3.1.10 \(\int \frac {(c+d \sec (e+f x))^3}{a+b \cos (e+f x)} \, dx\) [10]

Optimal. Leaf size=170 \[ \frac {2 (a c-b d)^3 \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a^3 \sqrt {a-b} \sqrt {a+b} f}+\frac {d^3 \tanh ^{-1}(\sin (e+f x))}{2 a f}+\frac {d \left (3 a^2 c^2-3 a b c d+b^2 d^2\right ) \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {d^2 (3 a c-b d) \tan (e+f x)}{a^2 f}+\frac {d^3 \sec (e+f x) \tan (e+f x)}{2 a f} \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 10, 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)^3 \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a^3 f \sqrt {a-b} \sqrt {a+b}}+\frac {d^2 (3 a c-b d) \tan (e+f x)}{a^2 f}+\frac {d \left (3 a^2 c^2-3 a b c d+b^2 d^2\right ) \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {d^3 \tanh ^{-1}(\sin (e+f x))}{2 a f}+\frac {d^3 \tan (e+f x) \sec (e+f x)}{2 a f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*c - b*d)^3*ArcTan[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*f) + (d^3*Ar
cTanh[Sin[e + f*x]])/(2*a*f) + (d*(3*a^2*c^2 - 3*a*b*c*d + b^2*d^2)*ArcTanh[Sin[e + f*x]])/(a^3*f) + (d^2*(3*a
*c - b*d)*Tan[e + f*x])/(a^2*f) + (d^3*Sec[e + f*x]*Tan[e + f*x])/(2*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))^3}{a+b \cos (e+f x)} \, dx &=\int \frac {(d+c \cos (e+f x))^3 \sec ^3(e+f x)}{a+b \cos (e+f x)} \, dx\\ &=\int \left (\frac {(a c-b d)^3}{a^3 (a+b \cos (e+f x))}+\frac {d \left (3 a^2 c^2-3 a b c d+b^2 d^2\right ) \sec (e+f x)}{a^3}+\frac {d^2 (3 a c-b d) \sec ^2(e+f x)}{a^2}+\frac {d^3 \sec ^3(e+f x)}{a}\right ) \, dx\\ &=\frac {d^3 \int \sec ^3(e+f x) \, dx}{a}+\frac {(a c-b d)^3 \int \frac {1}{a+b \cos (e+f x)} \, dx}{a^3}+\frac {\left (d^2 (3 a c-b d)\right ) \int \sec ^2(e+f x) \, dx}{a^2}+\frac {\left (d \left (3 a^2 c^2-3 a b c d+b^2 d^2\right )\right ) \int \sec (e+f x) \, dx}{a^3}\\ &=\frac {d \left (3 a^2 c^2-3 a b c d+b^2 d^2\right ) \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {d^3 \sec (e+f x) \tan (e+f x)}{2 a f}+\frac {d^3 \int \sec (e+f x) \, dx}{2 a}+\frac {\left (2 (a c-b d)^3\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^3 f}-\frac {\left (d^2 (3 a c-b d)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{a^2 f}\\ &=\frac {2 (a c-b d)^3 \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a^3 \sqrt {a-b} \sqrt {a+b} f}+\frac {d^3 \tanh ^{-1}(\sin (e+f x))}{2 a f}+\frac {d \left (3 a^2 c^2-3 a b c d+b^2 d^2\right ) \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {d^2 (3 a c-b d) \tan (e+f x)}{a^2 f}+\frac {d^3 \sec (e+f x) \tan (e+f x)}{2 a f}\\ \end {align*}

