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

Optimal. Leaf size=117 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{a^3 f \sqrt{a+b}}+\frac{x \left (3 a^2-4 a b+8 b^2\right )}{8 a^3}+\frac{(3 a-4 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f} \]

[Out]

((3*a^2 - 4*a*b + 8*b^2)*x)/(8*a^3) - (b^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^3*Sqrt[a + b]*f)
+ ((3*a - 4*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f)

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Rubi [A]  time = 0.156664, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{a^3 f \sqrt{a+b}}+\frac{x \left (3 a^2-4 a b+8 b^2\right )}{8 a^3}+\frac{(3 a-4 b) \sin (e+f x) \cos (e+f x)}{8 a^2 f}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*a^2 - 4*a*b + 8*b^2)*x)/(8*a^3) - (b^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^3*Sqrt[a + b]*f)
+ ((3*a - 4*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f)

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

Mathematica [A]  time = 0.463789, size = 95, normalized size = 0.81 \[ \frac{4 \left (3 a^2-4 a b+8 b^2\right ) (e+f x)+a^2 \sin (4 (e+f x))-\frac{32 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{\sqrt{a+b}}+8 a (a-b) \sin (2 (e+f x))}{32 a^3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(3*a^2 - 4*a*b + 8*b^2)*(e + f*x) - (32*b^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/Sqrt[a + b] + 8
*a*(a - b)*Sin[2*(e + f*x)] + a^2*Sin[4*(e + f*x)])/(32*a^3*f)

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Maple [A]  time = 0.097, size = 194, normalized size = 1.7 \begin{align*}{\frac{3\, \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{8\,fa \left ( \left ( \tan \left ( fx+e \right ) \right ) ^{2}+1 \right ) ^{2}}}-{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{3}b}{2\,f{a}^{2} \left ( \left ( \tan \left ( fx+e \right ) \right ) ^{2}+1 \right ) ^{2}}}-{\frac{\tan \left ( fx+e \right ) b}{2\,f{a}^{2} \left ( \left ( \tan \left ( fx+e \right ) \right ) ^{2}+1 \right ) ^{2}}}+{\frac{5\,\tan \left ( fx+e \right ) }{8\,fa \left ( \left ( \tan \left ( fx+e \right ) \right ) ^{2}+1 \right ) ^{2}}}+{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ){b}^{2}}{f{a}^{3}}}+{\frac{3\,\arctan \left ( \tan \left ( fx+e \right ) \right ) }{8\,fa}}-{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) b}{2\,f{a}^{2}}}-{\frac{{b}^{3}}{f{a}^{3}}\arctan \left ({\tan \left ( fx+e \right ) b{\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \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),x)

[Out]

3/8/f/a/(tan(f*x+e)^2+1)^2*tan(f*x+e)^3-1/2/f/a^2/(tan(f*x+e)^2+1)^2*tan(f*x+e)^3*b-1/2/f/a^2/(tan(f*x+e)^2+1)
^2*tan(f*x+e)*b+5/8/f/a/(tan(f*x+e)^2+1)^2*tan(f*x+e)+1/f/a^3*arctan(tan(f*x+e))*b^2+3/8/f/a*arctan(tan(f*x+e)
)-1/2/f/a^2*arctan(tan(f*x+e))*b-1/f*b^3/a^3/((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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.59508, size = 811, normalized size = 6.93 \begin{align*} \left [\frac{2 \, b^{2} \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 ) +{\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} f x +{\left (2 \, a^{2} \cos \left (f x + e\right )^{3} +{\left (3 \, a^{2} - 4 \, a b\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, a^{3} f}, \frac{4 \, b^{2} \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 (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} f x +{\left (2 \, a^{2} \cos \left (f x + e\right )^{3} +{\left (3 \, a^{2} - 4 \, a b\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, a^{3} f}\right ] \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),x, algorithm="fricas")

[Out]

[1/8*(2*b^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*co
s(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + (3*a^2 - 4*a*b + 8*b^2)*f*x + (2*a^2*cos(f*x + e)^3 + (3*a^2 - 4
*a*b)*cos(f*x + e))*sin(f*x + e))/(a^3*f), 1/8*(4*b^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))) + (3*a^2 - 4*a*b + 8*b^2)*f*x + (2*a^2*cos(f*x + e)^3 + (3*a^
2 - 4*a*b)*cos(f*x + e))*sin(f*x + e))/(a^3*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),x)

[Out]

Timed out

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Giac [A]  time = 1.2972, size = 201, normalized size = 1.72 \begin{align*} -\frac{\frac{8 \,{\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 )} b^{3}}{\sqrt{a b + b^{2}} a^{3}} - \frac{{\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )}{\left (f x + e\right )}}{a^{3}} - \frac{3 \, a \tan \left (f x + e\right )^{3} - 4 \, b \tan \left (f x + e\right )^{3} + 5 \, a \tan \left (f x + e\right ) - 4 \, b \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{2} a^{2}}}{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),x, algorithm="giac")

[Out]

-1/8*(8*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b + b^2)))*b^3/(sqrt(a*b + b^2)*a^
3) - (3*a^2 - 4*a*b + 8*b^2)*(f*x + e)/a^3 - (3*a*tan(f*x + e)^3 - 4*b*tan(f*x + e)^3 + 5*a*tan(f*x + e) - 4*b
*tan(f*x + e))/((tan(f*x + e)^2 + 1)^2*a^2))/f