3.217 \(\int \frac{\cos ^4(e+f x)}{(a+b \sec ^2(e+f x))^3} \, dx\)

Optimal. Leaf size=269 \[ -\frac{3 b^{5/2} \left (21 a^2+36 a b+16 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{8 a^5 f (a+b)^{5/2}}+\frac{3 b (a+2 b) \left (a^2-4 a b-4 b^2\right ) \tan (e+f x)}{8 a^4 f (a+b)^2 \left (a+b \tan ^2(e+f x)+b\right )}+\frac{b \left (3 a^2-7 a b-12 b^2\right ) \tan (e+f x)}{8 a^3 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac{3 x \left (a^2-4 a b+16 b^2\right )}{8 a^5}+\frac{(3 a-8 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left (a+b \tan ^2(e+f x)+b\right )^2} \]

[Out]

(3*(a^2 - 4*a*b + 16*b^2)*x)/(8*a^5) - (3*b^(5/2)*(21*a^2 + 36*a*b + 16*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqr
t[a + b]])/(8*a^5*(a + b)^(5/2)*f) + ((3*a - 8*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^
2)^2) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(3*a^2 - 7*a*b - 12*b^2)*Tan[e
+ f*x])/(8*a^3*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + (3*b*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*Tan[e + f*x])/(
8*a^4*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))

________________________________________________________________________________________

Rubi [A]  time = 0.376786, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {4146, 414, 527, 522, 203, 205} \[ -\frac{3 b^{5/2} \left (21 a^2+36 a b+16 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{8 a^5 f (a+b)^{5/2}}+\frac{3 b (a+2 b) \left (a^2-4 a b-4 b^2\right ) \tan (e+f x)}{8 a^4 f (a+b)^2 \left (a+b \tan ^2(e+f x)+b\right )}+\frac{b \left (3 a^2-7 a b-12 b^2\right ) \tan (e+f x)}{8 a^3 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac{3 x \left (a^2-4 a b+16 b^2\right )}{8 a^5}+\frac{(3 a-8 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f \left (a+b \tan ^2(e+f x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a^2 - 4*a*b + 16*b^2)*x)/(8*a^5) - (3*b^(5/2)*(21*a^2 + 36*a*b + 16*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqr
t[a + b]])/(8*a^5*(a + b)^(5/2)*f) + ((3*a - 8*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^
2)^2) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(3*a^2 - 7*a*b - 12*b^2)*Tan[e
+ f*x])/(8*a^3*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + (3*b*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*Tan[e + f*x])/(
8*a^4*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))

Rule 4146

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]

Rule 414

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]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

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 522

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 203

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

Rule 205

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

Rubi steps

\begin{align*} \int \frac{\cos ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^3 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\cos ^3(e+f x) \sin (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-3 a+b-7 b x^2}{\left (1+x^2\right )^2 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{4 a f}\\ &=\frac{(3 a-8 b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{\cos ^3(e+f x) \sin (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{3 a^2+3 a b+8 b^2+5 (3 a-8 b) b x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{8 a^2 f}\\ &=\frac{(3 a-8 b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{\cos ^3(e+f x) \sin (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{b \left (3 a^2-7 a b-12 b^2\right ) \tan (e+f x)}{8 a^3 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{12 \left (a^3+5 a b^2+4 b^3\right )+12 b \left (3 a^2-7 a b-12 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{32 a^3 (a+b) f}\\ &=\frac{(3 a-8 b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{\cos ^3(e+f x) \sin (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{b \left (3 a^2-7 a b-12 b^2\right ) \tan (e+f x)}{8 a^3 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{3 b (a+2 b) \left (a^2-4 a b-4 b^2\right ) \tan (e+f x)}{8 a^4 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{24 \left (a^4-a^3 b+7 a^2 b^2+16 a b^3+8 b^4\right )+24 b (a+2 b) \left (a^2-4 a b-4 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{64 a^4 (a+b)^2 f}\\ &=\frac{(3 a-8 b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{\cos ^3(e+f x) \sin (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{b \left (3 a^2-7 a b-12 b^2\right ) \tan (e+f x)}{8 a^3 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{3 b (a+2 b) \left (a^2-4 a b-4 b^2\right ) \tan (e+f x)}{8 a^4 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac{\left (3 \left (a^2-4 a b+16 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{8 a^5 f}-\frac{\left (3 b^3 \left (21 a^2+36 a b+16 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a^5 (a+b)^2 f}\\ &=\frac{3 \left (a^2-4 a b+16 b^2\right ) x}{8 a^5}-\frac{3 b^{5/2} \left (21 a^2+36 a b+16 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{8 a^5 (a+b)^{5/2} f}+\frac{(3 a-8 b) \cos (e+f x) \sin (e+f x)}{8 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{\cos ^3(e+f x) \sin (e+f x)}{4 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{b \left (3 a^2-7 a b-12 b^2\right ) \tan (e+f x)}{8 a^3 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac{3 b (a+2 b) \left (a^2-4 a b-4 b^2\right ) \tan (e+f x)}{8 a^4 (a+b)^2 f \left (a+b+b \tan ^2(e+f x)\right )}\\ \end{align*}