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Mathematica [A]
time = 1.38, size = 335, normalized size = 1.97 \begin {gather*} \frac {-\frac {8 (a c-b d)^3 \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}}-2 d \left (-6 a b c d+2 b^2 d^2+a^2 \left (6 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 )+2 d \left (-6 a b c d+2 b^2 d^2+a^2 \left (6 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}{\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 (3 a c-b d) \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 {a^2 d^3}{\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 (3 a c-b d) \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 )}}{4 a^3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.44, size = 291, normalized size = 1.71

method	result	size
derivativedivides	\(\frac {\frac {2 \left (c^{3} a^{3}-3 a^{2} b \,c^{2} d +3 a \,b^{2} c \,d^{2}-b^{3} d^{3}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {d^{3}}{2 a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {d \left (6 a^{2} c^{2}+d^{2} a^{2}-6 c d a b +2 b^{2} d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 a^{3}}-\frac {d^{2} \left (6 a c -d a -2 b d \right )}{2 a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {d^{3}}{2 a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {d \left (6 a^{2} c^{2}+d^{2} a^{2}-6 c d a b +2 b^{2} d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 a^{3}}-\frac {d^{2} \left (6 a c -d a -2 b d \right )}{2 a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{f}\)	\(291\)
default	\(\frac {\frac {2 \left (c^{3} a^{3}-3 a^{2} b \,c^{2} d +3 a \,b^{2} c \,d^{2}-b^{3} d^{3}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {d^{3}}{2 a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {d \left (6 a^{2} c^{2}+d^{2} a^{2}-6 c d a b +2 b^{2} d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 a^{3}}-\frac {d^{2} \left (6 a c -d a -2 b d \right )}{2 a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {d^{3}}{2 a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {d \left (6 a^{2} c^{2}+d^{2} a^{2}-6 c d a b +2 b^{2} d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 a^{3}}-\frac {d^{2} \left (6 a c -d a -2 b d \right )}{2 a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{f}\)	\(291\)
risch	\(-\frac {i d^{2} \left (a d \,{\mathrm e}^{3 i \left (f x +e \right )}-6 a c \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b d \,{\mathrm e}^{2 i \left (f x +e \right )}-d \,{\mathrm e}^{i \left (f x +e \right )} a -6 a c +2 b d \right )}{a^{2} f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}+\frac {3 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c^{2}}{a f}+\frac {d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{2 a f}-\frac {3 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c b}{a^{2} f}+\frac {d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) b^{2}}{a^{3} f}-\frac {3 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c^{2}}{a f}-\frac {d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{2 a f}+\frac {3 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c b}{a^{2} f}-\frac {d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) b^{2}}{a^{3} f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) c^{3}}{\sqrt {-a^{2}+b^{2}}\, f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) b \,c^{2} d}{\sqrt {-a^{2}+b^{2}}\, f a}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2} c \,d^{2}}{\sqrt {-a^{2}+b^{2}}\, f \,a^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{3} d^{3}}{\sqrt {-a^{2}+b^{2}}\, f \,a^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) c^{3}}{\sqrt {-a^{2}+b^{2}}\, f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) b \,c^{2} d}{\sqrt {-a^{2}+b^{2}}\, f a}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) b^{2} c \,d^{2}}{\sqrt {-a^{2}+b^{2}}\, f \,a^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) b^{3} d^{3}}{\sqrt {-a^{2}+b^{2}}\, f \,a^{3}}\)	\(892\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*sec(f*x+e))^3/(a+b*cos(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

1/f*(2*(a^3*c^3-3*a^2*b*c^2*d+3*a*b^2*c*d^2-b^3*d^3)/a^3/((a-b)*(a+b))^(1/2)*arctan((a-b)*tan(1/2*f*x+1/2*e)/(
(a-b)*(a+b))^(1/2))+1/2*d^3/a/(tan(1/2*f*x+1/2*e)-1)^2-1/2*d*(6*a^2*c^2+a^2*d^2-6*a*b*c*d+2*b^2*d^2)/a^3*ln(ta
n(1/2*f*x+1/2*e)-1)-1/2*d^2*(6*a*c-a*d-2*b*d)/a^2/(tan(1/2*f*x+1/2*e)-1)-1/2*d^3/a/(tan(1/2*f*x+1/2*e)+1)^2+1/
2*d*(6*a^2*c^2+a^2*d^2-6*a*b*c*d+2*b^2*d^2)/a^3*ln(tan(1/2*f*x+1/2*e)+1)-1/2*d^2*(6*a*c-a*d-2*b*d)/a^2/(tan(1/
2*f*x+1/2*e)+1))

