3.3.18 \(\int \frac {\cos ^6(e+f x)}{(a+b \sec ^2(e+f x))^3} \, dx\) [218]

Optimal. Leaf size=352 \[ \frac {\left (5 a^3-18 a^2 b+48 a b^2-160 b^3\right ) x}{16 a^6}+\frac {b^{7/2} \left (99 a^2+176 a b+80 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^6 (a+b)^{5/2} f}+\frac {\left (15 a^2-34 a b+80 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {5 (a-2 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (5 a^4-8 a^3 b+17 a^2 b^2+116 a b^3+80 b^4\right ) \tan (e+f x)}{16 a^5 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )} \]

[Out]

1/16*(5*a^3-18*a^2*b+48*a*b^2-160*b^3)*x/a^6+1/8*b^(7/2)*(99*a^2+176*a*b+80*b^2)*arctan(b^(1/2)*tan(f*x+e)/(a+
b)^(1/2))/a^6/(a+b)^(5/2)/f+1/48*(15*a^2-34*a*b+80*b^2)*cos(f*x+e)*sin(f*x+e)/a^3/f/(a+b+b*tan(f*x+e)^2)^2+5/2
4*(a-2*b)*cos(f*x+e)^3*sin(f*x+e)/a^2/f/(a+b+b*tan(f*x+e)^2)^2+1/6*cos(f*x+e)^5*sin(f*x+e)/a/f/(a+b+b*tan(f*x+
e)^2)^2+1/48*b*(15*a^3-29*a^2*b+64*a*b^2+120*b^3)*tan(f*x+e)/a^4/(a+b)/f/(a+b+b*tan(f*x+e)^2)^2+1/16*b*(5*a^4-
8*a^3*b+17*a^2*b^2+116*a*b^3+80*b^4)*tan(f*x+e)/a^5/(a+b)^2/f/(a+b+b*tan(f*x+e)^2)

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Rubi [A]
time = 0.34, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4231, 425, 541, 536, 209, 211} \begin {gather*} \frac {5 (a-2 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac {b^{7/2} \left (99 a^2+176 a b+80 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^6 f (a+b)^{5/2}}+\frac {\left (15 a^2-34 a b+80 b^2\right ) \sin (e+f x) \cos (e+f x)}{48 a^3 f \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac {x \left (5 a^3-18 a^2 b+48 a b^2-160 b^3\right )}{16 a^6}+\frac {b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\right ) \tan (e+f x)}{48 a^4 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac {b \left (5 a^4-8 a^3 b+17 a^2 b^2+116 a b^3+80 b^4\right ) \tan (e+f x)}{16 a^5 f (a+b)^2 \left (a+b \tan ^2(e+f x)+b\right )}+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 a f \left (a+b \tan ^2(e+f x)+b\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]

[Out]

((5*a^3 - 18*a^2*b + 48*a*b^2 - 160*b^3)*x)/(16*a^6) + (b^(7/2)*(99*a^2 + 176*a*b + 80*b^2)*ArcTan[(Sqrt[b]*Ta
n[e + f*x])/Sqrt[a + b]])/(8*a^6*(a + b)^(5/2)*f) + ((15*a^2 - 34*a*b + 80*b^2)*Cos[e + f*x]*Sin[e + f*x])/(48
*a^3*f*(a + b + b*Tan[e + f*x]^2)^2) + (5*(a - 2*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*(a + b + b*Tan[e +
f*x]^2)^2) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(15*a^3 - 29*a^2*b + 64*a
*b^2 + 120*b^3)*Tan[e + f*x])/(48*a^4*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(5*a^4 - 8*a^3*b + 17*a^2*b
^2 + 116*a*b^3 + 80*b^4)*Tan[e + f*x])/(16*a^5*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 211

