3.1.9 \(\int \frac {(c+d \sec (e+f x))^4}{a+b \cos (e+f x)} \, dx\) [9]

Optimal. Leaf size=247 \[ \frac {2 (a c-b d)^4 \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a^4 \sqrt {a-b} \sqrt {a+b} f}+\frac {d^3 (4 a c-b d) \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}+\frac {d (2 a c-b d) \left (2 a^2 c^2-2 a b c d+b^2 d^2\right ) \tanh ^{-1}(\sin (e+f x))}{a^4 f}+\frac {d^4 \tan (e+f x)}{a f}+\frac {d^2 \left (6 a^2 c^2-4 a b c d+b^2 d^2\right ) \tan (e+f x)}{a^3 f}+\frac {d^3 (4 a c-b d) \sec (e+f x) \tan (e+f x)}{2 a^2 f}+\frac {d^4 \tan ^3(e+f x)}{3 a f} \]

[Out]

1/2*d^3*(4*a*c-b*d)*arctanh(sin(f*x+e))/a^2/f+d*(2*a*c-b*d)*(2*a^2*c^2-2*a*b*c*d+b^2*d^2)*arctanh(sin(f*x+e))/
a^4/f+2*(a*c-b*d)^4*arctan((a-b)^(1/2)*tan(1/2*f*x+1/2*e)/(a+b)^(1/2))/a^4/f/(a-b)^(1/2)/(a+b)^(1/2)+d^4*tan(f
*x+e)/a/f+d^2*(6*a^2*c^2-4*a*b*c*d+b^2*d^2)*tan(f*x+e)/a^3/f+1/2*d^3*(4*a*c-b*d)*sec(f*x+e)*tan(f*x+e)/a^2/f+1
/3*d^4*tan(f*x+e)^3/a/f

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Rubi [A]
time = 0.29, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2907, 3031, 2738, 211, 3855, 3852, 8, 3853} \begin {gather*} \frac {2 (a c-b d)^4 \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a^4 f \sqrt {a-b} \sqrt {a+b}}+\frac {d^3 (4 a c-b d) \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}+\frac {d^3 (4 a c-b d) \tan (e+f x) \sec (e+f x)}{2 a^2 f}+\frac {d (2 a c-b d) \left (2 a^2 c^2-2 a b c d+b^2 d^2\right ) \tanh ^{-1}(\sin (e+f x))}{a^4 f}+\frac {d^2 \left (6 a^2 c^2-4 a b c d+b^2 d^2\right ) \tan (e+f x)}{a^3 f}+\frac {d^4 \tan ^3(e+f x)}{3 a f}+\frac {d^4 \tan (e+f x)}{a f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*c - b*d)^4*ArcTan[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*f) + (d^3*(4
*a*c - b*d)*ArcTanh[Sin[e + f*x]])/(2*a^2*f) + (d*(2*a*c - b*d)*(2*a^2*c^2 - 2*a*b*c*d + b^2*d^2)*ArcTanh[Sin[
e + f*x]])/(a^4*f) + (d^4*Tan[e + f*x])/(a*f) + (d^2*(6*a^2*c^2 - 4*a*b*c*d + b^2*d^2)*Tan[e + f*x])/(a^3*f) +
(d^3*(4*a*c - b*d)*Sec[e + f*x]*Tan[e + f*x])/(2*a^2*f) + (d^4*Tan[e + f*x]^3)/(3*a*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 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 2907

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> In
t[(a + b*Sin[e + f*x])^m*((d + c*Sin[e + f*x])^n/Sin[e + f*x]^n), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && Int
egerQ[n]

Rule 3031

Int[((g_.)*sin[(e_.) + (f_.)*(x_)])^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTrig[(g*sin[e + f*x])^p*(a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[b*c - a*d, 0] && (IntegersQ[m, n] || IntegersQ[m, p
] || IntegersQ[n, p]) && NeQ[p, 2]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {(c+d \sec (e+f x))^4}{a+b \cos (e+f x)} \, dx &=\int \frac {(d+c \cos (e+f x))^4 \sec ^4(e+f x)}{a+b \cos (e+f x)} \, dx\\ &=\int \left (\frac {(a c-b d)^4}{a^4 (a+b \cos (e+f x))}+\frac {d (2 a c-b d) \left (2 a^2 c^2-2 a b c d+b^2 d^2\right ) \sec (e+f x)}{a^4}+\frac {d^2 \left (6 a^2 c^2-4 a b c d+b^2 d^2\right ) \sec ^2(e+f x)}{a^3}+\frac {d^3 (4 a c-b d) \sec ^3(e+f x)}{a^2}+\frac {d^4 \sec ^4(e+f x)}{a}\right ) \, dx\\ &=\frac {d^4 \int \sec ^4(e+f x) \, dx}{a}+\frac {(a c-b d)^4 \int \frac {1}{a+b \cos (e+f x)} \, dx}{a^4}+\frac {\left (d^3 (4 a c-b d)\right ) \int \sec ^3(e+f x) \, dx}{a^2}+\frac {\left (d^2 \left (6 a^2 c^2-4 a b c d+b^2 d^2\right )\right ) \int \sec ^2(e+f x) \, dx}{a^3}+\frac {\left (d (2 a c-b