3.5.96 \(\int \frac {\cos ^4(c+d x)}{a+b \sec (c+d x)} \, dx\) [496]

Optimal. Leaf size=193 \[ \frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac {2 b^5 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 \sqrt {a-b} \sqrt {a+b} d}-\frac {b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d} \]

[Out]

1/8*(3*a^4+4*a^2*b^2+8*b^4)*x/a^5-1/3*b*(2*a^2+3*b^2)*sin(d*x+c)/a^4/d+1/8*(3*a^2+4*b^2)*cos(d*x+c)*sin(d*x+c)
/a^3/d-1/3*b*cos(d*x+c)^2*sin(d*x+c)/a^2/d+1/4*cos(d*x+c)^3*sin(d*x+c)/a/d-2*b^5*arctanh((a-b)^(1/2)*tan(1/2*d
*x+1/2*c)/(a+b)^(1/2))/a^5/d/(a-b)^(1/2)/(a+b)^(1/2)

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Rubi [A]
time = 0.48, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3938, 4189, 4004, 3916, 2738, 214} \begin {gather*} -\frac {2 b^5 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d \sqrt {a-b} \sqrt {a+b}}-\frac {b \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}-\frac {b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2+4 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {x \left (3 a^4+4 a^2 b^2+8 b^4\right )}{8 a^5}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4/(a + b*Sec[c + d*x]),x]

[Out]

((3*a^4 + 4*a^2*b^2 + 8*b^4)*x)/(8*a^5) - (2*b^5*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*Sqr
t[a - b]*Sqrt[a + b]*d) - (b*(2*a^2 + 3*b^2)*Sin[c + d*x])/(3*a^4*d) + ((3*a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d
*x])/(8*a^3*d) - (b*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)

Rule 214

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

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3938

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[Cot[e + f*x
]*((d*Csc[e + f*x])^n/(a*f*n)), x] - Dist[1/(a*d*n), Int[((d*Csc[e + f*x])^(n + 1)/(a + b*Csc[e + f*x]))*Simp[
b*n - a*(n + 1)*Csc[e + f*x] - b*(n + 1)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 -
b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 4189

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1
)*((d*Csc[e + f*x])^n/(a*f*n)), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[
a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ
[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]

Rubi steps

\begin {align*} \int \frac {\cos ^4(c+d x)}{a+b \sec (c+d x)} \, dx &=\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\int \frac {\cos ^3(c+d x) \left (-4 b+3 a \sec (c+d x)+3 b \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 a}\\ &=-\frac {b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\int \frac {\cos ^2(c+d x) \left (-3 \left (3 a^2+4 b^2\right )-a b \sec (c+d x)+8 b^2 \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{12 a^2}\\ &=\frac {\left (3 a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\int \frac {\cos (c+d x) \left (-8 b \left (2 a^2+3 b^2\right )+a \left (9 a^2-4 b^2\right ) \sec (c+d x)+3 b \left (3 a^2+4 b^2\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{24 a^3}\\ &=-\frac {b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\int \frac {-3 \left (3 a^4+4 a^2 b^2+8 b^4\right )-3 a b \left (3 a^2+4 b^2\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{24 a^4}\\ &=\frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac {b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {b^5 \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^5}\\ &=\frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac {b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {b^4 \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^5}\\ &=\frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac {b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}\\ &=\frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac {2 b^5 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 \sqrt {a-b} \sqrt {a+b} d}-\frac {b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}\\ \end {align*}

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Mathematica [A]
time = 0.61, size = 153, normalized size = 0.79 \begin {gather*} \frac {12 \left (3 a^4+4 a^2 b^2+8 b^4\right ) (c+d x)+\frac {192 b^5 \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-24 a b \left (3 a^2+4 b^2\right ) \sin (c+d x)+24 a^2 \left (a^2+b^2\right ) \sin (2 (c+d x))-8 a^3 b \sin (3 (c+d x))+3 a^4 \sin (4 (c+d x))}{96 a^5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4/(a + b*Sec[c + d*x]),x]

[Out]

(12*(3*a^4 + 4*a^2*b^2 + 8*b^4)*(c + d*x) + (192*b^5*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqr
t[a^2 - b^2] - 24*a*b*(3*a^2 + 4*b^2)*Sin[c + d*x] + 24*a^2*(a^2 + b^2)*Sin[2*(c + d*x)] - 8*a^3*b*Sin[3*(c +
d*x)] + 3*a^4*Sin[4*(c + d*x)])/(96*a^5*d)