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

Rule 425

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

Rule 536

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

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 4231

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

Rubi steps

\begin {align*} \int \frac {\cos ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^4 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-5 a+b-9 b x^2}{\left (1+x^2\right )^3 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{6 a f}\\ &=\frac {5 (a-2 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\text {Subst}\left (\int \frac {15 a^2+a b+10 b^2+35 (a-2 b) b x^2}{\left (1+x^2\right )^2 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{24 a^2 f}\\ &=\frac {\left (15 a^2-34 a b+80 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {5 (a-2 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-15 a^3-21 a^2 b+26 a b^2+80 b^3-5 b \left (15 a^2-34 a b+80 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{48 a^3 f}\\ &=\frac {\left (15 a^2-34 a b+80 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {5 (a-2 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-12 \left (5 a^4+2 a^3 b+a^2 b^2-48 a b^3-40 b^4\right )-12 b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{192 a^4 (a+b) f}\\ &=\frac {\left (15 a^2-34 a b+80 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {5 (a-2 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (5 a^4-8 a^3 b+17 a^2 b^2+116 a b^3+80 b^4\right ) \tan (e+f x)}{16 a^5 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {-24 \left (5 a^5-3 a^4 b+9 a^3 b^2-65 a^2 b^3-156 a b^4-80 b^5\right )-24 b \left (5 a^4-8 a^3 b+17 a^2 b^2+116 a b^3+80 b^4\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{384 a^5 (a+b)^2 f}\\ &=\frac {\left (15 a^2-34 a b+80 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {5 (a-2 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (5 a^4-8 a^3 b+17 a^2 b^2+116 a b^3+80 b^4\right ) \tan (e+f x)}{16 a^5 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\left (b^4 \left (99 a^2+176 a b+80 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a^6 (a+b)^2 f}+\frac {\left (5 a^3-18 a^2 b+48 a b^2-160 b^3\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{16 a^6 f}\\ &=\frac {\left (5 a^3-18 a^2 b+48 a b^2-160 b^3\right ) x}{16 a^6}+\frac {b^{7/2} \left (99 a^2+176 a b+80 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^6 (a+b)^{5/2} f}+\frac {\left (15 a^2-34 a b+80 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {5 (a-2 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^5(e+f x) \sin (e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (15 a^3-29 a^2 b+64 a b^2+120 b^3\right ) \tan (e+f x)}{48 a^4 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {b \left (5 a^4-8 a^3 b+17 a^2 b^2+116 a b^3+80 b^4\right ) \tan (e+f x)}{16 a^5 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 6.69, size = 1770, normalized size = 5.03 \begin {gather*} \frac {\left (99 a^2+176 a b+80 b^2\right ) (a+2 b+a \cos (2 e+2 f x))^3 \sec ^6(e+f x) \left (-\frac {b^4 \text {ArcTan}\left (\sec (f x) \left (\frac {\cos (2 e)}{2 \sqrt {a+b} \sqrt {b \cos (4 e)-i b \sin (4 e)}}-\frac {i \sin (2 e)}{2 \sqrt {a+b} \sqrt {b \cos (4 e)-i b \sin (4 e)}}\right ) (-a \sin (f x)-2 b \sin (f x)+a \sin (2 e+f x))\right ) \cos (2 e)}{64 a^6 \sqrt {a+b} f \sqrt {b \cos (4 e)-i b \sin (4 e)}}+\frac {i b^4 \text {ArcTan}\left (\sec (f x) \left (\frac {\cos (2 e)}{2 \sqrt {a+b} \sqrt {b \cos (4 e)-i b \sin (4 e)}}-\frac {i \sin (2 e)}{2 \sqrt {a+b} \sqrt {b \cos (4 e)-i b \sin (4 e)}}\right ) (-a \sin (f x)-2 b \sin (f x)+a \sin (2 e+f x))\right ) \sin (2 e)}{64 a^6 \sqrt {a+b} f \sqrt {b \cos (4 e)-i b \sin (4 e)}}\right )}{(a+b)^2 \left (a+b \sec ^2(e+f x)\right )^3}+\frac {(a+2 b+a \cos (2 e+2 f x)) \sec (2 e) \sec ^6(e+f x) \left (720 a^7 f x \cos (2 e)+768 a^6 b f x \cos (2 e)+1296 a^5 b^2 f x \cos (2 e)-8352 a^4 b^3 f x \cos (2 e)-64128 a^3 b^4 f x \cos (2 e)-158976 a^2 b^5 f x \cos (2 e)-165888 a b^6 f x \cos (2 e)-61440 b^7 f x \cos (2 e)+480 a^7 f x \cos (2 f x)+192 a^6 b f x \cos (2 f x)+96 a^5 b^2 f x \cos (2 f x)-4608 a^4 b^3 f x \cos (2 f x)-41856 a^3 b^4 f x \cos (2 f x)-67584 a^2 b^5 f x \cos (2 f x)-30720 a b^6 f x \cos (2 f x)+480 a^7 f x \cos (4 e+2 f x)+192 a^6 b f x \cos (4 e+2 f x)+96 a^5 b^2 f x \cos (4 e+2 f x)-4608 a^4 b^3 f x \cos (4 e+2 f x)-41856 a^3 b^4 f x \cos (4 e+2 f x)-67584 a^2 b^5 f x \cos (4 e+2 f x)-30720 a b^6 f x \cos (4 e+2 f x)+120 a^7 f x \cos (2 e+4 f x)-192 a^6 b f x \cos (2 e+4 f x)+408 a^5 b^2 f x \cos (2 e+4 f x)-1968 a^4 b^3 f x \cos (2 e+4 f x)-6528 a^3 b^4 f x \cos (2 e+4 f x)-3840 a^2 b^5 f x \cos (2 e+4 f x)+120 a^7 f x \cos (6 e+4 f x)-192 a^6 b f x \cos (6 e+4 f x)+408 a^5 b^2 f x \cos (6 e+4 f x)-1968 a^4 b^3 f x \cos (6 e+4 f x)-6528 a^3 b^4 f x \cos (6 e+4 f x)-3840 a^2 b^5 f x \cos (6 e+4 f x)-6048 a^3 b^4 \sin (2 e)-21312 a^2 b^5 \sin (2 e)-29952 a b^6 \sin (2 e)-13824 b^7 \sin (2 e)+262 a^7 \sin (2 f x)+524 a^6 b \sin (2 f x)-26 a^5 b^2 \sin (2 f x)+1728 a^4 b^3 \sin (2 f x)+14976 a^3 b^4 \sin (2 f x)+28416 a^2 b^5 \sin (2 f x)+14592 a b^6 \sin (2 f x)+262 a^7 \sin (4 e+2 f x)+524 a^6 b \sin (4 e+2 f x)-26 a^5 b^2 \sin (4 e+2 f x)+1728 a^4 b^3 \sin (4 e+2 f x)+6912 a^3 b^4 \sin (4 e+2 f x)+5376 a^2 b^5 \sin (4 e+2 f x)+768 a b^6 \sin (4 e+2 f x)+238 a^7 \sin (2 e+4 f x)+304 a^6 b \sin (2 e+4 f x)-250 a^5 b^2 \sin (2 e+4 f x)+1556 a^4 b^3 \sin (2 e+4 f x)+5904 a^3 b^4 \sin (2 e+4 f x)+3744 a^2 b^5 \sin (2 e+4 f x)+238 a^7 \sin (6 e+4 f x)+304 a^6 b \sin (6 e+4 f x)-250 a^5 b^2 \sin (6 e+4 f x)+1556 a^4 b^3 \sin (6 e+4 f x)+3888 a^3 b^4 \sin (6 e+4 f x)+2016 a^2 b^5 \sin (6 e+4 f x)+87 a^7 \sin (4 e+6 f x)+46 a^6 b \sin (4 e+6 f x)-9 a^5 b^2 \sin (4 e+6 f x)+192 a^4 b^3 \sin (4 e+6 f x)+160 a^3 b^4 \sin (4 e+6 f x)+87 a^7 \sin (8 e+6 f x)+46 a^6 b \sin (8 e+6 f x)-9 a^5 b^2 \sin (8 e+6 f x)+192 a^4 b^3 \sin (8 e+6 f x)+160 a^3 b^4 \sin (8 e+6 f x)+13 a^7 \sin (6 e+8 f x)+16 a^6 b \sin (6 e+8 f x)-7 a^5 b^2 \sin (6 e+8 f x)-10 a^4 b^3 \sin (6 e+8 f x)+13 a^7 \sin (10 e+8 f x)+16 a^6 b \sin (10 e+8 f x)-7 a^5 b^2 \sin (10 e+8 f x)-10 a^4 b^3 \sin (10 e+8 f x)+a^7 \sin (8 e+10 f x)+2 a^6 b \sin (8 e+10 f x)+a^5 b^2 \sin (8 e+10 f x)+a^7 \sin (12 e+10 f x)+2 a^6 b \sin (12 e+10 f x)+a^5 b^2 \sin (12 e+10 f x)\right )}{12288 a^6 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3,x]

[Out]

