p/Sqrt[1 - ff^2*x^2]), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] &&  !In
tegerQ[p]

Rubi steps

\begin {align*} \int \frac {\sin ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1-x^2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {a \cos (e+f x) \sin (e+f x)}{3 b (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {a+(-2 a-3 b) x^2}{\sqrt {1-x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 b (a+b) f}\\ &=\frac {a \cos (e+f x) \sin (e+f x)}{3 b (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (a+2 b) \cos (e+f x) \sin (e+f x)}{3 b (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {a (a+3 b)-2 a (a+2 b) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a b (a+b)^2 f}\\ &=\frac {a \cos (e+f x) \sin (e+f x)}{3 b (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (a+2 b) \cos (e+f x) \sin (e+f x)}{3 b (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (2 (a+2 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 b^2 (a+b)^2 f}+\frac {\left ((2 a+3 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 b^2 (a+b) f}\\ &=\frac {a \cos (e+f x) \sin (e+f x)}{3 b (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (a+2 b) \cos (e+f x) \sin (e+f x)}{3 b (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (2 (a+2 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {b x^2}{a}}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 b^2 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\left ((2 a+3 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {b x^2}{a}}} \, dx,x,\sin (e+f x)\right )}{3 b^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}\\ &=\frac {a \cos (e+f x) \sin (e+f x)}{3 b (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (a+2 b) \cos (e+f x) \sin (e+f x)}{3 b (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a+2 b) \sqrt {\cos ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 b^2 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {(2 a+3 b) \sqrt {\cos ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 b^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 1.13, size = 182, normalized size = 0.68 \begin {gather*} -\frac {2 a^2 (a+2 b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )-a \left (2 a^2+5 a b+3 b^2\right ) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} F\left (e+f x\left |-\frac {b}{a}\right .\right )-\sqrt {2} b \left (-a^2-4 a b-2 b^2+b (a+2 b) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{3 b^2 (a+b)^2 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),x]

[Out]

-1/3*(2*a^2*(a + 2*b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] - a*(2*a^2 + 5*a*b +
3*b^2)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*(-a^2 - 4*a*b - 2*b^2
+ b*(a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(b^2*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(622\) vs. \(2(247)=494\).
time = 10.11, size = 623, normalized size = 2.32

method	result	size
default	\(\frac {\left (2 a \,b^{2}+4 b^{3}\right ) \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-a^{2} b -5 a \,b^{2}-4 b^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, b \left (2 \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+5 \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b +3 \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}-2 \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}-4 \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+7 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b +8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}+3 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{3}-2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}-6 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b -4 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}}{3 \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (a +b \right )^{2} b^{2} \cos \left (f x +e \right ) f}\)	\(623\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3*((2*a*b^2+4*b^3)*sin(f*x+e)*cos(f*x+e)^4+(-a^2*b-5*a*b^2-4*b^3)*cos(f*x+e)^2*sin(f*x+e)-(-b/a*cos(f*x+e)^2
+(a+b)/a)^(1/2)*(cos(f*x+e)^2)^(1/2)*b*(2*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a^2+5*EllipticF(sin(f*x+e),(-1/
a*b)^(1/2))*a*b+3*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b^2-2*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a^2-4*Ellipt
icE(sin(f*x+e),(-1/a*b)^(1/2))*a*b)*cos(f*x+e)^2+2*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*Elli
pticF(sin(f*x+e),(-1/a*b)^(1/2))*a^3+7*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticF(sin(f*
x+e),(-1/a*b)^(1/2))*a^2*b+8*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticF(sin(f*x+e),(-1/a
*b)^(1/2))*a*b^2+3*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))
*b^3-2*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a^3-6*(cos(
f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a^2*b-4*(cos(f*x+e)^2)^
(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a*b^2)/(a+b*sin(f*x+e)^2)^(3/2)/(
a+b)^2/b^2/cos(f*x+e)/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(sin(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(sin(f*x + e)^4/(b*sin(f*x + e)^2 + a)^(5/2), x)