Mathematica [C]  time = 6.57259, size = 1430, normalized size = 5.32 \[ \frac{\left (21 a^2+36 b a+16 b^2\right ) (\cos (2 e+2 f x) a+a+2 b)^3 \left (\frac{3 b^3 \tan ^{-1}\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^5 \sqrt{a+b} f \sqrt{b \cos (4 e)-i b \sin (4 e)}}-\frac{3 i b^3 \tan ^{-1}\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^5 \sqrt{a+b} f \sqrt{b \cos (4 e)-i b \sin (4 e)}}\right ) \sec ^6(e+f x)}{(a+b)^2 \left (b \sec ^2(e+f x)+a\right )^3}+\frac{(\cos (2 e+2 f x) a+a+2 b) \sec (2 e) \left (144 f x \cos (2 e) a^6+96 f x \cos (2 f x) a^6+96 f x \cos (4 e+2 f x) a^6+24 f x \cos (2 e+4 f x) a^6+24 f x \cos (6 e+4 f x) a^6+44 \sin (2 f x) a^6+44 \sin (4 e+2 f x) a^6+38 \sin (2 e+4 f x) a^6+38 \sin (6 e+4 f x) a^6+12 \sin (4 e+6 f x) a^6+12 \sin (8 e+6 f x) a^6+\sin (6 e+8 f x) a^6+\sin (10 e+8 f x) a^6+96 b f x \cos (2 e) a^5-48 b f x \cos (2 e+4 f x) a^5-48 b f x \cos (6 e+4 f x) a^5+104 b \sin (2 f x) a^5+104 b \sin (4 e+2 f x) a^5+60 b \sin (2 e+4 f x) a^5+60 b \sin (6 e+4 f x) a^5+8 b \sin (4 e+6 f x) a^5+8 b \sin (8 e+6 f x) a^5+2 b \sin (6 e+8 f x) a^5+2 b \sin (10 e+8 f x) a^5+912 b^2 f x \cos (2 e) a^4+480 b^2 f x \cos (2 f x) a^4+480 b^2 f x \cos (4 e+2 f x) a^4+216 b^2 f x \cos (2 e+4 f x) a^4+216 b^2 f x \cos (6 e+4 f x) a^4-180 b^2 \sin (2 f x) a^4-180 b^2 \sin (4 e+2 f x) a^4-170 b^2 \sin (2 e+4 f x) a^4-170 b^2 \sin (6 e+4 f x) a^4-20 b^2 \sin (4 e+6 f x) a^4-20 b^2 \sin (8 e+6 f x) a^4+b^2 \sin (6 e+8 f x) a^4+b^2 \sin (10 e+8 f x) a^4+6720 b^3 f x \cos (2 e) a^3+4416 b^3 f x \cos (2 f x) a^3+4416 b^3 f x \cos (4 e+2 f x) a^3+672 b^3 f x \cos (2 e+4 f x) a^3+672 b^3 f x \cos (6 e+4 f x) a^3+816 b^3 \sin (2 e) a^3-1696 b^3 \sin (2 f x) a^3-608 b^3 \sin (4 e+2 f x) a^3-640 b^3 \sin (2 e+4 f x) a^3-368 b^3 \sin (6 e+4 f x) a^3-16 b^3 \sin (4 e+6 f x) a^3-16 b^3 \sin (8 e+6 f x) a^3+16512 b^4 f x \cos (2 e) a^2+6912 b^4 f x \cos (2 f x) a^2+6912 b^4 f x \cos (4 e+2 f x) a^2+384 b^4 f x \cos (2 e+4 f x) a^2+384 b^4 f x \cos (6 e+4 f x) a^2+2848 b^4 \sin (2 e) a^2-3264 b^4 \sin (2 f x) a^2-192 b^4 \sin (4 e+2 f x) a^2-400 b^4 \sin (2 e+4 f x) a^2-176 b^4 \sin (6 e+4 f x) a^2+16896 b^5 f x \cos (2 e) a+3072 b^5 f x \cos (2 f x) a+3072 b^5 f x \cos (4 e+2 f x) a+3968 b^5 \sin (2 e) a-1664 b^5 \sin (2 f x) a+128 b^5 \sin (4 e+2 f x) a+6144 b^6 f x \cos (2 e)+1792 b^6 \sin (2 e)\right ) \sec ^6(e+f x)}{2048 a^5 (a+b)^2 f \left (b \sec ^2(e+f x)+a\right )^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((21*a^2 + 36*a*b + 16*b^2)*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((3*b^3*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])/(64*a^5*Sqrt[a + b]*f*Sqrt[b*Cos[4*e] - I*b*S