d) \left (2 a^2 c^2-2 a b c d+b^2 d^2\right )\right ) \int \sec (e+f x) \, dx}{a^4}\\ &=\frac {d (2 a c-b d) \left (2 a^2 c^2-2 a b c d+b^2 d^2\right ) \tanh ^{-1}(\sin (e+f x))}{a^4 f}+\frac {d^3 (4 a c-b d) \sec (e+f x) \tan (e+f x)}{2 a^2 f}+\frac {\left (d^3 (4 a c-b d)\right ) \int \sec (e+f x) \, dx}{2 a^2}-\frac {d^4 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{a f}+\frac {\left (2 (a c-b d)^4\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{a^4 f}-\frac {\left (d^2 \left (6 a^2 c^2-4 a b c d+b^2 d^2\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{a^3 f}\\ &=\frac {2 (a c-b d)^4 \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a^4 \sqrt {a-b} \sqrt {a+b} f}+\frac {d^3 (4 a c-b d) \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}+\frac {d (2 a c-b d) \left (2 a^2 c^2-2 a b c d+b^2 d^2\right ) \tanh ^{-1}(\sin (e+f x))}{a^4 f}+\frac {d^4 \tan (e+f x)}{a f}+\frac {d^2 \left (6 a^2 c^2-4 a b c d+b^2 d^2\right ) \tan (e+f x)}{a^3 f}+\frac {d^3 (4 a c-b d) \sec (e+f x) \tan (e+f x)}{2 a^2 f}+\frac {d^4 \tan ^3(e+f x)}{3 a f}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(526\) vs. \(2(247)=494\).
time = 4.04, size = 526, normalized size = 2.13 \begin {gather*} \frac {-\frac {24 (a c-b d)^4 \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-6 d \left (8 a b^2 c d^2-2 b^3 d^3+4 a^3 c \left (2 c^2+d^2\right )-a^2 b d \left (12 c^2+d^2\right )\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-6 d \left (-8 a b^2 c d^2+2 b^3 d^3-4 a^3 c \left (2 c^2+d^2\right )+a^2 b d \left (12 c^2+d^2\right )\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+\frac {a^2 d^3 (-3 b d+a (12 c+d))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {2 a^3 d^4 \sin \left (\frac {1}{2} (e+f x)\right )}{\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {4 a d^2 \left (-12 a b c d+3 b^2 d^2+2 a^2 \left (9 c^2+d^2\right )\right ) \sin \left (\frac {1}{2} (e+f x)\right )}{\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )}+\frac {2 a^3 d^4 \sin \left (\frac {1}{2} (e+f x)\right )}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {a^2 d^3 (-3 b d+a (12 c+d))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {4 a d^2 \left (-12 a b c d+3 b^2 d^2+2 a^2 \left (9 c^2+d^2\right )\right ) \sin \left (\frac {1}{2} (e+f x)\right )}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}}{12 a^4 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-24*(a*c - b*d)^4*ArcTanh[((a - b)*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - 6*d*(8*a*b^2*c*d^
2 - 2*b^3*d^3 + 4*a^3*c*(2*c^2 + d^2) - a^2*b*d*(12*c^2 + d^2))*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 6*d
*(-8*a*b^2*c*d^2 + 2*b^3*d^3 - 4*a^3*c*(2*c^2 + d^2) + a^2*b*d*(12*c^2 + d^2))*Log[Cos[(e + f*x)/2] + Sin[(e +
f*x)/2]] + (a^2*d^3*(-3*b*d + a*(12*c + d)))/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (2*a^3*d^4*Sin[(e + f*
x)/2])/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + (4*a*d^2*(-12*a*b*c*d + 3*b^2*d^2 + 2*a^2*(9*c^2 + d^2))*Sin[
(e + f*x)/2])/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) + (2*a^3*d^4*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2])^3 - (a^2*d^3*(-3*b*d + a*(12*c + d)))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (4*a*d^2*(-12*a*b*c
*d + 3*b^2*d^2 + 2*a^2*(9*c^2 + d^2))*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(12*a^4*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(483\) vs. \(2(232)=464\).
