\(\int \frac {(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^2} \, dx\) [192]

Optimal result

Integrand size = 25, antiderivative size = 133 \[ \int \frac {(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^2} \, dx=\frac {a^2 x}{c^2}+\frac {2 (b c-a d) \left (2 a c^2-b c d-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^2 (c-d)^{3/2} (c+d)^{3/2} f}+\frac {(b c-a d)^2 \sin (e+f x)}{c \left (c^2-d^2\right ) f (d+c \cos (e+f x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4026, 2869, 2814, 2738, 214} \[ \int \frac {(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^2} \, dx=\frac {a^2 x}{c^2}+\frac {2 (b c-a d) \left (2 a c^2-a d^2-b c d\right ) \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^2 f (c-d)^{3/2} (c+d)^{3/2}}+\frac {(b c-a d)^2 \sin (e+f x)}{c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)} \]

[In]

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

[Out]

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

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 2814

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

Rule 2869

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

Rule 4026

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

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

Mathematica [A] (verified)

Time = 1.32 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^2} \, dx=\frac {a^2 (e+f x)+\frac {2 \left (-2 a b c^3+b^2 c^2 d+a^2 \left (2 c^2 d-d^3\right )\right ) \text {arctanh}\left (\frac {(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{3/2}}+\frac {c (b c-a d)^2 \sin (e+f x)}{(c-d) (c+d) (d+c \cos (e+f x))}}{c^2 f} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.47

method	result	size
derivativedivides	\(\frac {\frac {2 a^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{2}}+\frac {-\frac {2 \left (a^{2} d^{2}-2 d a b c +b^{2} c^{2}\right ) c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-d^{2}\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )}-\frac {2 \left (2 a^{2} c^{2} d -a^{2} d^{3}-2 a b \,c^{3}+b^{2} c^{2} d \right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \left (c -d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{c^{2}}}{f}\)	\(195\)
default	\(\frac {\frac {2 a^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{2}}+\frac {-\frac {2 \left (a^{2} d^{2}-2 d a b c +b^{2} c^{2}\right ) c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-d^{2}\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )}-\frac {2 \left (2 a^{2} c^{2} d -a^{2} d^{3}-2 a b \,c^{3}+b^{2} c^{2} d \right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \left (c -d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{c^{2}}}{f}\)	\(195\)
risch	\(\frac {a^{2} x}{c^{2}}+\frac {2 i \left (a^{2} d^{2}-2 d a b c +b^{2} c^{2}\right ) \left (d \,{\mathrm e}^{i \left (f x +e \right )}+c \right )}{c^{2} \left (c^{2}-d^{2}\right ) f \left ({\mathrm e}^{2 i \left (f x +e \right )} c +2 d \,{\mathrm e}^{i \left (f x +e \right )}+c \right )}+\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {-i c^{2}+i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{c \sqrt {c^{2}-d^{2}}}\right ) a^{2} d}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {-i c^{2}+i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{c \sqrt {c^{2}-d^{2}}}\right ) a^{2} d^{3}}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f \,c^{2}}-\frac {2 c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {-i c^{2}+i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{c \sqrt {c^{2}-d^{2}}}\right ) a b}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {-i c^{2}+i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{c \sqrt {c^{2}-d^{2}}}\right ) b^{2} d}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}-\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) a^{2} d}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) a^{2} d^{3}}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f \,c^{2}}+\frac {2 c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) a b}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) b^{2} d}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right ) f}\)	\(769\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (124) = 248\).

