3.211 \(\int \frac{\cos ^5(e+f x)}{(a+b \sec ^2(e+f x))^3} \, dx\)

Optimal. Leaf size=214 \[ -\frac{b^5 \sin (e+f x)}{4 a^5 f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^2}+\frac{b^4 (20 a+17 b) \sin (e+f x)}{8 a^5 f (a+b)^2 \left (-a \sin ^2(e+f x)+a+b\right )}+\frac{\left (a^2-3 a b+6 b^2\right ) \sin (e+f x)}{a^5 f}-\frac{b^3 \left (80 a^2+140 a b+63 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right )}{8 a^{11/2} f (a+b)^{5/2}}-\frac{(2 a-3 b) \sin ^3(e+f x)}{3 a^4 f}+\frac{\sin ^5(e+f x)}{5 a^3 f} \]

[Out]

-(b^3*(80*a^2 + 140*a*b + 63*b^2)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*a^(11/2)*(a + b)^(5/2)*f) +
((a^2 - 3*a*b + 6*b^2)*Sin[e + f*x])/(a^5*f) - ((2*a - 3*b)*Sin[e + f*x]^3)/(3*a^4*f) + Sin[e + f*x]^5/(5*a^3*
f) - (b^5*Sin[e + f*x])/(4*a^5*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) + (b^4*(20*a + 17*b)*Sin[e + f*x])/(8*a
^5*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))

________________________________________________________________________________________

Rubi [A]  time = 0.258366, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4147, 390, 1157, 385, 208} \[ -\frac{b^5 \sin (e+f x)}{4 a^5 f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^2}+\frac{b^4 (20 a+17 b) \sin (e+f x)}{8 a^5 f (a+b)^2 \left (-a \sin ^2(e+f x)+a+b\right )}+\frac{\left (a^2-3 a b+6 b^2\right ) \sin (e+f x)}{a^5 f}-\frac{b^3 \left (80 a^2+140 a b+63 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right )}{8 a^{11/2} f (a+b)^{5/2}}-\frac{(2 a-3 b) \sin ^3(e+f x)}{3 a^4 f}+\frac{\sin ^5(e+f x)}{5 a^3 f} \]

Antiderivative was successfully verified.

[In]

Int[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]

[Out]

-(b^3*(80*a^2 + 140*a*b + 63*b^2)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*a^(11/2)*(a + b)^(5/2)*f) +
((a^2 - 3*a*b + 6*b^2)*Sin[e + f*x])/(a^5*f) - ((2*a - 3*b)*Sin[e + f*x]^3)/(3*a^4*f) + Sin[e + f*x]^5/(5*a^3*
f) - (b^5*Sin[e + f*x])/(4*a^5*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) + (b^4*(20*a + 17*b)*Sin[e + f*x])/(8*a
^5*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))

Rule 4147

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^
((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n
/2] && IntegerQ[p]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 385