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Maple [A]
time = 0.16, size = 256, normalized size = 1.33

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (\left (-\frac {5}{8} a^{4}-b \,a^{3}-\frac {1}{2} b^{2} a^{2}-b^{3} a \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{8} a^{4}-\frac {5}{3} b \,a^{3}-3 b^{3} a -\frac {1}{2} b^{2} a^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{8} a^{4}+\frac {1}{2} b^{2} a^{2}-\frac {5}{3} b \,a^{3}-3 b^{3} a \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{8} a^{4}+\frac {1}{2} b^{2} a^{2}-b \,a^{3}-b^{3} a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\left (3 a^{4}+4 b^{2} a^{2}+8 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{5}}-\frac {2 b^{5} \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5} \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\)	\(256\)
default	\(\frac {\frac {\frac {2 \left (\left (-\frac {5}{8} a^{4}-b \,a^{3}-\frac {1}{2} b^{2} a^{2}-b^{3} a \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{8} a^{4}-\frac {5}{3} b \,a^{3}-3 b^{3} a -\frac {1}{2} b^{2} a^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{8} a^{4}+\frac {1}{2} b^{2} a^{2}-\frac {5}{3} b \,a^{3}-3 b^{3} a \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{8} a^{4}+\frac {1}{2} b^{2} a^{2}-b \,a^{3}-b^{3} a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\left (3 a^{4}+4 b^{2} a^{2}+8 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{5}}-\frac {2 b^{5} \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5} \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\)	\(256\)
risch	\(\frac {3 x}{8 a}+\frac {x \,b^{2}}{2 a^{3}}+\frac {x \,b^{4}}{a^{5}}+\frac {3 i b \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{2} d}+\frac {i b^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 a^{4} d}-\frac {3 i b \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{2} d}-\frac {i b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{4} d}+\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{5}}-\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{5}}+\frac {\sin \left (4 d x +4 c \right )}{32 a d}-\frac {b \sin \left (3 d x +3 c \right )}{12 a^{2} d}+\frac {\sin \left (2 d x +2 c \right )}{4 a d}+\frac {\sin \left (2 d x +2 c \right ) b^{2}}{4 a^{3} d}\)	\(326\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4/(a+b*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(2/a^5*(((-5/8*a^4-b*a^3-1/2*b^2*a^2-b^3*a)*tan(1/2*d*x+1/2*c)^7+(3/8*a^4-5/3*b*a^3-3*b^3*a-1/2*b^2*a^2)*t
an(1/2*d*x+1/2*c)^5+(-3/8*a^4+1/2*b^2*a^2-5/3*b*a^3-3*b^3*a)*tan(1/2*d*x+1/2*c)^3+(5/8*a^4+1/2*b^2*a^2-b*a^3-b
^3*a)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^4+1/8*(3*a^4+4*a^2*b^2+8*b^4)*arctan(tan(1/2*d*x+1/2*c)))-2
*b^5/a^5/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+b*sec(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

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Fricas [A]
time = 3.58, size = 482, normalized size = 2.50 \begin {gather*} \left [\frac {12 \, \sqrt {a^{2} - b^{2}} b^{5} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + 3 \, {\left (3 \, a^{6} + a^{4} b^{2} + 4 \, a^{2} b^{4} - 8 \, b^{6}\right )} d x - {\left (16 \, a^{5} b + 8 \, a^{3} b^{3} - 24 \, a b^{5} - 6 \, {\left (a^{6} - a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (a^{5} b - a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (3 \, a^{6} + a^{4} b^{2} - 4 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{7} - a^{5} b^{2}\right )} d}, -\frac {24 \, \sqrt {-a^{2} + b^{2}} b^{5} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - 3 \, {\left (3 \, a^{6} + a^{4} b^{2} + 4 \, a^{2} b^{4} - 8 \, b^{6}\right )} d x + {\left (16 \, a^{5} b + 8 \, a^{3} b^{3} - 24 \, a b^{5} - 6 \, {\left (a^{6} - a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (a^{5} b - a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (3 \, a^{6} + a^{4} b^{2} - 4 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{7} - a^{5} b^{2}\right )} d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+b*sec(d*x+c)),x, algorithm="fricas")