Time = 0.32 (sec) , antiderivative size = 671, normalized size of antiderivative = 5.05 \[ \int \frac {(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^2} \, dx=\left [\frac {2 \, {\left (a^{2} c^{5} - 2 \, a^{2} c^{3} d^{2} + a^{2} c d^{4}\right )} f x \cos \left (f x + e\right ) + 2 \, {\left (a^{2} c^{4} d - 2 \, a^{2} c^{2} d^{3} + a^{2} d^{5}\right )} f x + {\left (2 \, a b c^{3} d + a^{2} d^{4} - {\left (2 \, a^{2} + b^{2}\right )} c^{2} d^{2} + {\left (2 \, a b c^{4} + a^{2} c d^{3} - {\left (2 \, a^{2} + b^{2}\right )} c^{3} d\right )} \cos \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \log \left (\frac {2 \, c d \cos \left (f x + e\right ) - {\left (c^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt {c^{2} - d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right ) + 2 \, c^{2} - d^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c d \cos \left (f x + e\right ) + d^{2}}\right ) + 2 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + 2 \, a b c^{2} d^{3} - a^{2} c d^{4} + {\left (a^{2} - b^{2}\right )} c^{3} d^{2}\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{7} - 2 \, c^{5} d^{2} + c^{3} d^{4}\right )} f \cos \left (f x + e\right ) + {\left (c^{6} d - 2 \, c^{4} d^{3} + c^{2} d^{5}\right )} f\right )}}, \frac {{\left (a^{2} c^{5} - 2 \, a^{2} c^{3} d^{2} + a^{2} c d^{4}\right )} f x \cos \left (f x + e\right ) + {\left (a^{2} c^{4} d - 2 \, a^{2} c^{2} d^{3} + a^{2} d^{5}\right )} f x + {\left (2 \, a b c^{3} d + a^{2} d^{4} - {\left (2 \, a^{2} + b^{2}\right )} c^{2} d^{2} + {\left (2 \, a b c^{4} + a^{2} c d^{3} - {\left (2 \, a^{2} + b^{2}\right )} c^{3} d\right )} \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \arctan \left (-\frac {\sqrt {-c^{2} + d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )}}{{\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}\right ) + {\left (b^{2} c^{5} - 2 \, a b c^{4} d + 2 \, a b c^{2} d^{3} - a^{2} c d^{4} + {\left (a^{2} - b^{2}\right )} c^{3} d^{2}\right )} \sin \left (f x + e\right )}{{\left (c^{7} - 2 \, c^{5} d^{2} + c^{3} d^{4}\right )} f \cos \left (f x + e\right ) + {\left (c^{6} d - 2 \, c^{4} d^{3} + c^{2} d^{5}\right )} f}\right ] \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^2} \, dx=\int \frac {\left (a + b \sec {\left (e + f x \right )}\right )^{2}}{\left (c + d \sec {\left (e + f x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*sec(f*x+e))^2/(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*c^2-4*d^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^2} \, dx=\frac {\frac {{\left (f x + e\right )} a^{2}}{c^{2}} + \frac {2 \, {\left (2 \, a b c^{3} - 2 \, a^{2} c^{2} d - b^{2} c^{2} d + a^{2} d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, c + 2 \, d\right ) + \arctan \left (-\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (c^{4} - c^{2} d^{2}\right )} \sqrt {-c^{2} + d^{2}}} - \frac {2 \, {\left (b^{2} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a b c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (c^{3} - c d^{2}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}}}{f} \]

[In]

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

[Out]

((f*x + e)*a^2/c^2 + 2*(2*a*b*c^3 - 2*a^2*c^2*d - b^2*c^2*d + a^2*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)))/((c^4 - c^2*d^2)*sqr
t(-c^2 + d^2)) - 2*(b^2*c^2*tan(1/2*f*x + 1/2*e) - 2*a*b*c*d*tan(1/2*f*x + 1/2*e) + a^2*d^2*tan(1/2*f*x + 1/2*
e))/((c^3 - c*d^2)*(c*tan(1/2*f*x + 1/2*e)^2 - d*tan(1/2*f*x + 1/2*e)^2 - c - d)))/f

Mupad [B] (verification not implemented)

