3.1.14 \(\int \frac {1}{(a+b \cos (e+f x)) (c+d \sec (e+f x))^2} \, dx\) [14]

Optimal. Leaf size=187 \[ \frac {2 a^2 \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b} (a c-b d)^2 f}-\frac {2 d \left (2 a c^2-b c d-a d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{(c-d)^{3/2} (c+d)^{3/2} (a c-b d)^2 f}+\frac {d^2 \sin (e+f x)}{(a c-b d) \left (c^2-d^2\right ) f (d+c \cos (e+f x))} \]

[Out]

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

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Rubi [A]
time = 0.45, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2907, 3135, 3080, 2738, 211, 214} \begin {gather*} \frac {2 a^2 \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{f \sqrt {a-b} \sqrt {a+b} (a c-b d)^2}+\frac {d^2 \sin (e+f x)}{f \left (c^2-d^2\right ) (a c-b d) (c \cos (e+f x)+d)}-\frac {2 d \left (2 a c^2-a d^2-b c d\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{f (c-d)^{3/2} (c+d)^{3/2} (a c-b d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^2),x]

[Out]

(2*a^2*ArcTan[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a*c - b*d)^2*f) - (2*d*(2
*a*c^2 - b*c*d - a*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(3/2)*(c + d)^(3/2)*(a*c
- b*d)^2*f) + (d^2*Sin[e + f*x])/((a*c - b*d)*(c^2 - d^2)*f*(d + c*Cos[e + f*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 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 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 3080

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3135

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c
+ d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C
)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n +
3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && L
tQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rubi steps

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

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Mathematica [A]
time = 0.89, size = 205, normalized size = 1.10 \begin {gather*} \frac {(d+c \cos (e+f x)) \sec ^2(e+f x) \left (-\frac {2 a^2 \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {-a^2+b^2}}\right ) (d+c \cos (e+f x))}{\sqrt {-a^2+b^2}}-\frac {2 d \left (b c d+a \left (-2 c^2+d^2\right )\right ) \tanh ^{-1}\left (\frac {(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right ) (d+c \cos (e+f x))}{\left (c^2-d^2\right )^{3/2}}+\frac {d^2 (a c-b d) \sin (e+f x)}{(c-d) (c+d)}\right )}{(a c-b d)^2 f (c+d \sec (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^2),x]

[Out]

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

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Maple [A]
time = 1.13, size = 210, normalized size = 1.12

method	result	size
derivativedivides	\(\frac {\frac {2 a^{2} \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a c -b d \right )^{2} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {2 d \left (-\frac {d \left (a c -b d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-d^{2}\right ) \left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )}-\frac {\left (2 a \,c^{2}-d^{2} a -b c d \right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \left (c -d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (a c -b d \right )^{2}}}{f}\)	\(210\)
default	\(\frac {\frac {2 a^{2} \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a c -b d \right )^{2} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {2 d \left (-\frac {d \left (a c -b d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-d^{2}\right ) \left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )}-\frac {\left (2 a \,c^{2}-d^{2} a -b c d \right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \left (c -d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (a c -b d \right )^{2}}}{f}\)	\(210\)
risch	\(\frac {2 i d^{2} \left (d \,{\mathrm e}^{i \left (f x +e \right )}+c \right )}{c \left (c^{2}-d^{2}\right ) \left (a c -b d \right ) f \left (c \,{\mathrm e}^{2 i \left (f x +e \right )}+2 d \,{\mathrm e}^{i \left (f x +e \right )}+c \right )}-\frac {a^{2} \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 )}{\sqrt {-a^{2}+b^{2}}\, \left (a c -b d \right )^{2} f}+\frac {a^{2} \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 )}{\sqrt {-a^{2}+b^{2}}\, \left (a c -b d \right )^{2} f}+\frac {2 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-d \sqrt {c^{2}-d^{2}}}{\sqrt {c^{2}-d^{2}}\, c}\right ) a \,c^{2}}{\sqrt {c^{2}-d^{2}}\, \left (a c -b d \right )^{2} \left (c +d \right ) \left (c -d \right ) f}-\frac {d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-d \sqrt {c^{2}-d^{2}}}{\sqrt {c^{2}-d^{2}}\, c}\right ) a}{\sqrt {c^{2}-d^{2}}\, \left (a c -b d \right )^{2} \left (c +d \right ) \left (c -d \right ) f}-\frac {d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-d \sqrt {c^{2}-d^{2}}}{\sqrt {c^{2}-d^{2}}\, c}\right ) b c}{\sqrt {c^{2}-d^{2}}\, \left (a c -b d \right )^{2} \left (c +d \right ) \left (c -d \right ) f}-\frac {2 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+d \sqrt {c^{2}-d^{2}}}{\sqrt {c^{2}-d^{2}}\, c}\right ) a \,c^{2}}{\sqrt {c^{2}-d^{2}}\, \left (a c -b d \right )^{2} \left (c +d \right ) \left (c -d \right ) f}+\frac {d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+d \sqrt {c^{2}-d^{2}}}{\sqrt {c^{2}-d^{2}}\, c}\right ) a}{\sqrt {c^{2}-d^{2}}\, \left (a c -b d \right )^{2} \left (c +d \right ) \left (c -d \right ) f}+\frac {d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+d \sqrt {c^{2}-d^{2}}}{\sqrt {c^{2}-d^{2}}\, c}\right ) b c}{\sqrt {c^{2}-d^{2}}\, \left (a c -b d \right )^{2} \left (c +d \right ) \left (c -d \right ) f}\)	\(810\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cos(f*x+e))/(c+d*sec(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