[Out]

[1/24*(12*sqrt(a^2 - b^2)*b^5*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*co
s(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + 3*(3*a^6 + a^4*
b^2 + 4*a^2*b^4 - 8*b^6)*d*x - (16*a^5*b + 8*a^3*b^3 - 24*a*b^5 - 6*(a^6 - a^4*b^2)*cos(d*x + c)^3 + 8*(a^5*b
- a^3*b^3)*cos(d*x + c)^2 - 3*(3*a^6 + a^4*b^2 - 4*a^2*b^4)*cos(d*x + c))*sin(d*x + c))/((a^7 - a^5*b^2)*d), -
1/24*(24*sqrt(-a^2 + b^2)*b^5*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - 3*(3
*a^6 + a^4*b^2 + 4*a^2*b^4 - 8*b^6)*d*x + (16*a^5*b + 8*a^3*b^3 - 24*a*b^5 - 6*(a^6 - a^4*b^2)*cos(d*x + c)^3
+ 8*(a^5*b - a^3*b^3)*cos(d*x + c)^2 - 3*(3*a^6 + a^4*b^2 - 4*a^2*b^4)*cos(d*x + c))*sin(d*x + c))/((a^7 - a^5
*b^2)*d)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{4}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4/(a+b*sec(d*x+c)),x)

[Out]

Integral(cos(c + d*x)**4/(a + b*sec(c + d*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (174) = 348\).
time = 0.49, size = 393, normalized size = 2.04 \begin {gather*} -\frac {\frac {48 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )} b^{5}}{\sqrt {-a^{2} + b^{2}} a^{5}} - \frac {3 \, {\left (3 \, a^{4} + 4 \, a^{2} b^{2} + 8 \, b^{4}\right )} {\left (d x + c\right )}}{a^{5}} + \frac {2 \, {\left (15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{4}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+b*sec(d*x+c)),x, algorithm="giac")

[Out]

-1/24*(48*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x
+ 1/2*c))/sqrt(-a^2 + b^2)))*b^5/(sqrt(-a^2 + b^2)*a^5) - 3*(3*a^4 + 4*a^2*b^2 + 8*b^4)*(d*x + c)/a^5 + 2*(15*
a^3*tan(1/2*d*x + 1/2*c)^7 + 24*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 12*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 24*b^3*tan(1/
2*d*x + 1/2*c)^7 - 9*a^3*tan(1/2*d*x + 1/2*c)^5 + 40*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 12*a*b^2*tan(1/2*d*x + 1/2
*c)^5 + 72*b^3*tan(1/2*d*x + 1/2*c)^5 + 9*a^3*tan(1/2*d*x + 1/2*c)^3 + 40*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 12*a*
b^2*tan(1/2*d*x + 1/2*c)^3 + 72*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*a^3*tan(1/2*d*x + 1/2*c) + 24*a^2*b*tan(1/2*d*
x + 1/2*c) - 12*a*b^2*tan(1/2*d*x + 1/2*c) + 24*b^3*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a^4)
)/d

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Mupad [B]
time = 3.42, size = 2678, normalized size = 13.88 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^4/(a + b/cos(c + d*x)),x)

[Out]

