3.4.40 \(\int \cos ^4(e+f x) (a+b \sin ^2(e+f x))^{3/2} \, dx\) [340]

Optimal. Leaf size=321 \[ -\frac {\left (a^2-9 a b-2 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}+\frac {2 (4 a+b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos ^5(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {2 (a-b) \left (a^2+6 a b+b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {a (a+b) \left (2 a^2+9 a b-b^2\right ) \sqrt {\cos ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{35 b^2 f \sqrt {a+b \sin ^2(e+f x)}} \]

[Out]

-1/35*(a^2-9*a*b-2*b^2)*cos(f*x+e)*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/b/f+2/35*(4*a+b)*cos(f*x+e)^3*sin(f*x+e
)*(a+b*sin(f*x+e)^2)^(1/2)/f-1/7*b*cos(f*x+e)^5*sin(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/f-2/35*(a-b)*(a^2+6*a*b+b^
2)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(cos(f*x+e)^2)^(1/2)*(a+b*sin(f*x+e)^2)^(1/2)/b^2/f/(1+b*sin(
f*x+e)^2/a)^(1/2)+1/35*a*(a+b)*(2*a^2+9*a*b-b^2)*EllipticF(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(cos(f*x+e)^2)^
(1/2)*(1+b*sin(f*x+e)^2/a)^(1/2)/b^2/f/(a+b*sin(f*x+e)^2)^(1/2)

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Rubi [A]
time = 0.26, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3271, 427, 542, 538, 437, 435, 432, 430} \begin {gather*} \frac {a (a+b) \left (2 a^2+9 a b-b^2\right ) \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 )}{35 b^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a-b) \left (a^2+6 a b+b^2\right ) \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 )}{35 b^2 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {\left (a^2-9 a b-2 b^2\right ) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}-\frac {b \sin (e+f x) \cos ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}+\frac {2 (4 a+b) \sin (e+f x) \cos ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2),x]

[Out]

-1/35*((a^2 - 9*a*b - 2*b^2)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(b*f) + (2*(4*a + b)*Cos[e
+ f*x]^3*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(35*f) - (b*Cos[e + f*x]^5*Sin[e + f*x]*Sqrt[a + b*Sin[e + f
*x]^2])/(7*f) - (2*(a - b)*(a^2 + 6*a*b + b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Se
c[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(35*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(a + b)*(2*a^2 + 9*a*b -
b^2)*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]
)/(35*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])

Rule 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[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 542

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

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 \cos ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx &=\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \left (1-x^2\right )^{3/2} \left (a+b x^2\right )^{3/2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac {b \cos ^5(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\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 (7 a+b)-2 b (4 a+b) x^2\right )}{\sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{7 f}\\ &=\frac {2 (4 a+b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos ^5(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {1-x^2} \left (-3 a b (9 a+b)+3 b \left (a^2-9 a b-2 b^2\right ) x^2\right )}{\sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{35 b f}\\ &=-\frac {\left (a^2-9 a b-2 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}+\frac {2 (4 a+b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos ^5(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-3 a b \left (a^2+18 a b+b^2\right )+6 (a-b) b \left (a^2+6 a b+b^2\right ) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{105 b^2 f}\\ &=-\frac {\left (a^2-9 a b-2 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}+\frac {2 (4 a+b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos ^5(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}+\frac {\left (a (a+b) \left (2 a^2+9 a b-b^2\right ) \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 )}{35 b^2 f}-\frac {\left (2 (a-b) \left (a^2+6 a b+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 )}{35 b^2 f}\\ &=-\frac {\left (a^2-9 a b-2 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}+\frac {2 (4 a+b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos ^5(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {\left (2 (a-b) \left (a^2+6 a b+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 )}{35 b^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\left (a (a+b) \left (2 a^2+9 a b-b^2\right ) \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 )}{35 b^2 f \sqrt {a+b \sin ^2(e+f x)}}\\ &=-\frac {\left (a^2-9 a b-2 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}+\frac {2 (4 a+b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos ^5(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {2 (a-b) \left (a^2+6 a b+b^2\right ) \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)}}{35 b^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {a (a+b) \left (2 a^2+9 a b-b^2\right ) \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}}}{35 b^2 f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 1.85, size = 247, normalized size = 0.77 \begin {gather*} \frac {-128 a \left (a^3+5 a^2 b-5 a b^2-b^3\right ) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )+64 a \left (2 a^3+11 a^2 b+8 a b^2-b^3\right ) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} F\left (e+f x\left |-\frac {b}{a}\right .\right )+\sqrt {2} b \left (-32 a^3+400 a^2 b+212 a b^2+30 b^3+b \left (144 a^2-192 a b-37 b^2\right ) \cos (2 (e+f x))+2 b^2 (-26 a+b) \cos (4 (e+f x))+5 b^3 \cos (6 (e+f x))\right ) \sin (2 (e+f x))}{2240 b^2 f \sqrt {2 a+b-b \cos (2 (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2),x]