time = 0.61, size = 484, normalized size = 1.96 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(2*(a^4*c^4-4*a^3*b*c^3*d+6*a^2*b^2*c^2*d^2-4*a*b^3*c*d^3+b^4*d^4)/a^4/((a-b)*(a+b))^(1/2)*arctan((a-b)*ta
n(1/2*f*x+1/2*e)/((a-b)*(a+b))^(1/2))-1/3*d^4/a/(tan(1/2*f*x+1/2*e)-1)^3-1/2*d*(8*a^3*c^3+4*a^3*c*d^2-12*a^2*b
*c^2*d-a^2*b*d^3+8*a*b^2*c*d^2-2*b^3*d^3)/a^4*ln(tan(1/2*f*x+1/2*e)-1)-1/2*d^2*(12*a^2*c^2-4*a^2*c*d+2*a^2*d^2
-8*a*b*c*d+a*b*d^2+2*b^2*d^2)/a^3/(tan(1/2*f*x+1/2*e)-1)+1/2*d^3*(4*a*c-a*d-b*d)/a^2/(tan(1/2*f*x+1/2*e)-1)^2-
1/3*d^4/a/(tan(1/2*f*x+1/2*e)+1)^3+1/2*d*(8*a^3*c^3+4*a^3*c*d^2-12*a^2*b*c^2*d-a^2*b*d^3+8*a*b^2*c*d^2-2*b^3*d
^3)/a^4*ln(tan(1/2*f*x+1/2*e)+1)-1/2*d^2*(12*a^2*c^2-4*a^2*c*d+2*a^2*d^2-8*a*b*c*d+a*b*d^2+2*b^2*d^2)/a^3/(tan
(1/2*f*x+1/2*e)+1)-1/2*d^3*(4*a*c-a*d-b*d)/a^2/(tan(1/2*f*x+1/2*e)+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((c+d*sec(f*x+e))^4/(a+b*cos(f*x+e)),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*b^2-4*a^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (240) = 480\).