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Fricas [C] Result contains complex when optimal does not.
time = 0.22, size = 1432, normalized size = 5.32 \begin {gather*} \frac {{\left (2 \, {\left ({\left (-i \, a b^{3} - 2 i \, b^{4}\right )} \cos \left (f x + e\right )^{4} - i \, a^{3} b - 4 i \, a^{2} b^{2} - 5 i \, a b^{3} - 2 i \, b^{4} - 2 \, {\left (-i \, a^{2} b^{2} - 3 i \, a b^{3} - 2 i \, b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} - {\left ({\left (2 i \, a^{2} b^{2} + 5 i \, a b^{3} + 2 i \, b^{4}\right )} \cos \left (f x + e\right )^{4} + 2 i \, a^{4} + 9 i \, a^{3} b + 14 i \, a^{2} b^{2} + 9 i \, a b^{3} + 2 i \, b^{4} + 2 \, {\left (-2 i \, a^{3} b - 7 i \, a^{2} b^{2} - 7 i \, a b^{3} - 2 i \, b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left ({\left (i \, a b^{3} + 2 i \, b^{4}\right )} \cos \left (f x + e\right )^{4} + i \, a^{3} b + 4 i \, a^{2} b^{2} + 5 i \, a b^{3} + 2 i \, b^{4} - 2 \, {\left (i \, a^{2} b^{2} + 3 i \, a b^{3} + 2 i \, b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} - {\left ({\left (-2 i \, a^{2} b^{2} - 5 i \, a b^{3} - 2 i \, b^{4}\right )} \cos \left (f x + e\right )^{4} - 2 i \, a^{4} - 9 i \, a^{3} b - 14 i \, a^{2} b^{2} - 9 i \, a b^{3} - 2 i \, b^{4} + 2 \, {\left (2 i \, a^{3} b + 7 i \, a^{2} b^{2} + 7 i \, a b^{3} + 2 i \, b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left ({\left (i \, a b^{3} + i \, b^{4}\right )} \cos \left (f x + e\right )^{4} + i \, a^{3} b + 3 i \, a^{2} b^{2} + 3 i \, a b^{3} + i \, b^{4} - 2 \, {\left (i \, a^{2} b^{2} + 2 i \, a b^{3} + i \, b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} - {\left ({\left (-2 i \, a^{2} b^{2} - 7 i \, a b^{3} - 3 i \, b^{4}\right )} \cos \left (f x + e\right )^{4} - 2 i \, a^{4} - 11 i \, a^{3} b - 19 i \, a^{2} b^{2} - 13 i \, a b^{3} - 3 i \, b^{4} + 2 \, {\left (2 i \, a^{3} b + 9 i \, a^{2} b^{2} + 10 i \, a b^{3} + 3 i \, b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left ({\left (-i \, a b^{3} - i \, b^{4}\right )} \cos \left (f x + e\right )^{4} - i \, a^{3} b - 3 i \, a^{2} b^{2} - 3 i \, a b^{3} - i \, b^{4} - 2 \, {\left (-i \, a^{2} b^{2} - 2 i \, a b^{3} - i \, b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} - {\left ({\left (2 i \, a^{2} b^{2} + 7 i \, a b^{3} + 3 i \, b^{4}\right )} \cos \left (f x + e\right )^{4} + 2 i \, a^{4} + 11 i \, a^{3} b + 19 i \, a^{2} b^{2} + 13 i \, a b^{3} + 3 i \, b^{4} + 2 \, {\left (-2 i \, a^{3} b - 9 i \, a^{2} b^{2} - 10 i \, a b^{3} - 3 i \, b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left (a b^{3} + 2 \, b^{4}\right )} \cos \left (f x + e\right )^{3} - {\left (a^{2} b^{2} + 5 \, a b^{3} + 4 \, b^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{3 \, {\left ({\left (a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{3} b^{4} + 3 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b^{3} + 4 \, a^{3} b^{4} + 6 \, a^{2} b^{5} + 4 \, a b^{6} + b^{7}\right )} f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