((99*a^2 + 176*a*b + 80*b^2)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(-1/64*(b^4*ArcTan[Sec[f*x]*(Cos[
2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*Sin[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin
[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Cos[2*e])/(a^6*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b
*Sin[4*e]]) + ((I/64)*b^4*ArcTan[Sec[f*x]*(Cos[2*e]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]) - ((I/2)*S
in[2*e])/(Sqrt[a + b]*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]]))*(-(a*Sin[f*x]) - 2*b*Sin[f*x] + a*Sin[2*e + f*x])]*Sin
[2*e])/(a^6*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*Sin[4*e]])))/((a + b)^2*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b
+ a*Cos[2*e + 2*f*x])*Sec[2*e]*Sec[e + f*x]^6*(720*a^7*f*x*Cos[2*e] + 768*a^6*b*f*x*Cos[2*e] + 1296*a^5*b^2*f*
x*Cos[2*e] - 8352*a^4*b^3*f*x*Cos[2*e] - 64128*a^3*b^4*f*x*Cos[2*e] - 158976*a^2*b^5*f*x*Cos[2*e] - 165888*a*b
^6*f*x*Cos[2*e] - 61440*b^7*f*x*Cos[2*e] + 480*a^7*f*x*Cos[2*f*x] + 192*a^6*b*f*x*Cos[2*f*x] + 96*a^5*b^2*f*x*
Cos[2*f*x] - 4608*a^4*b^3*f*x*Cos[2*f*x] - 41856*a^3*b^4*f*x*Cos[2*f*x] - 67584*a^2*b^5*f*x*Cos[2*f*x] - 30720
*a*b^6*f*x*Cos[2*f*x] + 480*a^7*f*x*Cos[4*e + 2*f*x] + 192*a^6*b*f*x*Cos[4*e + 2*f*x] + 96*a^5*b^2*f*x*Cos[4*e
+ 2*f*x] - 4608*a^4*b^3*f*x*Cos[4*e + 2*f*x] - 41856*a^3*b^4*f*x*Cos[4*e + 2*f*x] - 67584*a^2*b^5*f*x*Cos[4*e
+ 2*f*x] - 30720*a*b^6*f*x*Cos[4*e + 2*f*x] + 120*a^7*f*x*Cos[2*e + 4*f*x] - 192*a^6*b*f*x*Cos[2*e + 4*f*x] +
408*a^5*b^2*f*x*Cos[2*e + 4*f*x] - 1968*a^4*b^3*f*x*Cos[2*e + 4*f*x] - 6528*a^3*b^4*f*x*Cos[2*e + 4*f*x] - 38
40*a^2*b^5*f*x*Cos[2*e + 4*f*x] + 120*a^7*f*x*Cos[6*e + 4*f*x] - 192*a^6*b*f*x*Cos[6*e + 4*f*x] + 408*a^5*b^2*
f*x*Cos[6*e + 4*f*x] - 1968*a^4*b^3*f*x*Cos[6*e + 4*f*x] - 6528*a^3*b^4*f*x*Cos[6*e + 4*f*x] - 3840*a^2*b^5*f*
x*Cos[6*e + 4*f*x] - 6048*a^3*b^4*Sin[2*e] - 21312*a^2*b^5*Sin[2*e] - 29952*a*b^6*Sin[2*e] - 13824*b^7*Sin[2*e
] + 262*a^7*Sin[2*f*x] + 524*a^6*b*Sin[2*f*x] - 26*a^5*b^2*Sin[2*f*x] + 1728*a^4*b^3*Sin[2*f*x] + 14976*a^3*b^
4*Sin[2*f*x] + 28416*a^2*b^5*Sin[2*f*x] + 14592*a*b^6*Sin[2*f*x] + 262*a^7*Sin[4*e + 2*f*x] + 524*a^6*b*Sin[4*
e + 2*f*x] - 26*a^5*b^2*Sin[4*e + 2*f*x] + 1728*a^4*b^3*Sin[4*e + 2*f*x] + 6912*a^3*b^4*Sin[4*e + 2*f*x] + 537
6*a^2*b^5*Sin[4*e + 2*f*x] + 768*a*b^6*Sin[4*e + 2*f*x] + 238*a^7*Sin[2*e + 4*f*x] + 304*a^6*b*Sin[2*e + 4*f*x
] - 250*a^5*b^2*Sin[2*e + 4*f*x] + 1556*a^4*b^3*Sin[2*e + 4*f*x] + 5904*a^3*b^4*Sin[2*e + 4*f*x] + 3744*a^2*b^
5*Sin[2*e + 4*f*x] + 238*a^7*Sin[6*e + 4*f*x] + 304*a^6*b*Sin[6*e + 4*f*x] - 250*a^5*b^2*Sin[6*e + 4*f*x] + 15
56*a^4*b^3*Sin[6*e + 4*f*x] + 3888*a^3*b^4*Sin[6*e + 4*f*x] + 2016*a^2*b^5*Sin[6*e + 4*f*x] + 87*a^7*Sin[4*e +
6*f*x] + 46*a^6*b*Sin[4*e + 6*f*x] - 9*a^5*b^2*Sin[4*e + 6*f*x] + 192*a^4*b^3*Sin[4*e + 6*f*x] + 160*a^3*b^4*
Sin[4*e + 6*f*x] + 87*a^7*Sin[8*e + 6*f*x] + 46*a^6*b*Sin[8*e + 6*f*x] - 9*a^5*b^2*Sin[8*e + 6*f*x] + 192*a^4*
b^3*Sin[8*e + 6*f*x] + 160*a^3*b^4*Sin[8*e + 6*f*x] + 13*a^7*Sin[6*e + 8*f*x] + 16*a^6*b*Sin[6*e + 8*f*x] - 7*
a^5*b^2*Sin[6*e + 8*f*x] - 10*a^4*b^3*Sin[6*e + 8*f*x] + 13*a^7*Sin[10*e + 8*f*x] + 16*a^6*b*Sin[10*e + 8*f*x]
- 7*a^5*b^2*Sin[10*e + 8*f*x] - 10*a^4*b^3*Sin[10*e + 8*f*x] + a^7*Sin[8*e + 10*f*x] + 2*a^6*b*Sin[8*e + 10*f
*x] + a^5*b^2*Sin[8*e + 10*f*x] + a^7*Sin[12*e + 10*f*x] + 2*a^6*b*Sin[12*e + 10*f*x] + a^5*b^2*Sin[12*e + 10*
f*x]))/(12288*a^6*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3)