time = 102.64, size = 1075, normalized size = 4.35 \begin {gather*} \left [-\frac {6 \, {\left (a^{4} c^{4} - 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a b^{3} c d^{3} + b^{4} d^{4}\right )} \sqrt {-a^{2} + b^{2}} \cos \left (f x + e\right )^{3} \log \left (\frac {2 \, a b \cos \left (f x + e\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (f x + e\right ) + b\right )} \sin \left (f x + e\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (f x + e\right )^{2} + 2 \, a b \cos \left (f x + e\right ) + a^{2}}\right ) - 3 \, {\left (8 \, {\left (a^{5} - a^{3} b^{2}\right )} c^{3} d - 12 \, {\left (a^{4} b - a^{2} b^{3}\right )} c^{2} d^{2} + 4 \, {\left (a^{5} + a^{3} b^{2} - 2 \, a b^{4}\right )} c d^{3} - {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} d^{4}\right )} \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, {\left (8 \, {\left (a^{5} - a^{3} b^{2}\right )} c^{3} d - 12 \, {\left (a^{4} b - a^{2} b^{3}\right )} c^{2} d^{2} + 4 \, {\left (a^{5} + a^{3} b^{2} - 2 \, a b^{4}\right )} c d^{3} - {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} d^{4}\right )} \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (2 \, {\left (a^{5} - a^{3} b^{2}\right )} d^{4} + 2 \, {\left (18 \, {\left (a^{5} - a^{3} b^{2}\right )} c^{2} d^{2} - 12 \, {\left (a^{4} b - a^{2} b^{3}\right )} c d^{3} + {\left (2 \, a^{5} + a^{3} b^{2} - 3 \, a b^{4}\right )} d^{4}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (4 \, {\left (a^{5} - a^{3} b^{2}\right )} c d^{3} - {\left (a^{4} b - a^{2} b^{3}\right )} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, {\left (a^{6} - a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{3}}, \frac {12 \, {\left (a^{4} c^{4} - 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a b^{3} c d^{3} + b^{4} d^{4}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (f x + e\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right )^{3} + 3 \, {\left (8 \, {\left (a^{5} - a^{3} b^{2}\right )} c^{3} d - 12 \, {\left (a^{4} b - a^{2} b^{3}\right )} c^{2} d^{2} + 4 \, {\left (a^{5} + a^{3} b^{2} - 2 \, a b^{4}\right )} c d^{3} - {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} d^{4}\right )} \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (8 \, {\left (a^{5} - a^{3} b^{2}\right )} c^{3} d - 12 \, {\left (a^{4} b - a^{2} b^{3}\right )} c^{2} d^{2} + 4 \, {\left (a^{5} + a^{3} b^{2} - 2 \, a b^{4}\right )} c d^{3} - {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} d^{4}\right )} \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (2 \, {\left (a^{5} - a^{3} b^{2}\right )} d^{4} + 2 \, {\left (18 \, {\left (a^{5} - a^{3} b^{2}\right )} c^{2} d^{2} - 12 \, {\left (a^{4} b - a^{2} b^{3}\right )} c d^{3} + {\left (2 \, a^{5} + a^{3} b^{2} - 3 \, a b^{4}\right )} d^{4}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (4 \, {\left (a^{5} - a^{3} b^{2}\right )} c d^{3} - {\left (a^{4} b - a^{2} b^{3}\right )} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, {\left (a^{6} - a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*(6*(a^4*c^4 - 4*a^3*b*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c*d^3 + b^4*d^4)*sqrt(-a^2 + b^2)*cos(f*x + e
)^3*log((2*a*b*cos(f*x + e) + (2*a^2 - b^2)*cos(f*x + e)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(f*x + e) + b)*sin(f*x +
e) - a^2 + 2*b^2)/(b^2*cos(f*x + e)^2 + 2*a*b*cos(f*x + e) + a^2)) - 3*(8*(a^5 - a^3*b^2)*c^3*d - 12*(a^4*b -
a^2*b^3)*c^2*d^2 + 4*(a^5 + a^3*b^2 - 2*a*b^4)*c*d^3 - (a^4*b + a^2*b^3 - 2*b^5)*d^4)*cos(f*x + e)^3*log(sin(
f*x + e) + 1) + 3*(8*(a^5 - a^3*b^2)*c^3*d - 12*(a^4*b - a^2*b^3)*c^2*d^2 + 4*(a^5 + a^3*b^2 - 2*a*b^4)*c*d^3
- (a^4*b + a^2*b^3 - 2*b^5)*d^4)*cos(f*x + e)^3*log(-sin(f*x + e) + 1) - 2*(2*(a^5 - a^3*b^2)*d^4 + 2*(18*(a^5
- a^3*b^2)*c^2*d^2 - 12*(a^4*b - a^2*b^3)*c*d^3 + (2*a^5 + a^3*b^2 - 3*a*b^4)*d^4)*cos(f*x + e)^2 + 3*(4*(a^5
- a^3*b^2)*c*d^3 - (a^4*b - a^2*b^3)*d^4)*cos(f*x + e))*sin(f*x + e))/((a^6 - a^4*b^2)*f*cos(f*x + e)^3), 1/1
2*(12*(a^4*c^4 - 4*a^3*b*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c*d^3 + b^4*d^4)*sqrt(a^2 - b^2)*arctan(-(a*cos(f
*x + e) + b)/(sqrt(a^2 - b^2)*sin(f*x + e)))*cos(f*x + e)^3 + 3*(8*(a^5 - a^3*b^2)*c^3*d - 12*(a^4*b - a^2*b^3
)*c^2*d^2 + 4*(a^5 + a^3*b^2 - 2*a*b^4)*c*d^3 - (a^4*b + a^2*b^3 - 2*b^5)*d^4)*cos(f*x + e)^3*log(sin(f*x + e)
+ 1) - 3*(8*(a^5 - a^3*b^2)*c^3*d - 12*(a^4*b - a^2*b^3)*c^2*d^2 + 4*(a^5 + a^3*b^2 - 2*a*b^4)*c*d^3 - (a^4*b
+ a^2*b^3 - 2*b^5)*d^4)*cos(f*x + e)^3*log(-sin(f*x + e) + 1) + 2*(2*(a^5 - a^3*b^2)*d^4 + 2*(18*(a^5 - a^3*b
^2)*c^2*d^2 - 12*(a^4*b - a^2*b^3)*c*d^3 + (2*a^5 + a^3*b^2 - 3*a*b^4)*d^4)*cos(f*x + e)^2 + 3*(4*(a^5 - a^3*b
^2)*c*d^3 - (a^4*b - a^2*b^3)*d^4)*cos(f*x + e))*sin(f*x + e))/((a^6 - a^4*b^2)*f*cos(f*x + e)^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (240) = 480\).