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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))^3/(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. 349 vs. \(2 (163) = 326\).
time = 23.12, size = 771, normalized size = 4.54 \begin {gather*} \left [\frac {2 \, {\left (a^{3} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a b^{2} c d^{2} - b^{3} d^{3}\right )} \sqrt {-a^{2} + b^{2}} \cos \left (f x + e\right )^{2} \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 ) + {\left (6 \, {\left (a^{4} - a^{2} b^{2}\right )} c^{2} d - 6 \, {\left (a^{3} b - a b^{3}\right )} c d^{2} + {\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (6 \, {\left (a^{4} - a^{2} b^{2}\right )} c^{2} d - 6 \, {\left (a^{3} b - a b^{3}\right )} c d^{2} + {\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left ({\left (a^{4} - a^{2} b^{2}\right )} d^{3} + 2 \, {\left (3 \, {\left (a^{4} - a^{2} b^{2}\right )} c d^{2} - {\left (a^{3} b - a b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, {\left (a^{5} - a^{3} b^{2}\right )} f \cos \left (f x + e\right )^{2}}, \frac {4 \, {\left (a^{3} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a b^{2} c d^{2} - b^{3} d^{3}\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 )^{2} + {\left (6 \, {\left (a^{4} - a^{2} b^{2}\right )} c^{2} d - 6 \, {\left (a^{3} b - a b^{3}\right )} c d^{2} + {\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (6 \, {\left (a^{4} - a^{2} b^{2}\right )} c^{2} d - 6 \, {\left (a^{3} b - a b^{3}\right )} c d^{2} + {\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left ({\left (a^{4} - a^{2} b^{2}\right )} d^{3} + 2 \, {\left (3 \, {\left (a^{4} - a^{2} b^{2}\right )} c d^{2} - {\left (a^{3} b - a b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, {\left (a^{5} - a^{3} b^{2}\right )} f \cos \left (f x + e\right )^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sec(f*x+e))^3/(a+b*cos(f*x+e)),x, algorithm="fricas")

[Out]

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

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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 )^{3}}{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))**3/(a+b*cos(f*x+e)),x)

[Out]

Integral((c + d*sec(e + f*x))**3/(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. 352 vs. \(2 (163) = 326\).
time = 0.55, size = 352, normalized size = 2.07 \begin {gather*} \frac {\frac {{\left (6 \, a^{2} c^{2} d - 6 \, a b c d^{2} + a^{2} d^{3} + 2 \, b^{2} d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac {{\left (6 \, a^{2} c^{2} d - 6 \, a b c d^{2} + a^{2} d^{3} + 2 \, b^{2} d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} - \frac {4 \, {\left (a^{3} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a b^{2} c d^{2} - b^{3} d^{3}\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^{3}} - \frac {2 \, {\left (6 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, b d^{3} \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 )}^{2} a^{2}}}{2 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((6*a^2*c^2*d - 6*a*b*c*d^2 + a^2*d^3 + 2*b^2*d^3)*log(abs(tan(1/2*f*x + 1/2*e) + 1))/a^3 - (6*a^2*c^2*d -
6*a*b*c*d^2 + a^2*d^3 + 2*b^2*d^3)*log(abs(tan(1/2*f*x + 1/2*e) - 1))/a^3 - 4*(a^3*c^3 - 3*a^2*b*c^2*d + 3*a*
b^2*c*d^2 - b^3*d^3)*(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*t
an(1/2*f*x + 1/2*e))/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*a^3) - 2*(6*a*c*d^2*tan(1/2*f*x + 1/2*e)^3 - a*d^3*tan
(1/2*f*x + 1/2*e)^3 - 2*b*d^3*tan(1/2*f*x + 1/2*e)^3 - 6*a*c*d^2*tan(1/2*f*x + 1/2*e) - a*d^3*tan(1/2*f*x + 1/
2*e) + 2*b*d^3*tan(1/2*f*x + 1/2*e))/((tan(1/2*f*x + 1/2*e)^2 - 1)^2*a^2))/f