1/f*(2*a^2/(a*c-b*d)^2/((a-b)*(a+b))^(1/2)*arctan((a-b)*tan(1/2*f*x+1/2*e)/((a-b)*(a+b))^(1/2))+2*d/(a*c-b*d)^
2*(-d*(a*c-b*d)/(c^2-d^2)*tan(1/2*f*x+1/2*e)/(c*tan(1/2*f*x+1/2*e)^2-d*tan(1/2*f*x+1/2*e)^2-c-d)-(2*a*c^2-a*d^
2-b*c*d)/(c+d)/(c-d)/((c+d)*(c-d))^(1/2)*arctanh((c-d)*tan(1/2*f*x+1/2*e)/((c+d)*(c-d))^(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(1/(a+b*cos(f*x+e))/(c+d*sec(f*x+e))^2,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. 653 vs. \(2 (173) = 346\).
time = 85.13, size = 2883, normalized size = 15.42 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cos(f*x+e))/(c+d*sec(f*x+e))^2,x, algorithm="fricas")

[Out]

[-1/2*((a^2*c^4*d - 2*a^2*c^2*d^3 + a^2*d^5 + (a^2*c^5 - 2*a^2*c^3*d^2 + a^2*c*d^4)*cos(f*x + e))*sqrt(-a^2 +
b^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)) - (2*(a^3 - a*b^2)*c^2*d^2 - (a^2*b - b^3
)*c*d^3 - (a^3 - a*b^2)*d^4 + (2*(a^3 - a*b^2)*c^3*d - (a^2*b - b^3)*c^2*d^2 - (a^3 - a*b^2)*c*d^3)*cos(f*x +
e))*sqrt(c^2 - d^2)*log((2*c*d*cos(f*x + e) - (c^2 - 2*d^2)*cos(f*x + e)^2 - 2*sqrt(c^2 - d^2)*(d*cos(f*x + e)
+ c)*sin(f*x + e) + 2*c^2 - d^2)/(c^2*cos(f*x + e)^2 + 2*c*d*cos(f*x + e) + d^2)) - 2*((a^3 - a*b^2)*c^3*d^2
- (a^2*b - b^3)*c^2*d^3 - (a^3 - a*b^2)*c*d^4 + (a^2*b - b^3)*d^5)*sin(f*x + e))/(((a^4 - a^2*b^2)*c^7 - 2*(a^
3*b - a*b^3)*c^6*d - (2*a^4 - 3*a^2*b^2 + b^4)*c^5*d^2 + 4*(a^3*b - a*b^3)*c^4*d^3 + (a^4 - 3*a^2*b^2 + 2*b^4)
*c^3*d^4 - 2*(a^3*b - a*b^3)*c^2*d^5 + (a^2*b^2 - b^4)*c*d^6)*f*cos(f*x + e) + ((a^4 - a^2*b^2)*c^6*d - 2*(a^3
*b - a*b^3)*c^5*d^2 - (2*a^4 - 3*a^2*b^2 + b^4)*c^4*d^3 + 4*(a^3*b - a*b^3)*c^3*d^4 + (a^4 - 3*a^2*b^2 + 2*b^4
)*c^2*d^5 - 2*(a^3*b - a*b^3)*c*d^6 + (a^2*b^2 - b^4)*d^7)*f), -1/2*(2*(2*(a^3 - a*b^2)*c^2*d^2 - (a^2*b - b^3
)*c*d^3 - (a^3 - a*b^2)*d^4 + (2*(a^3 - a*b^2)*c^3*d - (a^2*b - b^3)*c^2*d^2 - (a^3 - a*b^2)*c*d^3)*cos(f*x +
e))*sqrt(-c^2 + d^2)*arctan(-sqrt(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*sin(f*x + e))) + (a^2*c^4*d -
2*a^2*c^2*d^3 + a^2*d^5 + (a^2*c^5 - 2*a^2*c^3*d^2 + a^2*c*d^4)*cos(f*x + e))*sqrt(-a^2 + b^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)) - 2*((a^3 - a*b^2)*c^3*d^2 - (a^2*b - b^3)*c^2*d^3 - (a^3 - a
*b^2)*c*d^4 + (a^2*b - b^3)*d^5)*sin(f*x + e))/(((a^4 - a^2*b^2)*c^7 - 2*(a^3*b - a*b^3)*c^6*d - (2*a^4 - 3*a^
2*b^2 + b^4)*c^5*d^2 + 4*(a^3*b - a*b^3)*c^4*d^3 + (a^4 - 3*a^2*b^2 + 2*b^4)*c^3*d^4 - 2*(a^3*b - a*b^3)*c^2*d