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^5}{\left (a+b-a x^2\right )^3} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2-3 a b+6 b^2}{a^5}-\frac{(2 a-3 b) x^2}{a^4}+\frac{x^4}{a^3}-\frac{b^3 \left (10 a^2+15 a b+6 b^2\right )-5 a b^3 (4 a+3 b) x^2+10 a^2 b^3 x^4}{a^5 \left (a+b-a x^2\right )^3}\right ) \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\left (a^2-3 a b+6 b^2\right ) \sin (e+f x)}{a^5 f}-\frac{(2 a-3 b) \sin ^3(e+f x)}{3 a^4 f}+\frac{\sin ^5(e+f x)}{5 a^3 f}-\frac{\operatorname{Subst}\left (\int \frac{b^3 \left (10 a^2+15 a b+6 b^2\right )-5 a b^3 (4 a+3 b) x^2+10 a^2 b^3 x^4}{\left (a+b-a x^2\right )^3} \, dx,x,\sin (e+f x)\right )}{a^5 f}\\ &=\frac{\left (a^2-3 a b+6 b^2\right ) \sin (e+f x)}{a^5 f}-\frac{(2 a-3 b) \sin ^3(e+f x)}{3 a^4 f}+\frac{\sin ^5(e+f x)}{5 a^3 f}-\frac{b^5 \sin (e+f x)}{4 a^5 (a+b) f \left (a+b-a \sin ^2(e+f x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{-b^3 \left (40 a^2+60 a b+23 b^2\right )+40 a b^3 (a+b) x^2}{\left (a+b-a x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{4 a^5 (a+b) f}\\ &=\frac{\left (a^2-3 a b+6 b^2\right ) \sin (e+f x)}{a^5 f}-\frac{(2 a-3 b) \sin ^3(e+f x)}{3 a^4 f}+\frac{\sin ^5(e+f x)}{5 a^3 f}-\frac{b^5 \sin (e+f x)}{4 a^5 (a+b) f \left (a+b-a \sin ^2(e+f x)\right )^2}+\frac{b^4 (20 a+17 b) \sin (e+f x)}{8 a^5 (a+b)^2 f \left (a+b-a \sin ^2(e+f x)\right )}-\frac{\left (b^3 \left (80 a^2+140 a b+63 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{8 a^5 (a+b)^2 f}\\ &=-\frac{b^3 \left (80 a^2+140 a b+63 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right )}{8 a^{11/2} (a+b)^{5/2} f}+\frac{\left (a^2-3 a b+6 b^2\right ) \sin (e+f x)}{a^5 f}-\frac{(2 a-3 b) \sin ^3(e+f x)}{3 a^4 f}+\frac{\sin ^5(e+f x)}{5 a^3 f}-\frac{b^5 \sin (e+f x)}{4 a^5 (a+b) f \left (a+b-a \sin ^2(e+f x)\right )^2}+\frac{b^4 (20 a+17 b) \sin (e+f x)}{8 a^5 (a+b)^2 f \left (a+b-a \sin ^2(e+f x)\right )}\\ \end{align*}

Mathematica [C]  time = 7.61834, size = 2670, normalized size = 12.48 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3,x]

[Out]