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Mupad [B]
time = 9.93, size = 2500, normalized size = 14.71 \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))^3/(a + b*cos(e + f*x)),x)

[Out]

((tan(e/2 + (f*x)/2)*(a*d^3 - 2*b*d^3 + 6*a*c*d^2))/a^2 + (tan(e/2 + (f*x)/2)^3*(a*d^3 + 2*b*d^3 - 6*a*c*d^2))
/a^2)/(f*(tan(e/2 + (f*x)/2)^4 - 2*tan(e/2 + (f*x)/2)^2 + 1)) + (atan(((((8*tan(e/2 + (f*x)/2)*(4*a^7*c^6 + a^
7*d^6 - 8*b^7*d^6 - 4*a^6*b*c^6 + 16*a*b^6*d^6 - 3*a^6*b*d^6 - 16*a^2*b^5*d^6 + 16*a^3*b^4*d^6 - 13*a^4*b^3*d^
6 + 7*a^5*b^2*d^6 + 12*a^7*c^2*d^4 + 36*a^7*c^4*d^2 - 96*a^2*b^5*c*d^5 + 84*a^3*b^4*c*d^5 - 60*a^4*b^3*c*d^5 +
36*a^5*b^2*c*d^5 + 24*a^5*b^2*c^5*d - 36*a^6*b*c^2*d^4 - 72*a^6*b*c^3*d^3 - 108*a^6*b*c^4*d^2 - 120*a^2*b^5*c
^2*d^4 + 240*a^3*b^4*c^2*d^4 + 152*a^3*b^4*c^3*d^3 - 192*a^4*b^3*c^2*d^4 - 296*a^4*b^3*c^3*d^3 - 96*a^4*b^3*c^
4*d^2 + 96*a^5*b^2*c^2*d^4 + 216*a^5*b^2*c^3*d^3 + 168*a^5*b^2*c^4*d^2 + 48*a*b^6*c*d^5 - 12*a^6*b*c*d^5 - 24*
a^6*b*c^5*d))/a^4 + (((8*(4*a^10*c^3 + 2*a^10*d^3 - 8*a^9*b*c^3 - 2*a^9*b*d^3 + 12*a^10*c^2*d + 4*a^8*b^2*c^3
+ 4*a^6*b^4*d^3 - 6*a^7*b^3*d^3 + 2*a^8*b^2*d^3 - 12*a^7*b^3*c*d^2 + 24*a^8*b^2*c*d^2 + 12*a^8*b^2*c^2*d - 12*
a^9*b*c*d^2 - 24*a^9*b*c^2*d))/a^6 + (8*tan(e/2 + (f*x)/2)*(8*a^8*b + 8*a^6*b^3 - 16*a^7*b^2)*(a^2*(3*c^2*d +
d^3/2) + b^2*d^3 - 3*a*b*c*d^2))/a^7)*(a^2*(3*c^2*d + d^3/2) + b^2*d^3 - 3*a*b*c*d^2))/a^3)*(a^2*(3*c^2*d + d^
3/2) + b^2*d^3 - 3*a*b*c*d^2)*1i)/a^3 + (((8*tan(e/2 + (f*x)/2)*(4*a^7*c^6 + a^7*d^6 - 8*b^7*d^6 - 4*a^6*b*c^6
+ 16*a*b^6*d^6 - 3*a^6*b*d^6 - 16*a^2*b^5*d^6 + 16*a^3*b^4*d^6 - 13*a^4*b^3*d^6 + 7*a^5*b^2*d^6 + 12*a^7*c^2*
d^4 + 36*a^7*c^4*d^2 - 96*a^2*b^5*c*d^5 + 84*a^3*b^4*c*d^5 - 60*a^4*b^3*c*d^5 + 36*a^5*b^2*c*d^5 + 24*a^5*b^2*