(b^5*atan(((b^5*(a^2 - b^2)^(1/2)*((tan(c/2 + (d*x)/2)*(256*a*b^10 - 27*a^10*b + 9*a^11 - 128*b^11 - 256*a^2*b
^9 + 256*a^3*b^8 - 256*a^4*b^7 + 256*a^5*b^6 - 216*a^6*b^5 + 136*a^7*b^4 - 81*a^8*b^3 + 51*a^9*b^2))/(2*a^8) +
(b^5*((12*a^16 - 12*a^15*b + 32*a^10*b^6 - 48*a^11*b^5 + 16*a^12*b^4 - 4*a^13*b^3 + 4*a^14*b^2)/a^12 - (b^5*t
an(c/2 + (d*x)/2)*(a^2 - b^2)^(1/2)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2))/(2*a^8*(a^7 - a^5*b^2)))*(a^2
- b^2)^(1/2))/(a^7 - a^5*b^2))*1i)/(a^7 - a^5*b^2) + (b^5*(a^2 - b^2)^(1/2)*((tan(c/2 + (d*x)/2)*(256*a*b^10 -
27*a^10*b + 9*a^11 - 128*b^11 - 256*a^2*b^9 + 256*a^3*b^8 - 256*a^4*b^7 + 256*a^5*b^6 - 216*a^6*b^5 + 136*a^7
*b^4 - 81*a^8*b^3 + 51*a^9*b^2))/(2*a^8) - (b^5*((12*a^16 - 12*a^15*b + 32*a^10*b^6 - 48*a^11*b^5 + 16*a^12*b^
4 - 4*a^13*b^3 + 4*a^14*b^2)/a^12 + (b^5*tan(c/2 + (d*x)/2)*(a^2 - b^2)^(1/2)*(128*a^12*b + 128*a^10*b^3 - 256
*a^11*b^2))/(2*a^8*(a^7 - a^5*b^2)))*(a^2 - b^2)^(1/2))/(a^7 - a^5*b^2))*1i)/(a^7 - a^5*b^2))/((96*a*b^13 - 64
*b^14 - 96*a^2*b^12 + 104*a^3*b^11 - 104*a^4*b^10 + 88*a^5*b^9 - 48*a^6*b^8 + 33*a^7*b^7 - 18*a^8*b^6 + 9*a^9*
b^5)/a^12 - (b^5*(a^2 - b^2)^(1/2)*((tan(c/2 + (d*x)/2)*(256*a*b^10 - 27*a^10*b + 9*a^11 - 128*b^11 - 256*a^2*
b^9 + 256*a^3*b^8 - 256*a^4*b^7 + 256*a^5*b^6 - 216*a^6*b^5 + 136*a^7*b^4 - 81*a^8*b^3 + 51*a^9*b^2))/(2*a^8)
+ (b^5*((12*a^16 - 12*a^15*b + 32*a^10*b^6 - 48*a^11*b^5 + 16*a^12*b^4 - 4*a^13*b^3 + 4*a^14*b^2)/a^12 - (b^5*
tan(c/2 + (d*x)/2)*(a^2 - b^2)^(1/2)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2))/(2*a^8*(a^7 - a^5*b^2)))*(a^2
- b^2)^(1/2))/(a^7 - a^5*b^2)))/(a^7 - a^5*b^2) + (b^5*(a^2 - b^2)^(1/2)*((tan(c/2 + (d*x)/2)*(256*a*b^10 - 2
7*a^10*b + 9*a^11 - 128*b^11 - 256*a^2*b^9 + 256*a^3*b^8 - 256*a^4*b^7 + 256*a^5*b^6 - 216*a^6*b^5 + 136*a^7*b
^4 - 81*a^8*b^3 + 51*a^9*b^2))/(2*a^8) - (b^5*((12*a^16 - 12*a^15*b + 32*a^10*b^6 - 48*a^11*b^5 + 16*a^12*b^4
- 4*a^13*b^3 + 4*a^14*b^2)/a^12 + (b^5*tan(c/2 + (d*x)/2)*(a^2 - b^2)^(1/2)*(128*a^12*b + 128*a^10*b^3 - 256*a
^11*b^2))/(2*a^8*(a^7 - a^5*b^2)))*(a^2 - b^2)^(1/2))/(a^7 - a^5*b^2)))/(a^7 - a^5*b^2)))*(a^2 - b^2)^(1/2)*2i
)/(d*(a^7 - a^5*b^2)) - (atan(((((((12*a^16 - 12*a^15*b + 32*a^10*b^6 - 48*a^11*b^5 + 16*a^12*b^4 - 4*a^13*b^3
+ 4*a^14*b^2)/a^12 - (tan(c/2 + (d*x)/2)*(a^4*3i + b^4*8i + a^2*b^2*4i)*(128*a^12*b + 128*a^10*b^3 - 256*a^11