^5 + (a^2*b^2 - b^4)*c*d^6)*f*cos(f*x + e) + ((a^4 - a^2*b^2)*c^6*d - 2*(a^3*b - a*b^3)*c^5*d^2 - (2*a^4 - 3*a
^2*b^2 + b^4)*c^4*d^3 + 4*(a^3*b - a*b^3)*c^3*d^4 + (a^4 - 3*a^2*b^2 + 2*b^4)*c^2*d^5 - 2*(a^3*b - a*b^3)*c*d^
6 + (a^2*b^2 - b^4)*d^7)*f), 1/2*(2*(a^2*c^4*d - 2*a^2*c^2*d^3 + a^2*d^5 + (a^2*c^5 - 2*a^2*c^3*d^2 + a^2*c*d^
4)*cos(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*cos(f*x + e) + b)/(sqrt(a^2 - b^2)*sin(f*x + e))) + (2*(a^3 - a*b^
2)*c^2*d^2 - (a^2*b - b^3)*c*d^3 - (a^3 - a*b^2)*d^4 + (2*(a^3 - a*b^2)*c^3*d - (a^2*b - b^3)*c^2*d^2 - (a^3 -
a*b^2)*c*d^3)*cos(f*x + e))*sqrt(c^2 - d^2)*log((2*c*d*cos(f*x + e) - (c^2 - 2*d^2)*cos(f*x + e)^2 - 2*sqrt(c
^2 - d^2)*(d*cos(f*x + e) + c)*sin(f*x + e) + 2*c^2 - d^2)/(c^2*cos(f*x + e)^2 + 2*c*d*cos(f*x + e) + d^2)) +
2*((a^3 - a*b^2)*c^3*d^2 - (a^2*b - b^3)*c^2*d^3 - (a^3 - a*b^2)*c*d^4 + (a^2*b - b^3)*d^5)*sin(f*x + e))/(((a
^4 - a^2*b^2)*c^7 - 2*(a^3*b - a*b^3)*c^6*d - (2*a^4 - 3*a^2*b^2 + b^4)*c^5*d^2 + 4*(a^3*b - a*b^3)*c^4*d^3 +
(a^4 - 3*a^2*b^2 + 2*b^4)*c^3*d^4 - 2*(a^3*b - a*b^3)*c^2*d^5 + (a^2*b^2 - b^4)*c*d^6)*f*cos(f*x + e) + ((a^4
- a^2*b^2)*c^6*d - 2*(a^3*b - a*b^3)*c^5*d^2 - (2*a^4 - 3*a^2*b^2 + b^4)*c^4*d^3 + 4*(a^3*b - a*b^3)*c^3*d^4 +
(a^4 - 3*a^2*b^2 + 2*b^4)*c^2*d^5 - 2*(a^3*b - a*b^3)*c*d^6 + (a^2*b^2 - b^4)*d^7)*f), ((a^2*c^4*d - 2*a^2*c^
2*d^3 + a^2*d^5 + (a^2*c^5 - 2*a^2*c^3*d^2 + a^2*c*d^4)*cos(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*cos(f*x + e)
+ b)/(sqrt(a^2 - b^2)*sin(f*x + e))) - (2*(a^3 - a*b^2)*c^2*d^2 - (a^2*b - b^3)*c*d^3 - (a^3 - a*b^2)*d^4 + (2
*(a^3 - a*b^2)*c^3*d - (a^2*b - b^3)*c^2*d^2 - (a^3 - a*b^2)*c*d^3)*cos(f*x + e))*sqrt(-c^2 + d^2)*arctan(-sqr
t(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*sin(f*x + e))) + ((a^3 - a*b^2)*c^3*d^2 - (a^2*b - b^3)*c^2*d^
3 - (a^3 - a*b^2)*c*d^4 + (a^2*b - b^3)*d^5)*sin(f*x + e))/(((a^4 - a^2*b^2)*c^7 - 2*(a^3*b - a*b^3)*c^6*d - (
2*a^4 - 3*a^2*b^2 + b^4)*c^5*d^2 + 4*(a^3*b - a*b^3)*c^4*d^3 + (a^4 - 3*a^2*b^2 + 2*b^4)*c^3*d^4 - 2*(a^3*b -
a*b^3)*c^2*d^5 + (a^2*b^2 - b^4)*c*d^6)*f*cos(f*x + e) + ((a^4 - a^2*b^2)*c^6*d - 2*(a^3*b - a*b^3)*c^5*d^2 -
(2*a^4 - 3*a^2*b^2 + b^4)*c^4*d^3 + 4*(a^3*b - a*b^3)*c^3*d^4 + (a^4 - 3*a^2*b^2 + 2*b^4)*c^2*d^5 - 2*(a^3*b -
a*b^3)*c*d^6 + (a^2*b^2 - b^4)*d^7)*f)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \cos {\left (e + f x \right )}\right ) \left (c + d \sec {\left (e + f x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cos(f*x+e))/(c+d*sec(f*x+e))**2,x)