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Maple [A]
time = 0.24, size = 267, normalized size = 0.76

method	result	size
derivativedivides	\(\frac {\frac {b^{4} \left (\frac {\frac {a b \left (19 a +16 b \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{8 a^{2}+16 a b +8 b^{2}}+\frac {\left (21 a +16 b \right ) a \tan \left (f x +e \right )}{8 a +8 b}}{\left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {\left (99 a^{2}+176 a b +80 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {\left (a +b \right ) b}}\right )}{a^{6}}+\frac {\frac {\left (\frac {5}{16} a^{3}-\frac {9}{8} a^{2} b +3 a \,b^{2}\right ) \left (\tan ^{5}\left (f x +e \right )\right )+\left (6 a \,b^{2}+\frac {5}{6} a^{3}-3 a^{2} b \right ) \left (\tan ^{3}\left (f x +e \right )\right )+\left (-\frac {15}{8} a^{2} b +3 a \,b^{2}+\frac {11}{16} a^{3}\right ) \tan \left (f x +e \right )}{\left (\tan ^{2}\left (f x +e \right )+1\right )^{3}}+\frac {\left (5 a^{3}-18 a^{2} b +48 a \,b^{2}-160 b^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{16}}{a^{6}}}{f}\)	\(267\)
default	\(\frac {\frac {b^{4} \left (\frac {\frac {a b \left (19 a +16 b \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{8 a^{2}+16 a b +8 b^{2}}+\frac {\left (21 a +16 b \right ) a \tan \left (f x +e \right )}{8 a +8 b}}{\left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {\left (99 a^{2}+176 a b +80 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {\left (a +b \right ) b}}\right )}{a^{6}}+\frac {\frac {\left (\frac {5}{16} a^{3}-\frac {9}{8} a^{2} b +3 a \,b^{2}\right ) \left (\tan ^{5}\left (f x +e \right )\right )+\left (6 a \,b^{2}+\frac {5}{6} a^{3}-3 a^{2} b \right ) \left (\tan ^{3}\left (f x +e \right )\right )+\left (-\frac {15}{8} a^{2} b +3 a \,b^{2}+\frac {11}{16} a^{3}\right ) \tan \left (f x +e \right )}{\left (\tan ^{2}\left (f x +e \right )+1\right )^{3}}+\frac {\left (5 a^{3}-18 a^{2} b +48 a \,b^{2}-160 b^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{16}}{a^{6}}}{f}\)	\(267\)
risch	\(\frac {5 x}{16 a^{3}}-\frac {9 x b}{8 a^{4}}+\frac {3 x \,b^{2}}{a^{5}}-\frac {10 x \,b^{3}}{a^{6}}+\frac {3 i {\mathrm e}^{4 i \left (f x +e \right )} b}{64 a^{4} f}-\frac {15 i {\mathrm e}^{2 i \left (f x +e \right )}}{128 f \,a^{3}}+\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} b^{2}}{4 a^{5} f}-\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} b}{8 a^{4} f}-\frac {3 i {\mathrm e}^{4 i \left (f x +e \right )}}{128 a^{3} f}-\frac {3 i {\mathrm e}^{2 i \left (f x +e \right )} b^{2}}{4 a^{5} f}+\frac {15 i {\mathrm e}^{-2 i \left (f x +e \right )}}{128 f \,a^{3}}-\frac {i {\mathrm e}^{6 i \left (f x +e \right )}}{384 a^{3} f}-\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )} b}{64 a^{4} f}+\frac {i b^{4} \left (21 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}+64 a^{2} b \,{\mathrm e}^{6 i \left (f x +e \right )}+40 a \,b^{2} {\mathrm e}^{6 i \left (f x +e \right )}+63 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+222 a^{2} b \,{\mathrm e}^{4 i \left (f x +e \right )}+312 a \,b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+144 b^{3} {\mathrm e}^{4 i \left (f x +e \right )}+63 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}+176 a^{2} b \,{\mathrm e}^{2 i \left (f x +e \right )}+104 a \,b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+21 a^{3}+18 a^{2} b \right )}{4 a^{6} \left (a +b \right )^{2} f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )^{2}}+\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )}}{128 a^{3} f}+\frac {3 i {\mathrm e}^{2 i \left (f x +e \right )} b}{8 a^{4} f}+\frac {i {\mathrm e}^{-6 i \left (f x +e \right )}}{384 a^{3} f}+\frac {99 \sqrt {-\left (a +b \right ) b}\, b^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-\left (a +b \right ) b}-a -2 b}{a}\right )}{16 \left (a +b \right )^{3} f \,a^{4}}+\frac {11 \sqrt {-\left (a +b \right ) b}\, b^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-\left (a +b \right ) b}-a -2 b}{a}\right )}{\left (a +b \right )^{3} f \,a^{5}}+\frac {5 \sqrt {-\left (a +b \right ) b}\, b^{5} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-\left (a +b \right ) b}-a -2 b}{a}\right )}{\left (a +b \right )^{3} f \,a^{6}}-\frac {99 \sqrt {-\left (a +b \right ) b}\, b^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-\left (a +b \right ) b}+a +2 b}{a}\right )}{16 \left (a +b \right )^{3} f \,a^{4}}-\frac {11 \sqrt {-\left (a +b \right ) b}\, b^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-\left (a +b \right ) b}+a +2 b}{a}\right )}{\left (a +b \right )^{3} f \,a^{5}}-\frac {5 \sqrt {-\left (a +b \right ) b}\, b^{5} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-\left (a +b \right ) b}+a +2 b}{a}\right )}{\left (a +b \right )^{3} f \,a^{6}}\)	\(812\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/f*(b^4/a^6*((1/8*a*b*(19*a+16*b)/(a^2+2*a*b+b^2)*tan(f*x+e)^3+1/8*(21*a+16*b)*a/(a+b)*tan(f*x+e))/(a+b+b*tan
(f*x+e)^2)^2+1/8*(99*a^2+176*a*b+80*b^2)/(a^2+2*a*b+b^2)/((a+b)*b)^(1/2)*arctan(b*tan(f*x+e)/((a+b)*b)^(1/2)))
+1/a^6*(((5/16*a^3-9/8*a^2*b+3*a*b^2)*tan(f*x+e)^5+(6*a*b^2+5/6*a^3-3*a^2*b)*tan(f*x+e)^3+(-15/8*a^2*b+3*a*b^2
+11/16*a^3)*tan(f*x+e))/(tan(f*x+e)^2+1)^3+1/16*(5*a^3-18*a^2*b+48*a*b^2-160*b^3)*arctan(tan(f*x+e))))