c^5*d - 36*a^6*b*c^2*d^4 - 72*a^6*b*c^3*d^3 - 108*a^6*b*c^4*d^2 - 120*a^2*b^5*c^2*d^4 + 240*a^3*b^4*c^2*d^4 +
152*a^3*b^4*c^3*d^3 - 192*a^4*b^3*c^2*d^4 - 296*a^4*b^3*c^3*d^3 - 96*a^4*b^3*c^4*d^2 + 96*a^5*b^2*c^2*d^4 + 21
6*a^5*b^2*c^3*d^3 + 168*a^5*b^2*c^4*d^2 + 48*a*b^6*c*d^5 - 12*a^6*b*c*d^5 - 24*a^6*b*c^5*d))/a^4 - (((8*(4*a^1
0*c^3 + 2*a^10*d^3 - 8*a^9*b*c^3 - 2*a^9*b*d^3 + 12*a^10*c^2*d + 4*a^8*b^2*c^3 + 4*a^6*b^4*d^3 - 6*a^7*b^3*d^3
+ 2*a^8*b^2*d^3 - 12*a^7*b^3*c*d^2 + 24*a^8*b^2*c*d^2 + 12*a^8*b^2*c^2*d - 12*a^9*b*c*d^2 - 24*a^9*b*c^2*d))/
a^6 - (8*tan(e/2 + (f*x)/2)*(8*a^8*b + 8*a^6*b^3 - 16*a^7*b^2)*(a^2*(3*c^2*d + d^3/2) + b^2*d^3 - 3*a*b*c*d^2)
)/a^7)*(a^2*(3*c^2*d + d^3/2) + b^2*d^3 - 3*a*b*c*d^2))/a^3)*(a^2*(3*c^2*d + d^3/2) + b^2*d^3 - 3*a*b*c*d^2)*1
i)/a^3)/((16*(4*b^8*d^9 - 6*a*b^7*d^9 - 12*a^8*c^8*d + 6*a^2*b^6*d^9 - 5*a^3*b^5*d^9 + 2*a^4*b^4*d^9 - a^5*b^3
*d^9 + a^8*c^3*d^6 + 12*a^8*c^5*d^4 - 2*a^8*c^6*d^3 + 36*a^8*c^7*d^2 + 48*a^2*b^6*c*d^8 - 36*a^3*b^5*c*d^8 + 2
7*a^4*b^4*c*d^8 - 6*a^5*b^3*c*d^8 + 3*a^6*b^2*c*d^8 - 3*a^7*b*c^2*d^7 - 2*a^7*b*c^3*d^6 - 48*a^7*b*c^4*d^5 - 1
2*a^7*b*c^5*d^4 - 178*a^7*b*c^6*d^3 + 12*a^7*b*c^7*d^2 + 144*a^2*b^6*c^2*d^7 - 4*a^2*b^6*c^3*d^6 - 174*a^3*b^5
*c^2*d^7 - 324*a^3*b^5*c^3*d^6 + 24*a^3*b^5*c^4*d^5 + 90*a^4*b^4*c^2*d^7 + 364*a^4*b^4*c^3*d^6 + 432*a^4*b^4*c
^4*d^5 - 60*a^4*b^4*c^5*d^4 - 63*a^5*b^3*c^2*d^7 - 112*a^5*b^3*c^3*d^6 - 474*a^5*b^3*c^4*d^5 - 324*a^5*b^3*c^5
*d^4 + 76*a^5*b^3*c^6*d^3 + 6*a^6*b^2*c^2*d^7 + 77*a^6*b^2*c^3*d^6 + 66*a^6*b^2*c^4*d^5 + 384*a^6*b^2*c^5*d^4
+ 104*a^6*b^2*c^6*d^3 - 48*a^6*b^2*c^7*d^2 - 36*a*b^7*c*d^8 + 12*a^7*b*c^8*d))/a^6 - (((8*tan(e/2 + (f*x)/2)*(
4*a^7*c^6 + a^7*d^6 - 8*b^7*d^6 - 4*a^6*b*c^6 + 16*a*b^6*d^6 - 3*a^6*b*d^6 - 16*a^2*b^5*d^6 + 16*a^3*b^4*d^6 -