in[4*e]]) - (((3*I)/64)*b^3*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])]*S
in[2*e])/(a^5*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*(144*a^6*f*x*Cos[2*e] + 96*a^5*b*f*x*Cos[2*e] + 912*a^4*b^2*f*
x*Cos[2*e] + 6720*a^3*b^3*f*x*Cos[2*e] + 16512*a^2*b^4*f*x*Cos[2*e] + 16896*a*b^5*f*x*Cos[2*e] + 6144*b^6*f*x*
Cos[2*e] + 96*a^6*f*x*Cos[2*f*x] + 480*a^4*b^2*f*x*Cos[2*f*x] + 4416*a^3*b^3*f*x*Cos[2*f*x] + 6912*a^2*b^4*f*x
*Cos[2*f*x] + 3072*a*b^5*f*x*Cos[2*f*x] + 96*a^6*f*x*Cos[4*e + 2*f*x] + 480*a^4*b^2*f*x*Cos[4*e + 2*f*x] + 441
6*a^3*b^3*f*x*Cos[4*e + 2*f*x] + 6912*a^2*b^4*f*x*Cos[4*e + 2*f*x] + 3072*a*b^5*f*x*Cos[4*e + 2*f*x] + 24*a^6*
f*x*Cos[2*e + 4*f*x] - 48*a^5*b*f*x*Cos[2*e + 4*f*x] + 216*a^4*b^2*f*x*Cos[2*e + 4*f*x] + 672*a^3*b^3*f*x*Cos[
2*e + 4*f*x] + 384*a^2*b^4*f*x*Cos[2*e + 4*f*x] + 24*a^6*f*x*Cos[6*e + 4*f*x] - 48*a^5*b*f*x*Cos[6*e + 4*f*x]
+ 216*a^4*b^2*f*x*Cos[6*e + 4*f*x] + 672*a^3*b^3*f*x*Cos[6*e + 4*f*x] + 384*a^2*b^4*f*x*Cos[6*e + 4*f*x] + 816
*a^3*b^3*Sin[2*e] + 2848*a^2*b^4*Sin[2*e] + 3968*a*b^5*Sin[2*e] + 1792*b^6*Sin[2*e] + 44*a^6*Sin[2*f*x] + 104*
a^5*b*Sin[2*f*x] - 180*a^4*b^2*Sin[2*f*x] - 1696*a^3*b^3*Sin[2*f*x] - 3264*a^2*b^4*Sin[2*f*x] - 1664*a*b^5*Sin
[2*f*x] + 44*a^6*Sin[4*e + 2*f*x] + 104*a^5*b*Sin[4*e + 2*f*x] - 180*a^4*b^2*Sin[4*e + 2*f*x] - 608*a^3*b^3*Si
n[4*e + 2*f*x] - 192*a^2*b^4*Sin[4*e + 2*f*x] + 128*a*b^5*Sin[4*e + 2*f*x] + 38*a^6*Sin[2*e + 4*f*x] + 60*a^5*
b*Sin[2*e + 4*f*x] - 170*a^4*b^2*Sin[2*e + 4*f*x] - 640*a^3*b^3*Sin[2*e + 4*f*x] - 400*a^2*b^4*Sin[2*e + 4*f*x
] + 38*a^6*Sin[6*e + 4*f*x] + 60*a^5*b*Sin[6*e + 4*f*x] - 170*a^4*b^2*Sin[6*e + 4*f*x] - 368*a^3*b^3*Sin[6*e +
4*f*x] - 176*a^2*b^4*Sin[6*e + 4*f*x] + 12*a^6*Sin[4*e + 6*f*x] + 8*a^5*b*Sin[4*e + 6*f*x] - 20*a^4*b^2*Sin[4
*e + 6*f*x] - 16*a^3*b^3*Sin[4*e + 6*f*x] + 12*a^6*Sin[8*e + 6*f*x] + 8*a^5*b*Sin[8*e + 6*f*x] - 20*a^4*b^2*Si
n[8*e + 6*f*x] - 16*a^3*b^3*Sin[8*e + 6*f*x] + a^6*Sin[6*e + 8*f*x] + 2*a^5*b*Sin[6*e + 8*f*x] + a^4*b^2*Sin[6
*e + 8*f*x] + a^6*Sin[10*e + 8*f*x] + 2*a^5*b*Sin[10*e + 8*f*x] + a^4*b^2*Sin[10*e + 8*f*x]))/(2048*a^5*(a + b
)^2*f*(a + b*Sec[e + f*x]^2)^3)