((5*a^2 - 18*a*b + 48*b^2)*Cos[f*x]*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*Sin[e])/(64*a^5*f*(a + b*S
ec[e + f*x]^2)^3) + ((-80*a^2*b^3 - 140*a*b^4 - 63*b^5)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(((I/1
28)*ArcTan[((-I)*a*Cos[e] - I*b*Cos[e] + I*a*Cos[3*e] + I*b*Cos[3*e] + a*Sin[e] + b*Sin[e] - Sqrt[a]*Sqrt[a +
b]*Cos[e - f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + Sqrt[a]*Sqrt[a + b]*Cos[3*e + f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] +
a*Sin[3*e] + b*Sin[3*e] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e - f*x] - (2*I)*Sqrt[a]*Sqrt
[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e + f*x] + I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[3*e +
f*x])/(a*Cos[e] + 3*b*Cos[e] + a*Cos[3*e] + b*Cos[3*e] + a*Cos[e + 2*f*x] + a*Cos[3*e + 2*f*x] - (3*I)*a*Sin[
e] - I*b*Sin[e] - I*a*Sin[3*e] - I*b*Sin[3*e] - I*a*Sin[e + 2*f*x] + I*a*Sin[3*e + 2*f*x])]*Cos[e])/(a^(11/2)*
Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) + (ArcTan[((-I)*a*Cos[e] - I*b*Cos[e] + I*a*Cos[3*e] + I*b*Cos[3*e]
+ a*Sin[e] + b*Sin[e] - Sqrt[a]*Sqrt[a + b]*Cos[e - f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + Sqrt[a]*Sqrt[a + b]*Co
s[3*e + f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + a*Sin[3*e] + b*Sin[3*e] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*S
in[2*e]]*Sin[e - f*x] - (2*I)*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e + f*x] + I*Sqrt[a]*Sqrt[a
+ b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[3*e + f*x])/(a*Cos[e] + 3*b*Cos[e] + a*Cos[3*e] + b*Cos[3*e] + a*Cos[e +
2*f*x] + a*Cos[3*e + 2*f*x] - (3*I)*a*Sin[e] - I*b*Sin[e] - I*a*Sin[3*e] - I*b*Sin[3*e] - I*a*Sin[e + 2*f*x] +
I*a*Sin[3*e + 2*f*x])]*Sin[e])/(128*a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((a + b)^2*(a + b*S
ec[e + f*x]^2)^3) + ((80*a^2*b^3 + 140*a*b^4 + 63*b^5)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((ArcTa
nh[(2*(a + b)*Sin[e])/((-2*I)*a*Cos[e] - (2*I)*b*Cos[e] - Sqrt[a]*Sqrt[a + b]*Cos[e - f*x]*Sqrt[Cos[2*e] - I*S
in[2*e]] + Sqrt[a]*Sqrt[a + b]*Cos[3*e + f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e
] - I*Sin[2*e]]*Sin[e - f*x] + I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[3*e + f*x])]*Cos[e])/(128
*a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - ((I/128)*ArcTanh[(2*(a + b)*Sin[e])/((-2*I)*a*Cos[e] -
(2*I)*b*Cos[e] - Sqrt[a]*Sqrt[a + b]*Cos[e - f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] + Sqrt[a]*Sqrt[a + b]*Cos[3*e +
f*x]*Sqrt[Cos[2*e] - I*Sin[2*e]] - I*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[e - f*x] + I*Sqrt[a]*
Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[3*e + f*x])]*Sin[e])/(a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin
[2*e]])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + ((-80*a^2*b^3 - 140*a*b^4 - 63*b^5)*(a + 2*b + a*Cos[2*e + 2*
f*x])^3*Sec[e + f*x]^6*((Cos[e]*Log[a + 2*a*Cos[2*e] + 2*b*Cos[2*e] - a*Cos[2*e + 2*f*x] - (2*I)*a*Sin[2*e] -
(2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos
[2*e] - I*Sin[2*e]]*Sin[2*e + f*x]])/(256*a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - ((I/256)*Log[a
+ 2*a*Cos[2*e] + 2*b*Cos[2*e] - a*Cos[2*e + 2*f*x] - (2*I)*a*Sin[2*e] - (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a +
b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[2*e + f*x]]*S
in[e])/(a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + ((80*a^2*
b^3 + 140*a*b^4 + 63*b^5)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((Cos[e]*Log[-a - 2*a*Cos[2*e] - 2*b
*Cos[2*e] + a*Cos[2*e + 2*f*x] + (2*I)*a*Sin[2*e] + (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I
*Sin[2*e]]*Sin[f*x] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[2*e + f*x]])/(256*a^(11/2)*Sqrt[a
+ b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - ((I/256)*Log[-a - 2*a*Cos[2*e] - 2*b*Cos[2*e] + a*Cos[2*e + 2*f*x] + (2*
I)*a*Sin[2*e] + (2*I)*b*Sin[2*e] + 2*Sqrt[a]*Sqrt[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[f*x] + 2*Sqrt[a]*Sqrt
[a + b]*Sqrt[Cos[2*e] - I*Sin[2*e]]*Sin[2*e + f*x]]*Sin[e])/(a^(11/2)*Sqrt[a + b]*f*Sqrt[Cos[2*e] - I*Sin[2*e]
])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + ((5*a - 12*b)*Cos[3*f*x]*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e +
f*x]^6*Sin[3*e])/(384*a^4*f*(a + b*Sec[e + f*x]^2)^3) + (Cos[5*f*x]*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f
*x]^6*Sin[5*e])/(640*a^3*f*(a + b*Sec[e + f*x]^2)^3) + ((5*a^2 - 18*a*b + 48*b^2)*Cos[e]*(a + 2*b + a*Cos[2*e
+ 2*f*x])^3*Sec[e + f*x]^6*Sin[f*x])/(64*a^5*f*(a + b*Sec[e + f*x]^2)^3) + ((5*a - 12*b)*Cos[3*e]*(a + 2*b + a
*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*Sin[3*f*x])/(384*a^4*f*(a + b*Sec[e + f*x]^2)^3) + (Cos[5*e]*(a + 2*b + a*
Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*Sin[5*f*x])/(640*a^3*f*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*Cos[2*e +
2*f*x])^2*Sec[e + f*x]^6*(20*a*b^4*Sin[e + f*x] + 17*b^5*Sin[e + f*x]))/(32*a^5*(a + b)^2*f*(a + b*Sec[e + f*x
]^2)^3) - (b^5*(a + 2*b + a*Cos[2*e + 2*f*x])*Sec[e + f*x]^5*Tan[e + f*x])/(8*a^5*(a + b)*f*(a + b*Sec[e + f*x
]^2)^3)

