3.4.68 \(\int \frac {\cos ^4(e+f x)}{(a+b \sin ^2(e+f x))^{5/2}} \, dx\) [368]

Optimal. Leaf size=223 \[ \frac {(a+b) \cos (e+f x) \sin (e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (a-b) \cos (e+f x) \sin (e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a-b) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a b^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(2 a-b) F\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a b^2 f \sqrt {a+b \sin ^2(e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 263, normalized size of antiderivative = 1.18, 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 = {3271, 424, 541, 538, 437, 435, 432, 430} \begin {gather*} -\frac {2 (a-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 a^2 b^2 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {2 (a-b) \sin (e+f x) \cos (e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(2 a-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 a b^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(a+b) \sin (e+f x) \cos (e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b)*Cos[e + f*x]*Sin[e + f*x])/(3*a*b*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (2*(a - b)*Cos[e + f*x]*Sin[e + f
*x])/(3*a^2*b*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*(a - 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*a^2*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((2*a - b)*Sqr
t[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*a*b^
2*f*Sqrt[a + b*Sin[e + f*x]^2])

Rule 424

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

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 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 3271

Int[cos[(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*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2
)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] &&  !IntegerQ
[p]

Rubi steps

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

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Mathematica [A]
time = 0.99, size = 171, normalized size = 0.77 \begin {gather*} \frac {-2 a^2 (a-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^2 (2 a-b) \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^2-2 a b-b^2+b (-a+b) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{3 a^2 b^2 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 10.00, size = 485, normalized size = 2.17

method	result	size
default	\(\frac {\left (2 a \,b^{2}-2 b^{3}\right ) \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-a^{2} b +a \,b^{2}+2 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}}\, a b \left (2 \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a -\EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -2 \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +2 \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) 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}+\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 -\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}-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}+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 \,b^{2}}{3 a^{2} \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} b^{2} \cos \left (f x +e \right ) f}\)	\(485\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(cos(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 = 0.23, size = 1343, normalized size = 6.02 \begin {gather*} \frac {{\left (2 \, {\left ({\left (-i \, a b^{3} + i \, b^{4}\right )} \cos \left (f x + e\right )^{4} - i \, a^{3} b - i \, a^{2} b^{2} + i \, a b^{3} + i \, b^{4} - 2 \, {\left (-i \, a^{2} b^{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} - i \, a b^{3} - i \, b^{4}\right )} \cos \left (f x + e\right )^{4} + 2 i \, a^{4} + 3 i \, a^{3} b - i \, a^{2} b^{2} - 3 i \, a b^{3} - i \, b^{4} + 2 \, {\left (-2 i \, a^{3} b - i \, a^{2} b^{2} + 2 i \, a b^{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}} 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 + i \, a^{2} b^{2} - i \, a b^{3} - i \, b^{4} - 2 \, {\left (i \, a^{2} b^{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} + i \, a b^{3} + i \, b^{4}\right )} \cos \left (f x + e\right )^{4} - 2 i \, a^{4} - 3 i \, a^{3} b + i \, a^{2} b^{2} + 3 i \, a b^{3} + i \, b^{4} + 2 \, {\left (2 i \, a^{3} b + i \, a^{2} b^{2} - 2 i \, a b^{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}} 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 - 3 i \, a b^{3} - 2 i \, b^{4} - 2 \, {\left (i \, a^{2} b^{2} - 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} - i \, a b^{3}\right )} \cos \left (f x + e\right )^{4} - 2 i \, a^{4} - 5 i \, a^{3} b - 4 i \, a^{2} b^{2} - i \, a b^{3} + 2 \, {\left (2 i \, a^{3} b + 3 i \, a^{2} b^{2} + i \, a b^{3}\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} + 2 i \, b^{4}\right )} \cos \left (f x + e\right )^{4} - i \, a^{3} b + 3 i \, a b^{3} + 2 i \, b^{4} - 2 \, {\left (-i \, a^{2} b^{2} + 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} + i \, a b^{3}\right )} \cos \left (f x + e\right )^{4} + 2 i \, a^{4} + 5 i \, a^{3} b + 4 i \, a^{2} b^{2} + i \, a b^{3} + 2 \, {\left (-2 i \, a^{3} b - 3 i \, a^{2} b^{2} - i \, a b^{3}\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} - b^{4}\right )} \cos \left (f x + e\right )^{3} - {\left (a^{2} b^{2} - a b^{3} - 2 \, 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 (a^{2} b^{5} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{3} b^{4} + a^{2} b^{5}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b^{3} + 2 \, a^{3} b^{4} + a^{2} b^{5}\right )} f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((2*((-I*a*b^3 + I*b^4)*cos(f*x + e)^4 - I*a^3*b - I*a^2*b^2 + I*a*b^3 + I*b^4 - 2*(-I*a^2*b^2 + I*b^4)*co
s(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - ((2*I*a^2*b^2 - I*a*b^3 - I*b^4)*cos(f*x + e)^4 + 2*I*a^4 + 3*I
*a^3*b - I*a^2*b^2 - 3*I*a*b^3 - I*b^4 + 2*(-2*I*a^3*b - I*a^2*b^2 + 2*I*a*b^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_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 + I*a^2*b^2 - I*a*b^3 - I*b^4 - 2*(I*a^2*b^2 - I*b^4)*cos(f*x + e)
^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - ((-2*I*a^2*b^2 + I*a*b^3 + I*b^4)*cos(f*x + e)^4 - 2*I*a^4 - 3*I*a^3*b +
I*a^2*b^2 + 3*I*a*b^3 + I*b^4 + 2*(2*I*a^3*b + I*a^2*b^2 - 2*I*a*b^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_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 - 3*I*a*b^3 - 2*I*b^4 - 2*(I*a^2*b^2 - 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 - I*a*b^3)*cos(f*x + e)^4 - 2*I*a^4 - 5*I*a^3*b - 4*I*a^2*b^2
- I*a*b^3 + 2*(2*I*a^3*b + 3*I*a^2*b^2 + I*a*b^3)*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 + 2*I*b^4)*cos(f*x + e)^4 -
I*a^3*b + 3*I*a*b^3 + 2*I*b^4 - 2*(-I*a^2*b^2 + 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 + I*a*b^3)*cos(f*x + e)^4 + 2*I*a^4 + 5*I*a^3*b + 4*I*a^2*b^2 + I*a*b^3 + 2*(-2*I*a^3*b -
3*I*a^2*b^2 - I*a*b^3)*cos(f*x + e)^2)*sqrt(-b))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_f(arc
sin(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 - b^4)*cos(f*x + e)^3 - (a^2*b^2 - a*b^3 - 2*b^4)*cos(f*x
+ e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/(a^2*b^5*f*cos(f*x + e)^4 - 2*(a^3*b^4 + a^2*b^5)*f*cos(f
*x + e)^2 + (a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*f)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

[Out]

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

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