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Maple [A]  time = 0.126, size = 470, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8/f/a^3/(tan(f*x+e)^2+1)^2*tan(f*x+e)^3-3/2/f/a^4/(tan(f*x+e)^2+1)^2*tan(f*x+e)^3*b-3/2/f/a^4/(tan(f*x+e)^2+
1)^2*tan(f*x+e)*b+5/8/f/a^3/(tan(f*x+e)^2+1)^2*tan(f*x+e)+6/f/a^5*arctan(tan(f*x+e))*b^2+3/8/f/a^3*arctan(tan(
f*x+e))-3/2/f/a^4*arctan(tan(f*x+e))*b-15/8/f*b^4/a^3/(a+b+b*tan(f*x+e)^2)^2/(a^2+2*a*b+b^2)*tan(f*x+e)^3-3/2/
f*b^5/a^4/(a+b+b*tan(f*x+e)^2)^2/(a^2+2*a*b+b^2)*tan(f*x+e)^3-17/8/f*b^3/a^3/(a+b+b*tan(f*x+e)^2)^2/(a+b)*tan(
f*x+e)-3/2/f*b^4/a^4/(a+b+b*tan(f*x+e)^2)^2/(a+b)*tan(f*x+e)-63/8/f/a^3*b^3/(a^2+2*a*b+b^2)/((a+b)*b)^(1/2)*ar
ctan(tan(f*x+e)*b/((a+b)*b)^(1/2))-27/2/f*b^4/a^4/(a^2+2*a*b+b^2)/((a+b)*b)^(1/2)*arctan(tan(f*x+e)*b/((a+b)*b
)^(1/2))-6/f*b^5/a^5/(a^2+2*a*b+b^2)/((a+b)*b)^(1/2)*arctan(tan(f*x+e)*b/((a+b)*b)^(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)^4/(a+b*sec(f*x+e)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.880947, size = 2531, normalized size = 9.41 \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)^4/(a+b*sec(f*x+e)^2)^3,x, algorithm="fricas")