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Maxima [A]
time = 0.50, size = 623, normalized size = 1.77 \begin {gather*} \frac {\frac {6 \, {\left (99 \, a^{2} b^{4} + 176 \, a b^{5} + 80 \, b^{6}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{{\left (a^{8} + 2 \, a^{7} b + a^{6} b^{2}\right )} \sqrt {{\left (a + b\right )} b}} + \frac {3 \, {\left (5 \, a^{4} b^{2} - 8 \, a^{3} b^{3} + 17 \, a^{2} b^{4} + 116 \, a b^{5} + 80 \, b^{6}\right )} \tan \left (f x + e\right )^{9} + 2 \, {\left (15 \, a^{5} b + 11 \, a^{4} b^{2} - 5 \, a^{3} b^{3} + 368 \, a^{2} b^{4} + 876 \, a b^{5} + 480 \, b^{6}\right )} \tan \left (f x + e\right )^{7} + {\left (15 \, a^{6} + 86 \, a^{5} b + 3 \, a^{4} b^{2} + 240 \, a^{3} b^{3} + 1982 \, a^{2} b^{4} + 3168 \, a b^{5} + 1440 \, b^{6}\right )} \tan \left (f x + e\right )^{5} + 2 \, {\left (20 \, a^{6} + 41 \, a^{5} b - 15 \, a^{4} b^{2} + 197 \, a^{3} b^{3} + 980 \, a^{2} b^{4} + 1236 \, a b^{5} + 480 \, b^{6}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (11 \, a^{6} + 14 \, a^{5} b - 6 \, a^{4} b^{2} + 56 \, a^{3} b^{3} + 221 \, a^{2} b^{4} + 236 \, a b^{5} + 80 \, b^{6}\right )} \tan \left (f x + e\right )}{{\left (a^{7} b^{2} + 2 \, a^{6} b^{3} + a^{5} b^{4}\right )} \tan \left (f x + e\right )^{10} + a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4} + {\left (2 \, a^{8} b + 9 \, a^{7} b^{2} + 12 \, a^{6} b^{3} + 5 \, a^{5} b^{4}\right )} \tan \left (f x + e\right )^{8} + {\left (a^{9} + 10 \, a^{8} b + 27 \, a^{7} b^{2} + 28 \, a^{6} b^{3} + 10 \, a^{5} b^{4}\right )} \tan \left (f x + e\right )^{6} + {\left (3 \, a^{9} + 18 \, a^{8} b + 37 \, a^{7} b^{2} + 32 \, a^{6} b^{3} + 10 \, a^{5} b^{4}\right )} \tan \left (f x + e\right )^{4} + {\left (3 \, a^{9} + 14 \, a^{8} b + 24 \, a^{7} b^{2} + 18 \, a^{6} b^{3} + 5 \, a^{5} b^{4}\right )} \tan \left (f x + e\right )^{2}} + \frac {3 \, {\left (5 \, a^{3} - 18 \, a^{2} b + 48 \, a b^{2} - 160 \, b^{3}\right )} {\left (f x + e\right )}}{a^{6}}}{48 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x, algorithm="maxima")

[Out]

1/48*(6*(99*a^2*b^4 + 176*a*b^5 + 80*b^6)*arctan(b*tan(f*x + e)/sqrt((a + b)*b))/((a^8 + 2*a^7*b + a^6*b^2)*sq
rt((a + b)*b)) + (3*(5*a^4*b^2 - 8*a^3*b^3 + 17*a^2*b^4 + 116*a*b^5 + 80*b^6)*tan(f*x + e)^9 + 2*(15*a^5*b + 1
1*a^4*b^2 - 5*a^3*b^3 + 368*a^2*b^4 + 876*a*b^5 + 480*b^6)*tan(f*x + e)^7 + (15*a^6 + 86*a^5*b + 3*a^4*b^2 + 2
40*a^3*b^3 + 1982*a^2*b^4 + 3168*a*b^5 + 1440*b^6)*tan(f*x + e)^5 + 2*(20*a^6 + 41*a^5*b - 15*a^4*b^2 + 197*a^
3*b^3 + 980*a^2*b^4 + 1236*a*b^5 + 480*b^6)*tan(f*x + e)^3 + 3*(11*a^6 + 14*a^5*b - 6*a^4*b^2 + 56*a^3*b^3 + 2
21*a^2*b^4 + 236*a*b^5 + 80*b^6)*tan(f*x + e))/((a^7*b^2 + 2*a^6*b^3 + a^5*b^4)*tan(f*x + e)^10 + a^9 + 4*a^8*
b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4 + (2*a^8*b + 9*a^7*b^2 + 12*a^6*b^3 + 5*a^5*b^4)*tan(f*x + e)^8 + (a^9 + 1
0*a^8*b + 27*a^7*b^2 + 28*a^6*b^3 + 10*a^5*b^4)*tan(f*x + e)^6 + (3*a^9 + 18*a^8*b + 37*a^7*b^2 + 32*a^6*b^3 +
10*a^5*b^4)*tan(f*x + e)^4 + (3*a^9 + 14*a^8*b + 24*a^7*b^2 + 18*a^6*b^3 + 5*a^5*b^4)*tan(f*x + e)^2) + 3*(5*
a^3 - 18*a^2*b + 48*a*b^2 - 160*b^3)*(f*x + e)/a^6)/f