Time = 22.90 (sec) , antiderivative size = 4934, normalized size of antiderivative = 37.10 \[ \int \frac {(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(2*a^2*atan(((a^2*((32*tan(e/2 + (f*x)/2)*(a^4*c^6 + 2*a^4*d^6 - 2*a^4*c*d^5 - 2*a^4*c^5*d + 4*a^2*b^2*c^6 - 5
*a^4*c^2*d^4 + 4*a^4*c^3*d^3 + 3*a^4*c^4*d^2 + b^4*c^4*d^2 + 4*a^3*b*c^3*d^3 - 2*a^2*b^2*c^2*d^4 + 4*a^2*b^2*c
^4*d^2 - 4*a*b^3*c^5*d - 8*a^3*b*c^5*d))/(c^4*d + c^5 - c^2*d^3 - c^3*d^2) + (a^2*((32*(2*a^2*c^8*d - a^2*c^9
+ b^2*c^8*d + a^2*c^4*d^5 - 3*a^2*c^6*d^3 + a^2*c^7*d^2 + b^2*c^5*d^4 - b^2*c^6*d^3 - b^2*c^7*d^2 - 2*a*b*c^9
+ 2*a*b*c^8*d - 2*a*b*c^6*d^3 + 2*a*b*c^7*d^2))/(c^5*d + c^6 - c^3*d^3 - c^4*d^2) - (a^2*tan(e/2 + (f*x)/2)*(2
*c^9*d - 2*c^4*d^6 + 2*c^5*d^5 + 4*c^6*d^4 - 4*c^7*d^3 - 2*c^8*d^2)*32i)/(c^2*(c^4*d + c^5 - c^2*d^3 - c^3*d^2
)))*1i)/c^2))/c^2 + (a^2*((32*tan(e/2 + (f*x)/2)*(a^4*c^6 + 2*a^4*d^6 - 2*a^4*c*d^5 - 2*a^4*c^5*d + 4*a^2*b^2*
c^6 - 5*a^4*c^2*d^4 + 4*a^4*c^3*d^3 + 3*a^4*c^4*d^2 + b^4*c^4*d^2 + 4*a^3*b*c^3*d^3 - 2*a^2*b^2*c^2*d^4 + 4*a^
2*b^2*c^4*d^2 - 4*a*b^3*c^5*d - 8*a^3*b*c^5*d))/(c^4*d + c^5 - c^2*d^3 - c^3*d^2) - (a^2*((32*(2*a^2*c^8*d - a
^2*c^9 + b^2*c^8*d + a^2*c^4*d^5 - 3*a^2*c^6*d^3 + a^2*c^7*d^2 + b^2*c^5*d^4 - b^2*c^6*d^3 - b^2*c^7*d^2 - 2*a
*b*c^9 + 2*a*b*c^8*d - 2*a*b*c^6*d^3 + 2*a*b*c^7*d^2))/(c^5*d + c^6 - c^3*d^3 - c^4*d^2) + (a^2*tan(e/2 + (f*x
)/2)*(2*c^9*d - 2*c^4*d^6 + 2*c^5*d^5 + 4*c^6*d^4 - 4*c^7*d^3 - 2*c^8*d^2)*32i)/(c^2*(c^4*d + c^5 - c^2*d^3 -
c^3*d^2)))*1i)/c^2))/c^2)/((64*(a^6*d^5 - 2*a^5*b*c^5 - a^6*c*d^4 + 2*a^6*c^4*d + 4*a^4*b^2*c^5 - 3*a^6*c^2*d^
3 + 2*a^6*c^3*d^2 - 4*a^3*b^3*c^4*d - a^4*b^2*c*d^4 + a^4*b^2*c^4*d + 2*a^5*b*c^2*d^3 + 2*a^5*b*c^3*d^2 + a^2*
b^4*c^3*d^2 - a^4*b^2*c^2*d^3 + 3*a^4*b^2*c^3*d^2 - 6*a^5*b*c^4*d))/(c^5*d + c^6 - c^3*d^3 - c^4*d^2) + (a^2*(
(32*tan(e/2 + (f*x)/2)*(a^4*c^6 + 2*a^4*d^6 - 2*a^4*c*d^5 - 2*a^4*c^5*d + 4*a^2*b^2*c^6 - 5*a^4*c^2*d^4 + 4*a^
4*c^3*d^3 + 3*a^4*c^4*d^2 + b^4*c^4*d^2 + 4*a^3*b*c^3*d^3 - 2*a^2*b^2*c^2*d^4 + 4*a^2*b^2*c^4*d^2 - 4*a*b^3*c^
5*d - 8*a^3*b*c^5*d))/(c^4*d + c^5 - c^2*d^3 - c^3*d^2) + (a^2*((32*(2*a^2*c^8*d - a^2*c^9 + b^2*c^8*d + a^2*c
^4*d^5 - 3*a^2*c^6*d^3 + a^2*c^7*d^2 + b^2*c^5*d^4 - b^2*c^6*d^3 - b^2*c^7*d^2 - 2*a*b*c^9 + 2*a*b*c^8*d - 2*a