________________________________________________________________________________________

Maple [A]  time = 0.118, size = 214, normalized size = 1. \begin{align*}{\frac{1}{f} \left ({\frac{1}{{a}^{5}} \left ({\frac{ \left ( \sin \left ( fx+e \right ) \right ) ^{5}{a}^{2}}{5}}-{\frac{2\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}{a}^{2}}{3}}+ \left ( \sin \left ( fx+e \right ) \right ) ^{3}ab+{a}^{2}\sin \left ( fx+e \right ) -3\,ab\sin \left ( fx+e \right ) +6\,{b}^{2}\sin \left ( fx+e \right ) \right ) }+{\frac{{b}^{3}}{{a}^{5}} \left ({\frac{1}{ \left ( -a-b+a \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}} \left ( -{\frac{ab \left ( 20\,a+17\,b \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{8\,{a}^{2}+16\,ab+8\,{b}^{2}}}+{\frac{ \left ( 20\,a+15\,b \right ) b\sin \left ( fx+e \right ) }{8\,a+8\,b}} \right ) }-{\frac{80\,{a}^{2}+140\,ab+63\,{b}^{2}}{8\,{a}^{2}+16\,ab+8\,{b}^{2}}{\it Artanh} \left ({\sin \left ( fx+e \right ) a{\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x)

[Out]

1/f*(1/a^5*(1/5*sin(f*x+e)^5*a^2-2/3*sin(f*x+e)^3*a^2+sin(f*x+e)^3*a*b+a^2*sin(f*x+e)-3*a*b*sin(f*x+e)+6*b^2*s
in(f*x+e))+1/a^5*b^3*((-1/8*a*b*(20*a+17*b)/(a^2+2*a*b+b^2)*sin(f*x+e)^3+5/8*(4*a+3*b)*b/(a+b)*sin(f*x+e))/(-a
-b+a*sin(f*x+e)^2)^2-1/8*(80*a^2+140*a*b+63*b^2)/(a^2+2*a*b+b^2)/((a+b)*a)^(1/2)*arctanh(a*sin(f*x+e)/((a+b)*a
)^(1/2))))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.845971, size = 2271, normalized size = 10.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x, algorithm="fricas")

[Out]