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Fricas [A]
time = 3.11, size = 1330, normalized size = 3.78 \begin {gather*} \left [\frac {6 \, {\left (5 \, a^{7} - 8 \, a^{6} b + 17 \, a^{5} b^{2} - 82 \, a^{4} b^{3} - 272 \, a^{3} b^{4} - 160 \, a^{2} b^{5}\right )} f x \cos \left (f x + e\right )^{4} + 12 \, {\left (5 \, a^{6} b - 8 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 82 \, a^{3} b^{4} - 272 \, a^{2} b^{5} - 160 \, a b^{6}\right )} f x \cos \left (f x + e\right )^{2} + 6 \, {\left (5 \, a^{5} b^{2} - 8 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 82 \, a^{2} b^{5} - 272 \, a b^{6} - 160 \, b^{7}\right )} f x + 3 \, {\left (99 \, a^{2} b^{5} + 176 \, a b^{6} + 80 \, b^{7} + {\left (99 \, a^{4} b^{3} + 176 \, a^{3} b^{4} + 80 \, a^{2} b^{5}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (99 \, a^{3} b^{4} + 176 \, a^{2} b^{5} + 80 \, a b^{6}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {b}{a + b}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {b}{a + b}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right ) + 2 \, {\left (8 \, {\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} \cos \left (f x + e\right )^{9} + 10 \, {\left (a^{7} - 3 \, a^{5} b^{2} - 2 \, a^{4} b^{3}\right )} \cos \left (f x + e\right )^{7} + {\left (15 \, a^{7} - 4 \, a^{6} b + 27 \, a^{5} b^{2} + 126 \, a^{4} b^{3} + 80 \, a^{3} b^{4}\right )} \cos \left (f x + e\right )^{5} + 2 \, {\left (15 \, a^{6} b - 19 \, a^{5} b^{2} + 43 \, a^{4} b^{3} + 266 \, a^{3} b^{4} + 180 \, a^{2} b^{5}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (5 \, a^{5} b^{2} - 8 \, a^{4} b^{3} + 17 \, a^{3} b^{4} + 116 \, a^{2} b^{5} + 80 \, a b^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{96 \, {\left ({\left (a^{10} + 2 \, a^{9} b + a^{8} b^{2}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{9} b + 2 \, a^{8} b^{2} + a^{7} b^{3}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{8} b^{2} + 2 \, a^{7} b^{3} + a^{6} b^{4}\right )} f\right )}}, \frac {3 \, {\left (5 \, a^{7} - 8 \, a^{6} b + 17 \, a^{5} b^{2} - 82 \, a^{4} b^{3} - 272 \, a^{3} b^{4} - 160 \, a^{2} b^{5}\right )} f x \cos \left (f x + e\right )^{4} + 6 \, {\left (5 \, a^{6} b - 8 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 82 \, a^{3} b^{4} - 272 \, a^{2} b^{5} - 160 \, a b^{6}\right )} f x \cos \left (f x + e\right )^{2} + 3 \, {\left (5 \, a^{5} b^{2} - 8 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 82 \, a^{2} b^{5} - 272 \, a b^{6} - 160 \, b^{7}\right )} f x - 3 \, {\left (99 \, a^{2} b^{5} + 176 \, a b^{6} + 80 \, b^{7} + {\left (99 \, a^{4} b^{3} + 176 \, a^{3} b^{4} + 80 \, a^{2} b^{5}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (99 \, a^{3} b^{4} + 176 \, a^{2} b^{5} + 80 \, a b^{6}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{a + b}} \arctan \left (\frac {{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {b}{a + b}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) + {\left (8 \, {\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} \cos \left (f x + e\right )^{9} + 10 \, {\left (a^{7} - 3 \, a^{5} b^{2} - 2 \, a^{4} b^{3}\right )} \cos \left (f x + e\right )^{7} + {\left (15 \, a^{7} - 4 \, a^{6} b + 27 \, a^{5} b^{2} + 126 \, a^{4} b^{3} + 80 \, a^{3} b^{4}\right )} \cos \left (f x + e\right )^{5} + 2 \, {\left (15 \, a^{6} b - 19 \, a^{5} b^{2} + 43 \, a^{4} b^{3} + 266 \, a^{3} b^{4} + 180 \, a^{2} b^{5}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (5 \, a^{5} b^{2} - 8 \, a^{4} b^{3} + 17 \, a^{3} b^{4} + 116 \, a^{2} b^{5} + 80 \, a b^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, {\left ({\left (a^{10} + 2 \, a^{9} b + a^{8} b^{2}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{9} b + 2 \, a^{8} b^{2} + a^{7} b^{3}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{8} b^{2} + 2 \, a^{7} b^{3} + a^{6} b^{4}\right )} f\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x, algorithm="fricas")

[Out]

[1/96*(6*(5*a^7 - 8*a^6*b + 17*a^5*b^2 - 82*a^4*b^3 - 272*a^3*b^4 - 160*a^2*b^5)*f*x*cos(f*x + e)^4 + 12*(5*a^
6*b - 8*a^5*b^2 + 17*a^4*b^3 - 82*a^3*b^4 - 272*a^2*b^5 - 160*a*b^6)*f*x*cos(f*x + e)^2 + 6*(5*a^5*b^2 - 8*a^4
*b^3 + 17*a^3*b^4 - 82*a^2*b^5 - 272*a*b^6 - 160*b^7)*f*x + 3*(99*a^2*b^5 + 176*a*b^6 + 80*b^7 + (99*a^4*b^3 +
176*a^3*b^4 + 80*a^2*b^5)*cos(f*x + e)^4 + 2*(99*a^3*b^4 + 176*a^2*b^5 + 80*a*b^6)*cos(f*x + e)^2)*sqrt(-b/(a
+ b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)
*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*c
os(f*x + e)^2 + b^2)) + 2*(8*(a^7 + 2*a^6*b + a^5*b^2)*cos(f*x + e)^9 + 10*(a^7 - 3*a^5*b^2 - 2*a^4*b^3)*cos(f
*x + e)^7 + (15*a^7 - 4*a^6*b + 27*a^5*b^2 + 126*a^4*b^3 + 80*a^3*b^4)*cos(f*x + e)^5 + 2*(15*a^6*b - 19*a^5*b
^2 + 43*a^4*b^3 + 266*a^3*b^4 + 180*a^2*b^5)*cos(f*x + e)^3 + 3*(5*a^5*b^2 - 8*a^4*b^3 + 17*a^3*b^4 + 116*a^2*
b^5 + 80*a*b^6)*cos(f*x + e))*sin(f*x + e))/((a^10 + 2*a^9*b + a^8*b^2)*f*cos(f*x + e)^4 + 2*(a^9*b + 2*a^8*b^
2 + a^7*b^3)*f*cos(f*x + e)^2 + (a^8*b^2 + 2*a^7*b^3 + a^6*b^4)*f), 1/48*(3*(5*a^7 - 8*a^6*b + 17*a^5*b^2 - 82
*a^4*b^3 - 272*a^3*b^4 - 160*a^2*b^5)*f*x*cos(f*x + e)^4 + 6*(5*a^6*b - 8*a^5*b^2 + 17*a^4*b^3 - 82*a^3*b^4 -
272*a^2*b^5 - 160*a*b^6)*f*x*cos(f*x + e)^2 + 3*(5*a^5*b^2 - 8*a^4*b^3 + 17*a^3*b^4 - 82*a^2*b^5 - 272*a*b^6 -
160*b^7)*f*x - 3*(99*a^2*b^5 + 176*a*b^6 + 80*b^7 + (99*a^4*b^3 + 176*a^3*b^4 + 80*a^2*b^5)*cos(f*x + e)^4 +
2*(99*a^3*b^4 + 176*a^2*b^5 + 80*a*b^6)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 -
b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) + (8*(a^7 + 2*a^6*b + a^5*b^2)*cos(f*x + e)^9 + 10*(a^7 - 3
*a^5*b^2 - 2*a^4*b^3)*cos(f*x + e)^7 + (15*a^7 - 4*a^6*b + 27*a^5*b^2 + 126*a^4*b^3 + 80*a^3*b^4)*cos(f*x + e)
^5 + 2*(15*a^6*b - 19*a^5*b^2 + 43*a^4*b^3 + 266*a^3*b^4 + 180*a^2*b^5)*cos(f*x + e)^3 + 3*(5*a^5*b^2 - 8*a^4*
b^3 + 17*a^3*b^4 + 116*a^2*b^5 + 80*a*b^6)*cos(f*x + e))*sin(f*x + e))/((a^10 + 2*a^9*b + a^8*b^2)*f*cos(f*x +
e)^4 + 2*(a^9*b + 2*a^8*b^2 + a^7*b^3)*f*cos(f*x + e)^2 + (a^8*b^2 + 2*a^7*b^3 + a^6*b^4)*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)**6/(a+b*sec(f*x+e)**2)**3,x)

