Integrand size = 23, antiderivative size = 72 \[ \int \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {1}{8} \left (3 a^2+8 a b+8 b^2\right ) x+\frac {a (3 a+8 b) \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a^2 \cos ^3(e+f x) \sin (e+f x)}{4 f} \] Output:
1/8*(3*a^2+8*a*b+8*b^2)*x+1/8*a*(3*a+8*b)*cos(f*x+e)*sin(f*x+e)/f+1/4*a^2* cos(f*x+e)^3*sin(f*x+e)/f
Time = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.81 \[ \int \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {4 \left (3 a^2+8 a b+8 b^2\right ) (e+f x)+8 a (a+2 b) \sin (2 (e+f x))+a^2 \sin (4 (e+f x))}{32 f} \] Input:
Integrate[Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2,x]
Output:
(4*(3*a^2 + 8*a*b + 8*b^2)*(e + f*x) + 8*a*(a + 2*b)*Sin[2*(e + f*x)] + a^ 2*Sin[4*(e + f*x)])/(32*f)
Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.39, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 4634, 315, 298, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \sec (e+f x)^2\right )^2}{\sec (e+f x)^4}dx\) |
\(\Big \downarrow \) 4634 |
\(\displaystyle \frac {\int \frac {\left (b \tan ^2(e+f x)+a+b\right )^2}{\left (\tan ^2(e+f x)+1\right )^3}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 315 |
\(\displaystyle \frac {\frac {1}{4} \int \frac {b (a+4 b) \tan ^2(e+f x)+(a+b) (3 a+4 b)}{\left (\tan ^2(e+f x)+1\right )^2}d\tan (e+f x)+\frac {a \tan (e+f x) \left (a+b \tan ^2(e+f x)+b\right )}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
\(\Big \downarrow \) 298 |
\(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (3 a^2+8 a b+8 b^2\right ) \int \frac {1}{\tan ^2(e+f x)+1}d\tan (e+f x)+\frac {3 a (a+2 b) \tan (e+f x)}{2 \left (\tan ^2(e+f x)+1\right )}\right )+\frac {a \tan (e+f x) \left (a+b \tan ^2(e+f x)+b\right )}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (3 a^2+8 a b+8 b^2\right ) \arctan (\tan (e+f x))+\frac {3 a (a+2 b) \tan (e+f x)}{2 \left (\tan ^2(e+f x)+1\right )}\right )+\frac {a \tan (e+f x) \left (a+b \tan ^2(e+f x)+b\right )}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
Input:
Int[Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2,x]
Output:
((a*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2))/(4*(1 + Tan[e + f*x]^2)^2) + (((3*a^2 + 8*a*b + 8*b^2)*ArcTan[Tan[e + f*x]])/2 + (3*a*(a + 2*b)*Tan[e + f*x])/(2*(1 + Tan[e + f*x]^2)))/4)/f
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_) )^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[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]
Time = 0.95 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.75
method | result | size |
parallelrisch | \(\frac {8 a \left (a +2 b \right ) \sin \left (2 f x +2 e \right )+a^{2} \sin \left (4 f x +4 e \right )+12 x \left (a^{2}+\frac {8}{3} a b +\frac {8}{3} b^{2}\right ) f}{32 f}\) | \(54\) |
risch | \(\frac {3 a^{2} x}{8}+a x b +x \,b^{2}+\frac {a^{2} \sin \left (4 f x +4 e \right )}{32 f}+\frac {\sin \left (2 f x +2 e \right ) a^{2}}{4 f}+\frac {\sin \left (2 f x +2 e \right ) a b}{2 f}\) | \(67\) |
derivativedivides | \(\frac {a^{2} \left (\frac {\left (\cos \left (f x +e \right )^{3}+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+2 a b \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+b^{2} \left (f x +e \right )}{f}\) | \(78\) |
default | \(\frac {a^{2} \left (\frac {\left (\cos \left (f x +e \right )^{3}+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+2 a b \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+b^{2} \left (f x +e \right )}{f}\) | \(78\) |