[Out]

Integral(1/((a + b*cos(e + f*x))*(c + d*sec(e + f*x))**2), x)

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Giac [A]
time = 0.50, size = 342, normalized size = 1.83 \begin {gather*} \frac {2 \, {\left (\frac {{\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 )} a^{2}}{{\left (a^{2} c^{2} - 2 \, a b c d + b^{2} d^{2}\right )} \sqrt {a^{2} - b^{2}}} - \frac {d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a c^{3} - b c^{2} d - a c d^{2} + b d^{3}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}} - \frac {{\left (2 \, a c^{2} d - b c d^{2} - a d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, c + 2 \, d\right ) + \arctan \left (-\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (a^{2} c^{4} - 2 \, a b c^{3} d - a^{2} c^{2} d^{2} + b^{2} c^{2} d^{2} + 2 \, a b c d^{3} - b^{2} d^{4}\right )} \sqrt {-c^{2} + d^{2}}}\right )}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cos(f*x+e))/(c+d*sec(f*x+e))^2,x, algorithm="giac")

[Out]

2*((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)))*a^2/((a^2*c^2 - 2*a*b*c*d + b^2*d^2)*sqrt(a^2 - b^2)) - d^2*tan(1/2*f*x + 1/2*e)/((a*c^3 -
b*c^2*d - a*c*d^2 + b*d^3)*(c*tan(1/2*f*x + 1/2*e)^2 - d*tan(1/2*f*x + 1/2*e)^2 - c - d)) - (2*a*c^2*d - b*c*d
^2 - a*d^3)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(-2*c + 2*d) + arctan(-(c*tan(1/2*f*x + 1/2*e) - d*tan(1/2*f*
x + 1/2*e))/sqrt(-c^2 + d^2)))/((a^2*c^4 - 2*a*b*c^3*d - a^2*c^2*d^2 + b^2*c^2*d^2 + 2*a*b*c*d^3 - b^2*d^4)*sq
rt(-c^2 + d^2)))/f