1/3*((2*((-I*a*b^3 - 2*I*b^4)*cos(f*x + e)^4 - I*a^3*b - 4*I*a^2*b^2 - 5*I*a*b^3 - 2*I*b^4 - 2*(-I*a^2*b^2 - 3
*I*a*b^3 - 2*I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - ((2*I*a^2*b^2 + 5*I*a*b^3 + 2*I*b^4)*cos(
f*x + e)^4 + 2*I*a^4 + 9*I*a^3*b + 14*I*a^2*b^2 + 9*I*a*b^3 + 2*I*b^4 + 2*(-2*I*a^3*b - 7*I*a^2*b^2 - 7*I*a*b^
3 - 2*I*b^4)*cos(f*x + e)^2)*sqrt(-b))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2
*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2
)*sqrt((a^2 + a*b)/b^2))/b^2) + (2*((I*a*b^3 + 2*I*b^4)*cos(f*x + e)^4 + I*a^3*b + 4*I*a^2*b^2 + 5*I*a*b^3 + 2
*I*b^4 - 2*(I*a^2*b^2 + 3*I*a*b^3 + 2*I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - ((-2*I*a^2*b^2 -
5*I*a*b^3 - 2*I*b^4)*cos(f*x + e)^4 - 2*I*a^4 - 9*I*a^3*b - 14*I*a^2*b^2 - 9*I*a*b^3 - 2*I*b^4 + 2*(2*I*a^3*b
+ 7*I*a^2*b^2 + 7*I*a*b^3 + 2*I*b^4)*cos(f*x + e)^2)*sqrt(-b))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*
elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a
*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (2*((I*a*b^3 + I*b^4)*cos(f*x + e)^4 + I*a^3*b + 3*I*
a^2*b^2 + 3*I*a*b^3 + I*b^4 - 2*(I*a^2*b^2 + 2*I*a*b^3 + I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)
- ((-2*I*a^2*b^2 - 7*I*a*b^3 - 3*I*b^4)*cos(f*x + e)^4 - 2*I*a^4 - 11*I*a^3*b - 19*I*a^2*b^2 - 13*I*a*b^3 - 3
*I*b^4 + 2*(2*I*a^3*b + 9*I*a^2*b^2 + 10*I*a*b^3 + 3*I*b^4)*cos(f*x + e)^2)*sqrt(-b))*sqrt((2*b*sqrt((a^2 + a*
b)/b^2) + 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*
x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (2*((-I*a*b^3 - I*b^4)*cos(f*x
+ e)^4 - I*a^3*b - 3*I*a^2*b^2 - 3*I*a*b^3 - I*b^4 - 2*(-I*a^2*b^2 - 2*I*a*b^3 - I*b^4)*cos(f*x + e)^2)*sqrt(-
b)*sqrt((a^2 + a*b)/b^2) - ((2*I*a^2*b^2 + 7*I*a*b^3 + 3*I*b^4)*cos(f*x + e)^4 + 2*I*a^4 + 11*I*a^3*b + 19*I*a
^2*b^2 + 13*I*a*b^3 + 3*I*b^4 + 2*(-2*I*a^3*b - 9*I*a^2*b^2 - 10*I*a*b^3 - 3*I*b^4)*cos(f*x + e)^2)*sqrt(-b))*
sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*
(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (2*(a*b
^3 + 2*b^4)*cos(f*x + e)^3 - (a^2*b^2 + 5*a*b^3 + 4*b^4)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x
+ e))/((a^2*b^5 + 2*a*b^6 + b^7)*f*cos(f*x + e)^4 - 2*(a^3*b^4 + 3*a^2*b^5 + 3*a*b^6 + b^7)*f*cos(f*x + e)^2
+ (a^4*b^3 + 4*a^3*b^4 + 6*a^2*b^5 + 4*a*b^6 + b^7)*f)

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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(sin(f*x+e)**4/(a+b*sin(f*x+e)**2)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8857 deep

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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(sin(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="giac")

[Out]

integrate(sin(f*x + e)^4/(b*sin(f*x + e)^2 + a)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\sin \left (e+f\,x\right )}^4}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(e + f*x)^4/(a + b*sin(e + f*x)^2)^(5/2),x)

[Out]

int(sin(e + f*x)^4/(a + b*sin(e + f*x)^2)^(5/2), x)

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