time = 0.54, size = 629, normalized size = 2.55 \begin {gather*} \frac {\frac {3 \, {\left (8 \, a^{3} c^{3} d - 12 \, a^{2} b c^{2} d^{2} + 4 \, a^{3} c d^{3} + 8 \, a b^{2} c d^{3} - a^{2} b d^{4} - 2 \, b^{3} d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{4}} - \frac {3 \, {\left (8 \, a^{3} c^{3} d - 12 \, a^{2} b c^{2} d^{2} + 4 \, a^{3} c d^{3} + 8 \, a b^{2} c d^{3} - a^{2} b d^{4} - 2 \, b^{3} d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{4}} - \frac {12 \, {\left (a^{4} c^{4} - 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a b^{3} c d^{3} + b^{4} d^{4}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{4}} - \frac {2 \, {\left (36 \, a^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, a^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 24 \, a b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, a b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, b^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 72 \, a^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 48 \, a b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 12 \, b^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 36 \, a^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, a^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 24 \, a b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, a b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, b^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} a^{3}}}{6 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*(8*a^3*c^3*d - 12*a^2*b*c^2*d^2 + 4*a^3*c*d^3 + 8*a*b^2*c*d^3 - a^2*b*d^4 - 2*b^3*d^4)*log(abs(tan(1/2*
f*x + 1/2*e) + 1))/a^4 - 3*(8*a^3*c^3*d - 12*a^2*b*c^2*d^2 + 4*a^3*c*d^3 + 8*a*b^2*c*d^3 - a^2*b*d^4 - 2*b^3*d
^4)*log(abs(tan(1/2*f*x + 1/2*e) - 1))/a^4 - 12*(a^4*c^4 - 4*a^3*b*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c*d^3 +
b^4*d^4)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*f*x + 1/2*e) - b*tan(1/2*f*x
+ 1/2*e))/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*a^4) - 2*(36*a^2*c^2*d^2*tan(1/2*f*x + 1/2*e)^5 - 12*a^2*c*d^3*ta
n(1/2*f*x + 1/2*e)^5 - 24*a*b*c*d^3*tan(1/2*f*x + 1/2*e)^5 + 6*a^2*d^4*tan(1/2*f*x + 1/2*e)^5 + 3*a*b*d^4*tan(
1/2*f*x + 1/2*e)^5 + 6*b^2*d^4*tan(1/2*f*x + 1/2*e)^5 - 72*a^2*c^2*d^2*tan(1/2*f*x + 1/2*e)^3 + 48*a*b*c*d^3*t
an(1/2*f*x + 1/2*e)^3 - 4*a^2*d^4*tan(1/2*f*x + 1/2*e)^3 - 12*b^2*d^4*tan(1/2*f*x + 1/2*e)^3 + 36*a^2*c^2*d^2*
tan(1/2*f*x + 1/2*e) + 12*a^2*c*d^3*tan(1/2*f*x + 1/2*e) - 24*a*b*c*d^3*tan(1/2*f*x + 1/2*e) + 6*a^2*d^4*tan(1
/2*f*x + 1/2*e) - 3*a*b*d^4*tan(1/2*f*x + 1/2*e) + 6*b^2*d^4*tan(1/2*f*x + 1/2*e))/((tan(1/2*f*x + 1/2*e)^2 -
1)^3*a^3))/f

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Mupad [B]