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Mupad [B]
time = 15.23, size = 2500, normalized size = 13.37 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c + d/cos(e + f*x))^2*(a + b*cos(e + f*x))),x)

[Out]

(a^2*atan(((a^2*(b^2 - a^2)^(1/2)*((32*tan(e/2 + (f*x)/2)*(a^5*c^6 + 2*a^5*d^6 - a^4*b*c^6 - 4*a^4*b*d^6 - 2*a
^5*c*d^5 - 2*a^5*c^5*d - a^2*b^3*d^6 + 3*a^3*b^2*d^6 - 5*a^5*c^2*d^4 + 4*a^5*c^3*d^3 + 3*a^5*c^4*d^2 - b^5*c^2
*d^4 + 3*a*b^4*c^2*d^4 + 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c*d^5 - 6*a^3*b^2*c*d^5 + 13*a^4*b*c^2*d^4 - 8*a^4*b*c^3*
d^3 - 11*a^4*b*c^4*d^2 + a^2*b^3*c^2*d^4 - 12*a^2*b^3*c^3*d^3 - 4*a^2*b^3*c^4*d^2 - 11*a^3*b^2*c^2*d^4 + 12*a^
3*b^2*c^3*d^3 + 12*a^3*b^2*c^4*d^2 - 2*a*b^4*c*d^5 + 4*a^4*b*c*d^5 + 2*a^4*b*c^5*d))/(a^2*c^5 - b^2*d^5 + a^2*
c^4*d - b^2*c*d^4 - a^2*c^2*d^3 - a^2*c^3*d^2 + b^2*c^2*d^3 + b^2*c^3*d^2 + 2*a*b*c*d^4 - 2*a*b*c^4*d + 2*a*b*
c^2*d^3 - 2*a*b*c^3*d^2) + (a^2*(b^2 - a^2)^(1/2)*((32*(2*a^6*b*c^9 - a^7*c^9 + a*b^6*d^9 + 2*a^7*c^8*d + b^7*
c*d^8 - a^5*b^2*c^9 - 2*a^2*b^5*d^9 + a^3*b^4*d^9 + a^7*c^4*d^5 - 3*a^7*c^6*d^3 + a^7*c^7*d^2 - b^7*c^2*d^7 -
b^7*c^3*d^6 + b^7*c^4*d^5 - 5*a*b^6*c^2*d^7 + 7*a*b^6*c^3*d^6 + 4*a*b^6*c^4*d^5 - 5*a*b^6*c^5*d^4 - 3*a^2*b^5*
c*d^8 + 8*a^3*b^4*c*d^8 - 4*a^4*b^3*c*d^8 + 5*a^4*b^3*c^8*d - 8*a^5*b^2*c^8*d - 4*a^6*b*c^3*d^6 - 2*a^6*b*c^4*
d^5 + 13*a^6*b*c^5*d^4 + a^6*b*c^6*d^3 - 11*a^6*b*c^7*d^2 + 13*a^2*b^5*c^2*d^7 + 7*a^2*b^5*c^3*d^6 - 21*a^2*b^
5*c^4*d^5 - 4*a^2*b^5*c^5*d^4 + 10*a^2*b^5*c^6*d^3 - a^3*b^4*c^2*d^7 - 31*a^3*b^4*c^3*d^6 + 4*a^3*b^4*c^4*d^5
+ 33*a^3*b^4*c^5*d^4 - 4*a^3*b^4*c^6*d^3 - 10*a^3*b^4*c^7*d^2 - 12*a^4*b^3*c^2*d^7 + 14*a^4*b^3*c^3*d^6 + 34*a
^4*b^3*c^4*d^5 - 21*a^4*b^3*c^5*d^4 - 27*a^4*b^3*c^6*d^3 + 11*a^4*b^3*c^7*d^2 + 6*a^5*b^2*c^2*d^7 + 8*a^5*b^2*