[Out]

Timed out

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Giac [A]
time = 0.50, size = 350, normalized size = 0.99 \begin {gather*} \frac {\frac {6 \, {\left (99 \, a^{2} b^{4} + 176 \, a b^{5} + 80 \, b^{6}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )}}{{\left (a^{8} + 2 \, a^{7} b + a^{6} b^{2}\right )} \sqrt {a b + b^{2}}} + \frac {6 \, {\left (19 \, a b^{5} \tan \left (f x + e\right )^{3} + 16 \, b^{6} \tan \left (f x + e\right )^{3} + 21 \, a^{2} b^{4} \tan \left (f x + e\right ) + 37 \, a b^{5} \tan \left (f x + e\right ) + 16 \, b^{6} \tan \left (f x + e\right )\right )}}{{\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} {\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{2}} + \frac {3 \, {\left (5 \, a^{3} - 18 \, a^{2} b + 48 \, a b^{2} - 160 \, b^{3}\right )} {\left (f x + e\right )}}{a^{6}} + \frac {15 \, a^{2} \tan \left (f x + e\right )^{5} - 54 \, a b \tan \left (f x + e\right )^{5} + 144 \, b^{2} \tan \left (f x + e\right )^{5} + 40 \, a^{2} \tan \left (f x + e\right )^{3} - 144 \, a b \tan \left (f x + e\right )^{3} + 288 \, b^{2} \tan \left (f x + e\right )^{3} + 33 \, a^{2} \tan \left (f x + e\right ) - 90 \, a b \tan \left (f x + e\right ) + 144 \, b^{2} \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{3} a^{5}}}{48 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x, algorithm="giac")

[Out]

1/48*(6*(99*a^2*b^4 + 176*a*b^5 + 80*b^6)*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*
b + b^2)))/((a^8 + 2*a^7*b + a^6*b^2)*sqrt(a*b + b^2)) + 6*(19*a*b^5*tan(f*x + e)^3 + 16*b^6*tan(f*x + e)^3 +
21*a^2*b^4*tan(f*x + e) + 37*a*b^5*tan(f*x + e) + 16*b^6*tan(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*(b*tan(f*x +
e)^2 + a + b)^2) + 3*(5*a^3 - 18*a^2*b + 48*a*b^2 - 160*b^3)*(f*x + e)/a^6 + (15*a^2*tan(f*x + e)^5 - 54*a*b*
tan(f*x + e)^5 + 144*b^2*tan(f*x + e)^5 + 40*a^2*tan(f*x + e)^3 - 144*a*b*tan(f*x + e)^3 + 288*b^2*tan(f*x + e
)^3 + 33*a^2*tan(f*x + e) - 90*a*b*tan(f*x + e) + 144*b^2*tan(f*x + e))/((tan(f*x + e)^2 + 1)^3*a^5))/f

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Mupad [B]
time = 10.11, size = 2500, normalized size = 7.10 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(e + f*x)^6/(a + b/cos(e + f*x)^2)^3,x)

[Out]