[Out]

(-128*a*(a^3 + 5*a^2*b - 5*a*b^2 - b^3)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 64
*a*(2*a^3 + 11*a^2*b + 8*a*b^2 - b^3)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + Sqrt
[2]*b*(-32*a^3 + 400*a^2*b + 212*a*b^2 + 30*b^3 + b*(144*a^2 - 192*a*b - 37*b^2)*Cos[2*(e + f*x)] + 2*b^2*(-26
*a + b)*Cos[4*(e + f*x)] + 5*b^3*Cos[6*(e + f*x)])*Sin[2*(e + f*x)])/(2240*b^2*f*Sqrt[2*a + b - b*Cos[2*(e + f
*x)]])

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Maple [A]
time = 8.43, size = 590, normalized size = 1.84

method	result	size
default	\(\frac {5 b^{4} \sin \left (f x +e \right ) \left (\cos ^{8}\left (f x +e \right )\right )+\left (-13 a \,b^{3}-7 b^{4}\right ) \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (9 a^{2} b^{2}+a \,b^{3}\right ) \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-a^{3} b +8 a^{2} b^{2}+11 a \,b^{3}+2 b^{4}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \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^{4}+11 \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} 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^{2} b^{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 \,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^{4}-10 \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} b +10 \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^{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 \,b^{3}}{35 b^{2} \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\)	\(590\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/35*(5*b^4*sin(f*x+e)*cos(f*x+e)^8+(-13*a*b^3-7*b^4)*cos(f*x+e)^6*sin(f*x+e)+(9*a^2*b^2+a*b^3)*cos(f*x+e)^4*s
in(f*x+e)+(-a^3*b+8*a^2*b^2+11*a*b^3+2*b^4)*cos(f*x+e)^2*sin(f*x+e)+2*(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^4+11*(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^3*b+8*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*Ellipt
icF(sin(f*x+e),(-1/a*b)^(1/2))*a^2*b^2-(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^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^4-10*(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*b+10*(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^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*b^3)/b^2/cos(f*
x+e)/(a+b*sin(f*x+e)^2)^(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(cos(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e)^2 + a)^(3/2)*cos(f*x + e)^4, x)

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Fricas [F]
time = 0.14, size = 44, normalized size = 0.14 \begin {gather*} {\rm integral}\left (-{\left (b \cos \left (f x + e\right )^{6} - {\left (a + b\right )} \cos \left (f x + e\right )^{4}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}, x\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)^(3/2),x, algorithm="fricas")

[Out]

integral(-(b*cos(f*x + e)^6 - (a + b)*cos(f*x + e)^4)*sqrt(-b*cos(f*x + e)^2 + a + b), x)

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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(cos(f*x+e)**4*(a+b*sin(f*x+e)**2)**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 7316 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(cos(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e)^2 + a)^(3/2)*cos(f*x + e)^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\cos \left (e+f\,x\right )}^4\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,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)^(3/2),x)

[Out]

int(cos(e + f*x)^4*(a + b*sin(e + f*x)^2)^(3/2), x)

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