13*a^4*b^3*d^6 + 7*a^5*b^2*d^6 + 12*a^7*c^2*d^4 + 36*a^7*c^4*d^2 - 96*a^2*b^5*c*d^5 + 84*a^3*b^4*c*d^5 - 60*a
^4*b^3*c*d^5 + 36*a^5*b^2*c*d^5 + 24*a^5*b^2*c^5*d - 36*a^6*b*c^2*d^4 - 72*a^6*b*c^3*d^3 - 108*a^6*b*c^4*d^2 -
120*a^2*b^5*c^2*d^4 + 240*a^3*b^4*c^2*d^4 + 152*a^3*b^4*c^3*d^3 - 192*a^4*b^3*c^2*d^4 - 296*a^4*b^3*c^3*d^3 -
96*a^4*b^3*c^4*d^2 + 96*a^5*b^2*c^2*d^4 + 216*a^5*b^2*c^3*d^3 + 168*a^5*b^2*c^4*d^2 + 48*a*b^6*c*d^5 - 12*a^6
*b*c*d^5 - 24*a^6*b*c^5*d))/a^4 + (((8*(4*a^10*c^3 + 2*a^10*d^3 - 8*a^9*b*c^3 - 2*a^9*b*d^3 + 12*a^10*c^2*d +
4*a^8*b^2*c^3 + 4*a^6*b^4*d^3 - 6*a^7*b^3*d^3 + 2*a^8*b^2*d^3 - 12*a^7*b^3*c*d^2 + 24*a^8*b^2*c*d^2 + 12*a^8*b
^2*c^2*d - 12*a^9*b*c*d^2 - 24*a^9*b*c^2*d))/a^6 + (8*tan(e/2 + (f*x)/2)*(8*a^8*b + 8*a^6*b^3 - 16*a^7*b^2)*(a
^2*(3*c^2*d + d^3/2) + b^2*d^3 - 3*a*b*c*d^2))/a^7)*(a^2*(3*c^2*d + d^3/2) + b^2*d^3 - 3*a*b*c*d^2))/a^3)*(a^2
*(3*c^2*d + d^3/2) + b^2*d^3 - 3*a*b*c*d^2))/a^3 + (((8*tan(e/2 + (f*x)/2)*(4*a^7*c^6 + a^7*d^6 - 8*b^7*d^6 -
4*a^6*b*c^6 + 16*a*b^6*d^6 - 3*a^6*b*d^6 - 16*a^2*b^5*d^6 + 16*a^3*b^4*d^6 - 13*a^4*b^3*d^6 + 7*a^5*b^2*d^6 +
12*a^7*c^2*d^4 + 36*a^7*c^4*d^2 - 96*a^2*b^5*c*d^5 + 84*a^3*b^4*c*d^5 - 60*a^4*b^3*c*d^5 + 36*a^5*b^2*c*d^5 +
24*a^5*b^2*c^5*d - 36*a^6*b*c^2*d^4 - 72*a^6*b*c^3*d^3 - 108*a^6*b*c^4*d^2 - 120*a^2*b^5*c^2*d^4 + 240*a^3*b^4
*c^2*d^4 + 152*a^3*b^4*c^3*d^3 - 192*a^4*b^3*c^2*d^4 - 296*a^4*b^3*c^3*d^3 - 96*a^4*b^3*c^4*d^2 + 96*a^5*b^2*c
^2*d^4 + 216*a^5*b^2*c^3*d^3 + 168*a^5*b^2*c^4*d^2 + 48*a*b^6*c*d^5 - 12*a^6*b*c*d^5 - 24*a^6*b*c^5*d))/a^4 -
(((8*(4*a^10*c^3 + 2*a^10*d^3 - 8*a^9*b*c^3 - 2*a^9*b*d^3 + 12*a^10*c^2*d + 4*a^8*b^2*c^3 + 4*a^6*b^4*d^3 - 6*
a^7*b^3*d^3 + 2*a^8*b^2*d^3 - 12*a^7*b^3*c*d^2 ...

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