c^3*d^6 - 21*a^5*b^2*c^4*d^5 - 16*a^5*b^2*c^5*d^4 + 23*a^5*b^2*c^6*d^3 + 9*a^5*b^2*c^7*d^2 - 2*a*b^6*c*d^8 + a
^6*b*c^8*d))/(a^3*c^6 + b^3*d^6 + a^3*c^5*d + b^3*c*d^5 - a^3*c^3*d^3 - a^3*c^4*d^2 - b^3*c^2*d^4 - b^3*c^3*d^
3 - 3*a*b^2*c^2*d^4 + 3*a*b^2*c^3*d^3 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + 3*a^2*b*c^3*d^3 - 3*a^2*b*c^4*d^2
- 3*a*b^2*c*d^5 - 3*a^2*b*c^5*d) + (32*a^2*tan(e/2 + (f*x)/2)*(b^2 - a^2)^(1/2)*(2*a^6*b*c^10 + 2*a*b^6*d^10 -
2*a^7*c^9*d - 2*b^7*c*d^9 + 2*a^4*b^3*c^10 - 4*a^5*b^2*c^10 - 4*a^2*b^5*d^10 + 2*a^3*b^4*d^10 + 2*a^7*c^4*d^6
- 2*a^7*c^5*d^5 - 4*a^7*c^6*d^4 + 4*a^7*c^7*d^3 + 2*a^7*c^8*d^2 + 2*b^7*c^2*d^8 + 4*b^7*c^3*d^7 - 4*b^7*c^4*d
^6 - 2*b^7*c^5*d^5 + 2*b^7*c^6*d^4 - 12*a*b^6*c^3*d^7 - 6*a*b^6*c^4*d^6 + 18*a*b^6*c^5*d^5 + 4*a*b^6*c^6*d^4 -
8*a*b^6*c^7*d^3 - 6*a^2*b^5*c*d^9 + 14*a^3*b^4*c*d^9 - 8*a^3*b^4*c^9*d - 8*a^4*b^3*c*d^9 + 14*a^4*b^3*c^9*d -
6*a^5*b^2*c^9*d - 8*a^6*b*c^3*d^7 + 4*a^6*b*c^4*d^6 + 18*a^6*b*c^5*d^5 - 6*a^6*b*c^6*d^4 - 12*a^6*b*c^7*d^3 +
2*a^2*b^5*c^2*d^8 + 16*a^2*b^5*c^3*d^7 + 20*a^2*b^5*c^4*d^6 - 14*a^2*b^5*c^5*d^5 - 30*a^2*b^5*c^6*d^4 + 4*a^2
*b^5*c^7*d^3 + 12*a^2*b^5*c^8*d^2 - 24*a^3*b^4*c^3*d^7 - 22*a^3*b^4*c^4*d^6 - 2*a^3*b^4*c^5*d^5 + 36*a^3*b^4*c
^6*d^4 + 20*a^3*b^4*c^7*d^3 - 16*a^3*b^4*c^8*d^2 - 16*a^4*b^3*c^2*d^8 + 20*a^4*b^3*c^3*d^7 + 36*a^4*b^3*c^4*d^
6 - 2*a^4*b^3*c^5*d^5 - 22*a^4*b^3*c^6*d^4 - 24*a^4*b^3*c^7*d^3 + 12*a^5*b^2*c^2*d^8 + 4*a^5*b^2*c^3*d^7 - 30*
a^5*b^2*c^4*d^6 - 14*a^5*b^2*c^5*d^5 + 20*a^5*b^2*c^6*d^4 + 16*a^5*b^2*c^7*d^3 + 2*a^5*b^2*c^8*d^2 + 2*a*b^6*c
*d^9 + 2*a^6*b*c^9*d))/((a^4*c^2 - b^4*d^2 - a^2*b^2*c^2 + a^2*b^2*d^2 + 2*a*b^3*c*d - 2*a^3*b*c*d)*(a^2*c^5 -
b^2*d^5 + a^2*c^4*d - b^2*c*d^4 - a^2*c^2*d^3 - a^2*c^3*d^2 + b^2*c^2*d^3 + b^2*c^3*d^2 + 2*a*b*c*d^4 - 2*a*b