norman | \(\frac {\left (-\frac {3}{8} a^{2}-a b -b^{2}\right ) x +\left (-\frac {9}{8} a^{2}-3 a b -3 b^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+\left (-\frac {9}{8} a^{2}-3 a b -3 b^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}+\left (-\frac {3}{8} a^{2}-a b -b^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (\frac {3}{8} a^{2}+a b +b^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}+\left (\frac {3}{8} a^{2}+a b +b^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}+\left (\frac {9}{8} a^{2}+3 a b +3 b^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (\frac {9}{8} a^{2}+3 a b +3 b^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+\frac {a \left (7 a -8 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{f}-\frac {a \left (5 a +8 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {a \left (5 a +8 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{4 f}+\frac {a \left (9 a +8 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{2 f}+\frac {a \left (9 a +8 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{2 f}-\frac {a \left (27 a -8 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{4 f}-\frac {a \left (27 a -8 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{4 f}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{3}}\) | \(405\) |
Input:
int(cos(f*x+e)^4*(a+b*sec(f*x+e)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/32*(8*a*(a+2*b)*sin(2*f*x+2*e)+a^2*sin(4*f*x+4*e)+12*x*(a^2+8/3*a*b+8/3* b^2)*f)/f
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.86 \[ \int \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} f x + {\left (2 \, a^{2} \cos \left (f x + e\right )^{3} + {\left (3 \, a^{2} + 8 \, a b\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, f} \] Input:
integrate(cos(f*x+e)^4*(a+b*sec(f*x+e)^2)^2,x, algorithm="fricas")
Output:
1/8*((3*a^2 + 8*a*b + 8*b^2)*f*x + (2*a^2*cos(f*x + e)^3 + (3*a^2 + 8*a*b) *cos(f*x + e))*sin(f*x + e))/f
\[ \int \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \cos ^{4}{\left (e + f x \right )}\, dx \] Input:
integrate(cos(f*x+e)**4*(a+b*sec(f*x+e)**2)**2,x)
Output:
Integral((a + b*sec(e + f*x)**2)**2*cos(e + f*x)**4, x)
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.21 \[ \int \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} {\left (f x + e\right )} + \frac {{\left (3 \, a^{2} + 8 \, a b\right )} \tan \left (f x + e\right )^{3} + {\left (5 \, a^{2} + 8 \, a b\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}}{8 \, f} \] Input:
integrate(cos(f*x+e)^4*(a+b*sec(f*x+e)^2)^2,x, algorithm="maxima")
Output:
1/8*((3*a^2 + 8*a*b + 8*b^2)*(f*x + e) + ((3*a^2 + 8*a*b)*tan(f*x + e)^3 + (5*a^2 + 8*a*b)*tan(f*x + e))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1))/f
Time = 0.14 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.21 \[ \int \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} {\left (f x + e\right )} + \frac {3 \, a^{2} \tan \left (f x + e\right )^{3} + 8 \, a b \tan \left (f x + e\right )^{3} + 5 \, a^{2} \tan \left (f x + e\right ) + 8 \, a b \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{2}}}{8 \, f} \] Input:
integrate(cos(f*x+e)^4*(a+b*sec(f*x+e)^2)^2,x, algorithm="giac")
Output:
1/8*((3*a^2 + 8*a*b + 8*b^2)*(f*x + e) + (3*a^2*tan(f*x + e)^3 + 8*a*b*tan (f*x + e)^3 + 5*a^2*tan(f*x + e) + 8*a*b*tan(f*x + e))/(tan(f*x + e)^2 + 1 )^2)/f
Time = 15.83 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.06 \[ \int \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=x\,\left (\frac {3\,a^2}{8}+a\,b+b^2\right )+\frac {\left (\frac {3\,a^2}{8}+b\,a\right )\,{\mathrm {tan}\left (e+f\,x\right )}^3+\left (\frac {5\,a^2}{8}+b\,a\right )\,\mathrm {tan}\left (e+f\,x\right )}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^4+2\,{\mathrm {tan}\left (e+f\,x\right )}^2+1\right )} \] Input:
int(cos(e + f*x)^4*(a + b/cos(e + f*x)^2)^2,x)
Output:
x*(a*b + (3*a^2)/8 + b^2) + (tan(e + f*x)*(a*b + (5*a^2)/8) + tan(e + f*x) ^3*(a*b + (3*a^2)/8))/(f*(2*tan(e + f*x)^2 + tan(e + f*x)^4 + 1))
Time = 0.15 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.08 \[ \int \cos ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {-2 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a^{2}+5 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a^{2}+8 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a b +3 a^{2} f x +8 a b f x +8 b^{2} f x}{8 f} \] Input:
int(cos(f*x+e)^4*(a+b*sec(f*x+e)^2)^2,x)
Output:
( - 2*cos(e + f*x)*sin(e + f*x)**3*a**2 + 5*cos(e + f*x)*sin(e + f*x)*a**2 + 8*cos(e + f*x)*sin(e + f*x)*a*b + 3*a**2*f*x + 8*a*b*f*x + 8*b**2*f*x)/ (8*f)