[1/240*(15*(80*a^2*b^5 + 140*a*b^6 + 63*b^7 + (80*a^4*b^3 + 140*a^3*b^4 + 63*a^2*b^5)*cos(f*x + e)^4 + 2*(80*a
^3*b^4 + 140*a^2*b^5 + 63*a*b^6)*cos(f*x + e)^2)*sqrt(a^2 + a*b)*log(-(a*cos(f*x + e)^2 + 2*sqrt(a^2 + a*b)*si
n(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + 2*(24*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cos(f*x + e)^8 + 6
4*a^6*b^2 - 48*a^5*b^3 + 192*a^4*b^4 + 1774*a^3*b^5 + 2415*a^2*b^6 + 945*a*b^7 + 8*(4*a^8 + 3*a^7*b - 15*a^6*b
^2 - 23*a^5*b^3 - 9*a^4*b^4)*cos(f*x + e)^6 + 8*(8*a^8 + 2*a^7*b + 21*a^6*b^2 + 131*a^5*b^3 + 167*a^4*b^4 + 63
*a^3*b^5)*cos(f*x + e)^4 + (128*a^7*b - 64*a^6*b^2 + 360*a^5*b^3 + 3044*a^4*b^4 + 4067*a^3*b^5 + 1575*a^2*b^6)
*cos(f*x + e)^2)*sin(f*x + e))/((a^11 + 3*a^10*b + 3*a^9*b^2 + a^8*b^3)*f*cos(f*x + e)^4 + 2*(a^10*b + 3*a^9*b
^2 + 3*a^8*b^3 + a^7*b^4)*f*cos(f*x + e)^2 + (a^9*b^2 + 3*a^8*b^3 + 3*a^7*b^4 + a^6*b^5)*f), 1/120*(15*(80*a^2
*b^5 + 140*a*b^6 + 63*b^7 + (80*a^4*b^3 + 140*a^3*b^4 + 63*a^2*b^5)*cos(f*x + e)^4 + 2*(80*a^3*b^4 + 140*a^2*b
^5 + 63*a*b^6)*cos(f*x + e)^2)*sqrt(-a^2 - a*b)*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) + (24*(a^8 + 3*a
^7*b + 3*a^6*b^2 + a^5*b^3)*cos(f*x + e)^8 + 64*a^6*b^2 - 48*a^5*b^3 + 192*a^4*b^4 + 1774*a^3*b^5 + 2415*a^2*b
^6 + 945*a*b^7 + 8*(4*a^8 + 3*a^7*b - 15*a^6*b^2 - 23*a^5*b^3 - 9*a^4*b^4)*cos(f*x + e)^6 + 8*(8*a^8 + 2*a^7*b
+ 21*a^6*b^2 + 131*a^5*b^3 + 167*a^4*b^4 + 63*a^3*b^5)*cos(f*x + e)^4 + (128*a^7*b - 64*a^6*b^2 + 360*a^5*b^3
+ 3044*a^4*b^4 + 4067*a^3*b^5 + 1575*a^2*b^6)*cos(f*x + e)^2)*sin(f*x + e))/((a^11 + 3*a^10*b + 3*a^9*b^2 + a
^8*b^3)*f*cos(f*x + e)^4 + 2*(a^10*b + 3*a^9*b^2 + 3*a^8*b^3 + a^7*b^4)*f*cos(f*x + e)^2 + (a^9*b^2 + 3*a^8*b^
3 + 3*a^7*b^4 + a^6*b^5)*f)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)**5/(a+b*sec(f*x+e)**2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.32138, size = 383, normalized size = 1.79 \begin{align*} \frac{\frac{15 \,{\left (80 \, a^{2} b^{3} + 140 \, a b^{4} + 63 \, b^{5}\right )} \arctan \left (\frac{a \sin \left (f x + e\right )}{\sqrt{-a^{2} - a b}}\right )}{{\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} \sqrt{-a^{2} - a b}} - \frac{15 \,{\left (20 \, a^{2} b^{4} \sin \left (f x + e\right )^{3} + 17 \, a b^{5} \sin \left (f x + e\right )^{3} - 20 \, a^{2} b^{4} \sin \left (f x + e\right ) - 35 \, a b^{5} \sin \left (f x + e\right ) - 15 \, b^{6} \sin \left (f x + e\right )\right )}}{{\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )}{\left (a \sin \left (f x + e\right )^{2} - a - b\right )}^{2}} + \frac{8 \,{\left (3 \, a^{12} \sin \left (f x + e\right )^{5} - 10 \, a^{12} \sin \left (f x + e\right )^{3} + 15 \, a^{11} b \sin \left (f x + e\right )^{3} + 15 \, a^{12} \sin \left (f x + e\right ) - 45 \, a^{11} b \sin \left (f x + e\right ) + 90 \, a^{10} b^{2} \sin \left (f x + e\right )\right )}}{a^{15}}}{120 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x, algorithm="giac")

[Out]

1/120*(15*(80*a^2*b^3 + 140*a*b^4 + 63*b^5)*arctan(a*sin(f*x + e)/sqrt(-a^2 - a*b))/((a^7 + 2*a^6*b + a^5*b^2)
*sqrt(-a^2 - a*b)) - 15*(20*a^2*b^4*sin(f*x + e)^3 + 17*a*b^5*sin(f*x + e)^3 - 20*a^2*b^4*sin(f*x + e) - 35*a*
b^5*sin(f*x + e) - 15*b^6*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*(a*sin(f*x + e)^2 - a - b)^2) + 8*(3*a^12*s
in(f*x + e)^5 - 10*a^12*sin(f*x + e)^3 + 15*a^11*b*sin(f*x + e)^3 + 15*a^12*sin(f*x + e) - 45*a^11*b*sin(f*x +
e) + 90*a^10*b^2*sin(f*x + e))/a^15)/f