*c^4*d + 2*a*b*c^2*d^3 - 2*a*b*c^3*d^2))))/(a^4*c^2 - b^4*d^2 - a^2*b^2*c^2 + a^2*b^2*d^2 + 2*a*b^3*c*d - 2*a^
3*b*c*d))*1i)/(a^4*c^2 - b^4*d^2 - a^2*b^2*c^2 + a^2*b^2*d^2 + 2*a*b^3*c*d - 2*a^3*b*c*d) + (a^2*(b^2 - a^2)^(
1/2)*((32*tan(e/2 + (f*x)/2)*(a^5*c^6 + 2*a^5*d^6 - a^4*b*c^6 - 4*a^4*b*d^6 - 2*a^5*c*d^5 - 2*a^5*c^5*d - a^2*
b^3*d^6 + 3*a^3*b^2*d^6 - 5*a^5*c^2*d^4 + 4*a^5*c^3*d^3 + 3*a^5*c^4*d^2 - b^5*c^2*d^4 + 3*a*b^4*c^2*d^4 + 4*a*
b^4*c^3*d^3 + 6*a^2*b^3*c*d^5 - 6*a^3*b^2*c*d^5 + 13*a^4*b*c^2*d^4 - 8*a^4*b*c^3*d^3 - 11*a^4*b*c^4*d^2 + a^2*
b^3*c^2*d^4 - 12*a^2*b^3*c^3*d^3 - 4*a^2*b^3*c^4*d^2 - 11*a^3*b^2*c^2*d^4 + 12*a^3*b^2*c^3*d^3 + 12*a^3*b^2*c^
4*d^2 - 2*a*b^4*c*d^5 + 4*a^4*b*c*d^5 + 2*a^4*b*c^5*d))/(a^2*c^5 - b^2*d^5 + a^2*c^4*d - b^2*c*d^4 - a^2*c^2*d
^3 - a^2*c^3*d^2 + b^2*c^2*d^3 + b^2*c^3*d^2 + 2*a*b*c*d^4 - 2*a*b*c^4*d + 2*a*b*c^2*d^3 - 2*a*b*c^3*d^2) - (a
^2*(b^2 - a^2)^(1/2)*((32*(2*a^6*b*c^9 - a^7*c^9 + a*b^6*d^9 + 2*a^7*c^8*d + b^7*c*d^8 - a^5*b^2*c^9 - 2*a^2*b
^5*d^9 + a^3*b^4*d^9 + a^7*c^4*d^5 - 3*a^7*c^6*d^3 + a^7*c^7*d^2 - b^7*c^2*d^7 - b^7*c^3*d^6 + b^7*c^4*d^5 - 5
*a*b^6*c^2*d^7 + 7*a*b^6*c^3*d^6 + 4*a*b^6*c^4*d^5 - 5*a*b^6*c^5*d^4 - 3*a^2*b^5*c*d^8 + 8*a^3*b^4*c*d^8 - 4*a
^4*b^3*c*d^8 + 5*a^4*b^3*c^8*d - 8*a^5*b^2*c^8*d - 4*a^6*b*c^3*d^6 - 2*a^6*b*c^4*d^5 + 13*a^6*b*c^5*d^4 + a^6*
b*c^6*d^3 - 11*a^6*b*c^7*d^2 + 13*a^2*b^5*c^2*d^7 + 7*a^2*b^5*c^3*d^6 - 21*a^2*b^5*c^4*d^5 - 4*a^2*b^5*c^5*d^4
+ 10*a^2*b^5*c^6*d^3 - a^3*b^4*c^2*d^7 - 31*a^3*b^4*c^3*d^6 + 4*a^3*b^4*c^4*d^5 + 33*a^3*b^4*c^5*d^4 - 4*a^3*
b^4*c^6*d^3 - 10*a^3*b^4*c^7*d^2 - 12*a^4*b^3*c^2*d^7 + 14*a^4*b^3*c^3*d^6 + 34*a^4*b^3*c^4*d^5 - 21*a^4*b^3*c
^5*d^4 - 27*a^4*b^3*c^6*d^3 + 11*a^4*b^3*c^7*d^...

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