[Out]

[1/32*(12*(a^6 - 2*a^5*b + 9*a^4*b^2 + 28*a^3*b^3 + 16*a^2*b^4)*f*x*cos(f*x + e)^4 + 24*(a^5*b - 2*a^4*b^2 + 9
*a^3*b^3 + 28*a^2*b^4 + 16*a*b^5)*f*x*cos(f*x + e)^2 + 12*(a^4*b^2 - 2*a^3*b^3 + 9*a^2*b^4 + 28*a*b^5 + 16*b^6
)*f*x + 3*(21*a^2*b^4 + 36*a*b^5 + 16*b^6 + (21*a^4*b^2 + 36*a^3*b^3 + 16*a^2*b^4)*cos(f*x + e)^4 + 2*(21*a^3*
b^3 + 36*a^2*b^4 + 16*a*b^5)*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*cos(f*x + e)^2 + b^2)) + 4*(2*(a^6 + 2*a^5*b + a^4*b^2)*
cos(f*x + e)^7 + (3*a^6 - 2*a^5*b - 13*a^4*b^2 - 8*a^3*b^3)*cos(f*x + e)^5 + (6*a^5*b - 10*a^4*b^2 - 55*a^3*b^
3 - 36*a^2*b^4)*cos(f*x + e)^3 + 3*(a^4*b^2 - 2*a^3*b^3 - 12*a^2*b^4 - 8*a*b^5)*cos(f*x + e))*sin(f*x + e))/((
a^9 + 2*a^8*b + a^7*b^2)*f*cos(f*x + e)^4 + 2*(a^8*b + 2*a^7*b^2 + a^6*b^3)*f*cos(f*x + e)^2 + (a^7*b^2 + 2*a^
6*b^3 + a^5*b^4)*f), 1/16*(6*(a^6 - 2*a^5*b + 9*a^4*b^2 + 28*a^3*b^3 + 16*a^2*b^4)*f*x*cos(f*x + e)^4 + 12*(a^
5*b - 2*a^4*b^2 + 9*a^3*b^3 + 28*a^2*b^4 + 16*a*b^5)*f*x*cos(f*x + e)^2 + 6*(a^4*b^2 - 2*a^3*b^3 + 9*a^2*b^4 +
28*a*b^5 + 16*b^6)*f*x + 3*(21*a^2*b^4 + 36*a*b^5 + 16*b^6 + (21*a^4*b^2 + 36*a^3*b^3 + 16*a^2*b^4)*cos(f*x +
e)^4 + 2*(21*a^3*b^3 + 36*a^2*b^4 + 16*a*b^5)*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))) + 2*(2*(a^6 + 2*a^5*b + a^4*b^2)*cos(f*x + e)^7 + (3
*a^6 - 2*a^5*b - 13*a^4*b^2 - 8*a^3*b^3)*cos(f*x + e)^5 + (6*a^5*b - 10*a^4*b^2 - 55*a^3*b^3 - 36*a^2*b^4)*cos
(f*x + e)^3 + 3*(a^4*b^2 - 2*a^3*b^3 - 12*a^2*b^4 - 8*a*b^5)*cos(f*x + e))*sin(f*x + e))/((a^9 + 2*a^8*b + a^7
*b^2)*f*cos(f*x + e)^4 + 2*(a^8*b + 2*a^7*b^2 + a^6*b^3)*f*cos(f*x + e)^2 + (a^7*b^2 + 2*a^6*b^3 + a^5*b^4)*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)**4/(a+b*sec(f*x+e)**2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.37784, size = 663, normalized size = 2.46 \begin{align*} -\frac{\frac{3 \,{\left (21 \, a^{2} b^{3} + 36 \, a b^{4} + 16 \, b^{5}\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^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} \sqrt{a b + b^{2}}} - \frac{3 \, a^{3} b^{2} \tan \left (f x + e\right )^{7} - 6 \, a^{2} b^{3} \tan \left (f x + e\right )^{7} - 36 \, a b^{4} \tan \left (f x + e\right )^{7} - 24 \, b^{5} \tan \left (f x + e\right )^{7} + 6 \, a^{4} b \tan \left (f x + e\right )^{5} - a^{3} b^{2} \tan \left (f x + e\right )^{5} - 73 \, a^{2} b^{3} \tan \left (f x + e\right )^{5} - 144 \, a b^{4} \tan \left (f x + e\right )^{5} - 72 \, b^{5} \tan \left (f x + e\right )^{5} + 3 \, a^{5} \tan \left (f x + e\right )^{3} + 10 \, a^{4} b \tan \left (f x + e\right )^{3} - 24 \, a^{3} b^{2} \tan \left (f x + e\right )^{3} - 136 \, a^{2} b^{3} \tan \left (f x + e\right )^{3} - 180 \, a b^{4} \tan \left (f x + e\right )^{3} - 72 \, b^{5} \tan \left (f x + e\right )^{3} + 5 \, a^{5} \tan \left (f x + e\right ) + 8 \, a^{4} b \tan \left (f x + e\right ) - 18 \, a^{3} b^{2} \tan \left (f x + e\right ) - 69 \, a^{2} b^{3} \tan \left (f x + e\right ) - 72 \, a b^{4} \tan \left (f x + e\right ) - 24 \, b^{5} \tan \left (f x + e\right )}{{\left (a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right )}{\left (b \tan \left (f x + e\right )^{4} + a \tan \left (f x + e\right )^{2} + 2 \, b \tan \left (f x + e\right )^{2} + a + b\right )}^{2}} - \frac{3 \,{\left (a^{2} - 4 \, a b + 16 \, b^{2}\right )}{\left (f x + e\right )}}{a^{5}}}{8 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^4/(a+b*sec(f*x+e)^2)^3,x, algorithm="giac")