time = 11.82, size = 2500, normalized size = 10.12 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan(((((((8*(4*a^13*c^4 - 8*a^12*b*c^4 - 2*a^12*b*d^4 + 8*a^13*c*d^3 + 16*a^13*c^3*d + 4*a^11*b^2*c^4 - 4*a^
8*b^5*d^4 + 6*a^9*b^4*d^4 - 2*a^10*b^3*d^4 + 2*a^11*b^2*d^4 + 16*a^9*b^4*c*d^3 - 24*a^10*b^3*c*d^3 + 8*a^11*b^
2*c*d^3 + 16*a^11*b^2*c^3*d - 24*a^12*b*c^2*d^2 - 24*a^10*b^3*c^2*d^2 + 48*a^11*b^2*c^2*d^2 - 8*a^12*b*c*d^3 -
32*a^12*b*c^3*d))/a^9 - (8*tan(e/2 + (f*x)/2)*(8*a^10*b + 8*a^8*b^3 - 16*a^9*b^2)*(a^2*((b*d^4)/2 + 6*b*c^2*d
^2) - a^3*(2*c*d^3 + 4*c^3*d) + b^3*d^4 - 4*a*b^2*c*d^3))/a^10)*(a^2*((b*d^4)/2 + 6*b*c^2*d^2) - a^3*(2*c*d^3
+ 4*c^3*d) + b^3*d^4 - 4*a*b^2*c*d^3))/a^4 - (8*tan(e/2 + (f*x)/2)*(4*a^9*c^8 - 8*b^9*d^8 - 4*a^8*b*c^8 + 16*a
*b^8*d^8 - 16*a^2*b^7*d^8 + 16*a^3*b^6*d^8 - 13*a^4*b^5*d^8 + 7*a^5*b^4*d^8 - 3*a^6*b^3*d^8 + a^7*b^2*d^8 + 16
*a^9*c^2*d^6 + 64*a^9*c^4*d^4 + 64*a^9*c^6*d^2 - 128*a^2*b^7*c*d^7 + 128*a^3*b^6*c*d^7 - 128*a^4*b^5*c*d^7 + 1
04*a^5*b^4*c*d^7 - 56*a^6*b^3*c*d^7 + 24*a^7*b^2*c*d^7 + 32*a^7*b^2*c^7*d - 48*a^8*b*c^2*d^6 - 112*a^8*b*c^3*d
^5 - 192*a^8*b*c^4*d^4 - 192*a^8*b*c^5*d^3 - 192*a^8*b*c^6*d^2 - 224*a^2*b^7*c^2*d^6 + 448*a^3*b^6*c^2*d^6 + 4
48*a^3*b^6*c^3*d^5 - 424*a^4*b^5*c^2*d^6 - 896*a^4*b^5*c^3*d^5 - 552*a^4*b^5*c^4*d^4 + 376*a^5*b^4*c^2*d^6 + 7
84*a^5*b^4*c^3*d^5 + 1096*a^5*b^4*c^4*d^4 + 416*a^5*b^4*c^5*d^3 - 280*a^6*b^3*c^2*d^6 - 560*a^6*b^3*c^3*d^5 -
880*a^6*b^3*c^4*d^4 - 800*a^6*b^3*c^5*d^3 - 176*a^6*b^3*c^6*d^2 + 136*a^7*b^2*c^2*d^6 + 336*a^7*b^2*c^3*d^5 +
464*a^7*b^2*c^4*d^4 + 576*a^7*b^2*c^5*d^3 + 304*a^7*b^2*c^6*d^2 + 64*a*b^8*c*d^7 - 8*a^8*b*c*d^7 - 32*a^8*b*c^
7*d))/a^6)*(a^2*((b*d^4)/2 + 6*b*c^2*d^2) - a^3*(2*c*d^3 + 4*c^3*d) + b^3*d^4 - 4*a*b^2*c*d^3)*1i)/a^4 - (((((
8*(4*a^13*c^4 - 8*a^12*b*c^4 - 2*a^12*b*d^4 + 8*a^13*c*d^3 + 16*a^13*c^3*d + 4*a^11*b^2*c^4 - 4*a^8*b^5*d^4 +
6*a^9*b^4*d^4 - 2*a^10*b^3*d^4 + 2*a^11*b^2*d^4 + 16*a^9*b^4*c*d^3 - 24*a^10*b^3*c*d^3 + 8*a^11*b^2*c*d^3 + 16
*a^11*b^2*c^3*d - 24*a^12*b*c^2*d^2 - 24*a^10*b^3*c^2*d^2 + 48*a^11*b^2*c^2*d^2 - 8*a^12*b*c*d^3 - 32*a^12*b*c
^3*d))/a^9 + (8*tan(e/2 + (f*x)/2)*(8*a^10*b + 8*a^8*b^3 - 16*a^9*b^2)*(a^2*((b*d^4)/2 + 6*b*c^2*d^2) - a^3*(2
*c*d^3 + 4*c^3*d) + b^3*d^4 - 4*a*b^2*c*d^3))/a^10)*(a^2*((b*d^4)/2 + 6*b*c^2*d^2) - a^3*(2*c*d^3 + 4*c^3*d) +
b^3*d^4 - 4*a*b^2*c*d^3))/a^4 + (8*tan(e/2 + (f*x)/2)*(4*a^9*c^8 - 8*b^9*d^8 - 4*a^8*b*c^8 + 16*a*b^8*d^8 - 1
6*a^2*b^7*d^8 + 16*a^3*b^6*d^8 - 13*a^4*b^5*d^8 + 7*a^5*b^4*d^8 - 3*a^6*b^3*d^8 + a^7*b^2*d^8 + 16*a^9*c^2*d^6
+ 64*a^9*c^4*d^4 + 64*a^9*c^6*d^2 - 128*a^2*b^7*c*d^7 + 128*a^3*b^6*c*d^7 - 128*a^4*b^5*c*d^7 + 104*a^5*b^4*c