*b^2))/(16*a^13))*(a^4*3i + b^4*8i + a^2*b^2*4i))/(8*a^5) + (tan(c/2 + (d*x)/2)*(256*a*b^10 - 27*a^10*b + 9*a^
11 - 128*b^11 - 256*a^2*b^9 + 256*a^3*b^8 - 256*a^4*b^7 + 256*a^5*b^6 - 216*a^6*b^5 + 136*a^7*b^4 - 81*a^8*b^3
+ 51*a^9*b^2))/(2*a^8))*(a^4*3i + b^4*8i + a^2*b^2*4i)*1i)/(8*a^5) - (((((12*a^16 - 12*a^15*b + 32*a^10*b^6 -
48*a^11*b^5 + 16*a^12*b^4 - 4*a^13*b^3 + 4*a^14*b^2)/a^12 + (tan(c/2 + (d*x)/2)*(a^4*3i + b^4*8i + a^2*b^2*4i
)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2))/(16*a^13))*(a^4*3i + b^4*8i + a^2*b^2*4i))/(8*a^5) - (tan(c/2 +
(d*x)/2)*(256*a*b^10 - 27*a^10*b + 9*a^11 - 128*b^11 - 256*a^2*b^9 + 256*a^3*b^8 - 256*a^4*b^7 + 256*a^5*b^6 -
216*a^6*b^5 + 136*a^7*b^4 - 81*a^8*b^3 + 51*a^9*b^2))/(2*a^8))*(a^4*3i + b^4*8i + a^2*b^2*4i)*1i)/(8*a^5))/((
((((12*a^16 - 12*a^15*b + 32*a^10*b^6 - 48*a^11*b^5 + 16*a^12*b^4 - 4*a^13*b^3 + 4*a^14*b^2)/a^12 - (tan(c/2 +
(d*x)/2)*(a^4*3i + b^4*8i + a^2*b^2*4i)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2))/(16*a^13))*(a^4*3i + b^4*
8i + a^2*b^2*4i))/(8*a^5) + (tan(c/2 + (d*x)/2)*(256*a*b^10 - 27*a^10*b + 9*a^11 - 128*b^11 - 256*a^2*b^9 + 25
6*a^3*b^8 - 256*a^4*b^7 + 256*a^5*b^6 - 216*a^6*b^5 + 136*a^7*b^4 - 81*a^8*b^3 + 51*a^9*b^2))/(2*a^8))*(a^4*3i
+ b^4*8i + a^2*b^2*4i))/(8*a^5) - (96*a*b^13 - 64*b^14 - 96*a^2*b^12 + 104*a^3*b^11 - 104*a^4*b^10 + 88*a^5*b
^9 - 48*a^6*b^8 + 33*a^7*b^7 - 18*a^8*b^6 + 9*a^9*b^5)/a^12 + (((((12*a^16 - 12*a^15*b + 32*a^10*b^6 - 48*a^11
*b^5 + 16*a^12*b^4 - 4*a^13*b^3 + 4*a^14*b^2)/a^12 + (tan(c/2 + (d*x)/2)*(a^4*3i + b^4*8i + a^2*b^2*4i)*(128*a
^12*b + 128*a^10*b^3 - 256*a^11*b^2))/(16*a^13))*(a^4*3i + b^4*8i + a^2*b^2*4i))/(8*a^5) - (tan(c/2 + (d*x)/2)
*(256*a*b^10 - 27*a^10*b + 9*a^11 - 128*b^11 - 256*a^2*b^9 + 256*a^3*b^8 - 256*a^4*b^7 + 256*a^5*b^6 - 216*a^6
*b^5 + 136*a^7*b^4 - 81*a^8*b^3 + 51*a^9*b^2))/(2*a^8))*(a^4*3i + b^4*8i + a^2*b^2*4i))/(8*a^5)))*(a^4*3i + b^
4*8i + a^2*b^2*4i)*1i)/(4*a^5*d) - ((tan(c/2 + (d*x)/2)^7*(4*a*b^2 + 8*a^2*b + 5*a^3 + 8*b^3))/(4*a^4) - (tan(
c/2 + (d*x)/2)*(4*a*b^2 - 8*a^2*b + 5*a^3 - 8*b^3))/(4*a^4) + (tan(c/2 + (d*x)/2)^3*(40*a^2*b - 12*a*b^2 + 9*a
^3 + 72*b^3))/(12*a^4) + (tan(c/2 + (d*x)/2)^5*(12*a*b^2 + 40*a^2*b - 9*a^3 + 72*b^3))/(12*a^4))/(d*(4*tan(c/2
+ (d*x)/2)^2 + 6*tan(c/2 + (d*x)/2)^4 + 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1))

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