((tan(e + f*x)*(156*a*b^4 + 3*a^4*b + 11*a^5 + 80*b^5 + 65*a^2*b^3 - 9*a^3*b^2))/(16*a^5*(a + b)) + (tan(e + f
*x)^7*(876*a*b^5 + 15*a^5*b + 480*b^6 + 368*a^2*b^4 - 5*a^3*b^3 + 11*a^4*b^2))/(24*a^5*(a + b)^2) + (tan(e + f
*x)^3*(1236*a*b^5 + 41*a^5*b + 20*a^6 + 480*b^6 + 980*a^2*b^4 + 197*a^3*b^3 - 15*a^4*b^2))/(24*a^5*(a + b)^2)
+ (tan(e + f*x)^5*(3168*a*b^5 + 86*a^5*b + 15*a^6 + 1440*b^6 + 1982*a^2*b^4 + 240*a^3*b^3 + 3*a^4*b^2))/(48*a^
5*(a + b)^2) + (b*tan(e + f*x)^9*(116*a*b^4 + 5*a^4*b + 80*b^5 + 17*a^2*b^3 - 8*a^3*b^2))/(16*a^5*(a + b)^2))/
(f*(2*a*b + tan(e + f*x)^6*(8*a*b + a^2 + 10*b^2) + a^2 + b^2 + tan(e + f*x)^8*(2*a*b + 5*b^2) + b^2*tan(e + f
*x)^10 + tan(e + f*x)^2*(8*a*b + 3*a^2 + 5*b^2) + tan(e + f*x)^4*(12*a*b + 3*a^2 + 10*b^2))) - (atan(-((((((20
*a^12*b^9 + 79*a^13*b^8 + (457*a^14*b^7)/4 + (277*a^15*b^6)/4 + (25*a^16*b^5)/2 - 2*a^17*b^4 - (7*a^18*b^3)/4
- (5*a^19*b^2)/4)/(4*a^18*b + a^19 + a^15*b^4 + 4*a^16*b^3 + 6*a^17*b^2) - (tan(e + f*x)*(a*b^2*48i - a^2*b*18
i + a^3*5i - b^3*160i)*(2048*a^12*b^7 + 9216*a^13*b^6 + 16384*a^14*b^5 + 14336*a^15*b^4 + 6144*a^16*b^3 + 1024
*a^17*b^2))/(4096*a^6*(4*a^13*b + a^14 + a^10*b^4 + 4*a^11*b^3 + 6*a^12*b^2)))*(a*b^2*48i - a^2*b*18i + a^3*5i
- b^3*160i))/(32*a^6) - (tan(e + f*x)*(199680*a*b^12 + 51200*b^13 + 287488*a^2*b^11 + 178560*a^3*b^10 + 39240
*a^4*b^9 - 36*a^5*b^8 - 1119*a^6*b^7 - 1092*a^7*b^6 + 234*a^8*b^5 - 80*a^9*b^4 + 25*a^10*b^3))/(128*(4*a^13*b
+ a^14 + a^10*b^4 + 4*a^11*b^3 + 6*a^12*b^2)))*(a*b^2*48i - a^2*b*18i + a^3*5i - b^3*160i)*1i)/(32*a^6) - ((((
(20*a^12*b^9 + 79*a^13*b^8 + (457*a^14*b^7)/4 + (277*a^15*b^6)/4 + (25*a^16*b^5)/2 - 2*a^17*b^4 - (7*a^18*b^3)
/4 - (5*a^19*b^2)/4)/(4*a^18*b + a^19 + a^15*b^4 + 4*a^16*b^3 + 6*a^17*b^2) + (tan(e + f*x)*(a*b^2*48i - a^2*b
*18i + a^3*5i - b^3*160i)*(2048*a^12*b^7 + 9216*a^13*b^6 + 16384*a^14*b^5 + 14336*a^15*b^4 + 6144*a^16*b^3 + 1
024*a^17*b^2))/(4096*a^6*(4*a^13*b + a^14 + a^10*b^4 + 4*a^11*b^3 + 6*a^12*b^2)))*(a*b^2*48i - a^2*b*18i + a^3
*5i - b^3*160i))/(32*a^6) + (tan(e + f*x)*(199680*a*b^12 + 51200*b^13 + 287488*a^2*b^11 + 178560*a^3*b^10 + 39
240*a^4*b^9 - 36*a^5*b^8 - 1119*a^6*b^7 - 1092*a^7*b^6 + 234*a^8*b^5 - 80*a^9*b^4 + 25*a^10*b^3))/(128*(4*a^13
*b + a^14 + a^10*b^4 + 4*a^11*b^3 + 6*a^12*b^2)))*(a*b^2*48i - a^2*b*18i + a^3*5i - b^3*160i)*1i)/(32*a^6))/((
3350*a*b^14 + 1000*b^15 + (7315*a^2*b^13)/2 + (4597*a^3*b^12)/4 - (4325*a^4*b^11)/32 + (10281*a^5*b^10)/128 +
(8973*a^6*b^9)/512 - (25551*a^7*b^8)/1024 + (4235*a^8*b^7)/512 - (2475*a^9*b^6)/1024)/(4*a^18*b + a^19 + a^15*
b^4 + 4*a^16*b^3 + 6*a^17*b^2) + (((((20*a^12*b^9 + 79*a^13*b^8 + (457*a^14*b^7)/4 + (277*a^15*b^6)/4 + (25*a^
16*b^5)/2 - 2*a^17*b^4 - (7*a^18*b^3)/4 - (5*a^19*b^2)/4)/(4*a^18*b + a^19 + a^15*b^4 + 4*a^16*b^3 + 6*a^17*b^
2) - (tan(e + f*x)*(a*b^2*48i - a^2*b*18i + a^3*5i - b^3*160i)*(2048*a^12*b^7 + 9216*a^13*b^6 + 16384*a^14*b^5
+ 14336*a^15*b^4 + 6144*a^16*b^3 + 1024*a^17*b^2))/(4096*a^6*(4*a^13*b + a^14 + a^10*b^4 + 4*a^11*b^3 + 6*a^1
2*b^2)))*(a*b^2*48i - a^2*b*18i + a^3*5i - b^3*160i))/(32*a^6) - (tan(e + f*x)*(199680*a*b^12 + 51200*b^13 + 2
87488*a^2*b^11 + 178560*a^3*b^10 + 39240*a^4*b^9 - 36*a^5*b^8 - 1119*a^6*b^7 - 1092*a^7*b^6 + 234*a^8*b^5 - 80
*a^9*b^4 + 25*a^10*b^3))/(128*(4*a^13*b + a^14 + a^10*b^4 + 4*a^11*b^3 + 6*a^12*b^2)))*(a*b^2*48i - a^2*b*18i
+ a^3*5i - b^3*160i))/(32*a^6) + (((((20*a^12*b^9 + 79*a^13*b^8 + (457*a^14*b^7)/4 + (277*a^15*b^6)/4 + (25*a^
16*b^5)/2 - 2*a^17*b^4 - (7*a^18*b^3)/4 - (5*a^19*b^2)/4)/(4*a^18*b + a^19 + a^15*b^4 + 4*a^16*b^3 + 6*a^17*b^
2) + (tan(e + f*x)*(a*b^2*48i - a^2*b*18i + a^3*5i - b^3*160i)*(2048*a^12*b^7 + 9216*a^13*b^6 + 16384*a^14*b^5
+ 14336*a^15*b^4 + 6144*a^16*b^3 + 1024*a^17*b^2))/(4096*a^6*(4*a^13*b + a^14 + a^10*b^4 + 4*a^11*b^3 + 6*a^1
2*b^2)))*(a*b^2*48i - a^2*b*18i + a^3*5i - b^3*160i))/(32*a^6) + (tan(e + f*x)*(199680*a*b^12 + 51200*b^13 + 2
87488*a^2*b^11 + 178560*a^3*b^10 + 39240*a^4*b^9 - 36*a^5*b^8 - 1119*a^6*b^7 - 1092*a^7*b^6 + 234*a^8*b^5 - 80
*a^9*b^4 + 25*a^10*b^3))/(128*(4*a^13*b + a^14 + a^10*b^4 + 4*a^11*b^3 + 6*a^12*b^2)))*(a*b^2*48i - a^2*b*18i
+ a^3*5i - b^3*160i))/(32*a^6)))*(a*b^2*48i - a^2*b*18i + a^3*5i - b^3*160i)*1i)/(16*a^6*f) - (atan((((-b^7*(a
+ b)^5)^(1/2)*((tan(e + f*x)*(199680*a*b^12 + 51200*b^13 + 287488*a^2*b^11 + 178560*a^3*b^10 + 39240*a^4*b^9
- 36*a^5*b^8 - 1119*a^6*b^7 - 1092*a^7*b^6 + 234*a^8*b^5 - 80*a^9*b^4 + 25*a^10*b^3))/(128*(4*a^13*b + a^14 +
a^10*b^4 + 4*a^11*b^3 + 6*a^12*b^2)) + (((20*a^12*b^9 + 79*a^13*b^8 + (457*a^14*b^7)/4 + (277*a^15*b^6)/4 + (2
5*a^16*b^5)/2 - 2*a^17*b^4 - (7*a^18*b^3)/4 - (5*a^19*b^2)/4)/(4*a^18*b + a^19 + a^15*b^4 + 4*a^16*b^3 + 6*a^1
7*b^2) + (tan(e + f*x)*(-b^7*(a + b)^5)^(1/2)*(11*a*b + (99*a^2)/16 + 5*b^2)*(2048*a^12*b^7 + 9216*a^13*b^6 +
16384*a^14*b^5 + 14336*a^15*b^4 + 6144*a^16*b^3 + 1024*a^17*b^2))/(128*(4*a^13*b + a^14 + a^10*b^4 + 4*a^11*b^
3 + 6*a^12*b^2)*(5*a^10*b + a^11 + a^6*b^5 + 5*a^7*b^4 + 10*a^8*b^3 + 10*a^9*b^2)))*(-b^7*(a + b)^5)^(1/2)*(11
*a*b + (99*a^2)/16 + 5*b^2))/(5*a^10*b + a^11 +...

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