[Out]

-1/8*(3*(21*a^2*b^3 + 36*a*b^4 + 16*b^5)*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b
+ b^2)))/((a^7 + 2*a^6*b + a^5*b^2)*sqrt(a*b + b^2)) - (3*a^3*b^2*tan(f*x + e)^7 - 6*a^2*b^3*tan(f*x + e)^7 -
36*a*b^4*tan(f*x + e)^7 - 24*b^5*tan(f*x + e)^7 + 6*a^4*b*tan(f*x + e)^5 - a^3*b^2*tan(f*x + e)^5 - 73*a^2*b^
3*tan(f*x + e)^5 - 144*a*b^4*tan(f*x + e)^5 - 72*b^5*tan(f*x + e)^5 + 3*a^5*tan(f*x + e)^3 + 10*a^4*b*tan(f*x
+ e)^3 - 24*a^3*b^2*tan(f*x + e)^3 - 136*a^2*b^3*tan(f*x + e)^3 - 180*a*b^4*tan(f*x + e)^3 - 72*b^5*tan(f*x +
e)^3 + 5*a^5*tan(f*x + e) + 8*a^4*b*tan(f*x + e) - 18*a^3*b^2*tan(f*x + e) - 69*a^2*b^3*tan(f*x + e) - 72*a*b^
4*tan(f*x + e) - 24*b^5*tan(f*x + e))/((a^6 + 2*a^5*b + a^4*b^2)*(b*tan(f*x + e)^4 + a*tan(f*x + e)^2 + 2*b*ta
n(f*x + e)^2 + a + b)^2) - 3*(a^2 - 4*a*b + 16*b^2)*(f*x + e)/a^5)/f