*d^7 - 56*a^6*b^3*c*d^7 + 24*a^7*b^2*c*d^7 + 32*a^7*b^2*c^7*d - 48*a^8*b*c^2*d^6 - 112*a^8*b*c^3*d^5 - 192*a^8
*b*c^4*d^4 - 192*a^8*b*c^5*d^3 - 192*a^8*b*c^6*d^2 - 224*a^2*b^7*c^2*d^6 + 448*a^3*b^6*c^2*d^6 + 448*a^3*b^6*c
^3*d^5 - 424*a^4*b^5*c^2*d^6 - 896*a^4*b^5*c^3*d^5 - 552*a^4*b^5*c^4*d^4 + 376*a^5*b^4*c^2*d^6 + 784*a^5*b^4*c
^3*d^5 + 1096*a^5*b^4*c^4*d^4 + 416*a^5*b^4*c^5*d^3 - 280*a^6*b^3*c^2*d^6 - 560*a^6*b^3*c^3*d^5 - 880*a^6*b^3*
c^4*d^4 - 800*a^6*b^3*c^5*d^3 - 176*a^6*b^3*c^6*d^2 + 136*a^7*b^2*c^2*d^6 + 336*a^7*b^2*c^3*d^5 + 464*a^7*b^2*
c^4*d^4 + 576*a^7*b^2*c^5*d^3 + 304*a^7*b^2*c^6*d^2 + 64*a*b^8*c*d^7 - 8*a^8*b*c*d^7 - 32*a^8*b*c^7*d))/a^6)*(
a^2*((b*d^4)/2 + 6*b*c^2*d^2) - a^3*(2*c*d^3 + 4*c^3*d) + b^3*d^4 - 4*a*b^2*c*d^3)*1i)/a^4)/((16*(4*b^11*d^12
- 6*a*b^10*d^12 + 16*a^11*c^11*d + 6*a^2*b^9*d^12 - 5*a^3*b^8*d^12 + 2*a^4*b^7*d^12 - a^5*b^6*d^12 - 16*a^11*c
^6*d^6 - 64*a^11*c^8*d^4 + 8*a^11*c^9*d^3 - 64*a^11*c^10*d^2 + 72*a^2*b^9*c*d^11 - 72*a^3*b^8*c*d^11 + 60*a^4*
b^7*c*d^11 - 24*a^5*b^6*c*d^11 + 12*a^6*b^5*c*d^11 + 72*a^10*b*c^5*d^7 + 32*a^10*b*c^6*d^6 + 368*a^10*b*c^7*d^
5 + 62*a^10*b*c^8*d^4 + 440*a^10*b*c^9*d^3 - 24*a^10*b*c^10*d^2 + 264*a^2*b^9*c^2*d^10 - 384*a^3*b^8*c^2*d^10
- 880*a^3*b^8*c^3*d^9 + 4*a^3*b^8*c^4*d^8 + 360*a^4*b^7*c^2*d^10 + 1216*a^4*b^7*c^3*d^9 + 1968*a^4*b^7*c^4*d^8
- 32*a^4*b^7*c^5*d^7 - 294*a^5*b^6*c^2*d^10 - 1008*a^5*b^6*c^3*d^9 - 2556*a^5*b^6*c^4*d^8 - 3072*a^5*b^6*c^5*
d^7 + 112*a^5*b^6*c^6*d^6 + 108*a^6*b^5*c^2*d^10 + 788*a^6*b^5*c^3*d^9 + 1756*a^6*b^5*c^4*d^8 + 3744*a^6*b^5*c
^5*d^7 + 3360*a^6*b^5*c^6*d^6 - 224*a^6*b^5*c^7*d^5 - 54*a^7*b^4*c^2*d^10 - 232*a^7*b^4*c^3*d^9 - 1301*a^7*b^4
*c^4*d^8 - 1952*a^7*b^4*c^5*d^7 - 3888*a^7*b^4*c^6*d^6 - 2496*a^7*b^4*c^7*d^5 + 276*a^7*b^4*c^8*d^4 + 116*a^8*
b^3*c^3*d^9 + 258*a^8*b^3*c^4*d^8 + 1384*a^8*b^3*c^5*d^7 + 1336*a^8*b^3*c^6*d^6 + 2848*a^8*b^3*c^7*d^5 + 1148*
a^8*b^3*c^8*d^4 - 208*a^8*b^3*c^9*d^3 - 129*a^9*b^2*c^4*d^8 - 144*a^9*b^2*c^5*d^7 - 936*a^9*b^2*c^6*d^6 - 496*
a^9*b^2*c^7*d^5 - 1422*a^9*b^2*c^8*d^4 - 240*a^9*b^2*c^9*d^3 + 88*a^9*b^2*c^10*d^2 - 48*a*b^10*c*d^11 - 16*a^1
0*b*c^11*d))/a^9 + (((((8*(4*a^13*c^4 - 8*a^12*b*c^4 - 2*a^12*b*d^4 + 8*a^13*c*d^3 + 16*a^13*c^3*d + 4*a^11*b^
2*c^4 - 4*a^8*b^5*d^4 + 6*a^9*b^4*d^4 - 2*a^10*b^3*d^4 + 2*a^11*b^2*d^4 + 16*a^9*b^4*c*d^3 - 24*a^10*b^3*c*d^3
+ 8*a^11*b^2*c*d^3 + 16*a^11*b^2*c^3*d - 24*a^12*b*c^2*d^2 - 24*a^10*b^3*c^2*d^2 + 48*a^11*b^2*c^2*d^2 - 8*a^
12*b*c*d^3 - 32*a^12*b*c^3*d))/a^9 - (8*tan(e/2...

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