*b*c^6*d^3 + 2*a*b*c^7*d^2))/(c^5*d + c^6 - c^3*d^3 - c^4*d^2) - (a^2*tan(e/2 + (f*x)/2)*(2*c^9*d - 2*c^4*d^6
+ 2*c^5*d^5 + 4*c^6*d^4 - 4*c^7*d^3 - 2*c^8*d^2)*32i)/(c^2*(c^4*d + c^5 - c^2*d^3 - c^3*d^2)))*1i)/c^2)*1i)/c^
2 - (a^2*((32*tan(e/2 + (f*x)/2)*(a^4*c^6 + 2*a^4*d^6 - 2*a^4*c*d^5 - 2*a^4*c^5*d + 4*a^2*b^2*c^6 - 5*a^4*c^2*
d^4 + 4*a^4*c^3*d^3 + 3*a^4*c^4*d^2 + b^4*c^4*d^2 + 4*a^3*b*c^3*d^3 - 2*a^2*b^2*c^2*d^4 + 4*a^2*b^2*c^4*d^2 -
4*a*b^3*c^5*d - 8*a^3*b*c^5*d))/(c^4*d + c^5 - c^2*d^3 - c^3*d^2) - (a^2*((32*(2*a^2*c^8*d - a^2*c^9 + b^2*c^8
*d + a^2*c^4*d^5 - 3*a^2*c^6*d^3 + a^2*c^7*d^2 + b^2*c^5*d^4 - b^2*c^6*d^3 - b^2*c^7*d^2 - 2*a*b*c^9 + 2*a*b*c
^8*d - 2*a*b*c^6*d^3 + 2*a*b*c^7*d^2))/(c^5*d + c^6 - c^3*d^3 - c^4*d^2) + (a^2*tan(e/2 + (f*x)/2)*(2*c^9*d -
2*c^4*d^6 + 2*c^5*d^5 + 4*c^6*d^4 - 4*c^7*d^3 - 2*c^8*d^2)*32i)/(c^2*(c^4*d + c^5 - c^2*d^3 - c^3*d^2)))*1i)/c
^2)*1i)/c^2)))/(c^2*f) + (atan((((a*d - b*c)*((c + d)^3*(c - d)^3)^(1/2)*((32*tan(e/2 + (f*x)/2)*(a^4*c^6 + 2*
a^4*d^6 - 2*a^4*c*d^5 - 2*a^4*c^5*d + 4*a^2*b^2*c^6 - 5*a^4*c^2*d^4 + 4*a^4*c^3*d^3 + 3*a^4*c^4*d^2 + b^4*c^4*
d^2 + 4*a^3*b*c^3*d^3 - 2*a^2*b^2*c^2*d^4 + 4*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d - 8*a^3*b*c^5*d))/(c^4*d + c^5 -
c^2*d^3 - c^3*d^2) + (((32*(2*a^2*c^8*d - a^2*c^9 + b^2*c^8*d + a^2*c^4*d^5 - 3*a^2*c^6*d^3 + a^2*c^7*d^2 + b
^2*c^5*d^4 - b^2*c^6*d^3 - b^2*c^7*d^2 - 2*a*b*c^9 + 2*a*b*c^8*d - 2*a*b*c^6*d^3 + 2*a*b*c^7*d^2))/(c^5*d + c^
6 - c^3*d^3 - c^4*d^2) - (32*tan(e/2 + (f*x)/2)*(a*d - b*c)*((c + d)^3*(c - d)^3)^(1/2)*(a*d^2 - 2*a*c^2 + b*c
*d)*(2*c^9*d - 2*c^4*d^6 + 2*c^5*d^5 + 4*c^6*d^4 - 4*c^7*d^3 - 2*c^8*d^2))/((c^4*d + c^5 - c^2*d^3 - c^3*d^2)*
(c^8 - c^2*d^6 + 3*c^4*d^4 - 3*c^6*d^2)))*(a*d - b*c)*((c + d)^3*(c - d)^3)^(1/2)*(a*d^2 - 2*a*c^2 + b*c*d))/(
c^8 - c^2*d^6 + 3*c^4*d^4 - 3*c^6*d^2))*(a*d^2 - 2*a*c^2 + b*c*d)*1i)/(c^8 - c^2*d^6 + 3*c^4*d^4 - 3*c^6*d^2)
+ ((a*d - b*c)*((c + d)^3*(c - d)^3)^(1/2)*((32*tan(e/2 + (f*x)/2)*(a^4*c^6 + 2*a^4*d^6 - 2*a^4*c*d^5 - 2*a^4*
c^5*d + 4*a^2*b^2*c^6 - 5*a^4*c^2*d^4 + 4*a^4*c^3*d^3 + 3*a^4*c^4*d^2 + b^4*c^4*d^2 + 4*a^3*b*c^3*d^3 - 2*a^2*
b^2*c^2*d^4 + 4*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d - 8*a^3*b*c^5*d))/(c^4*d + c^5 - c^2*d^3 - c^3*d^2) - (((32*(2
*a^2*c^8*d - a^2*c^9 + b^2*c^8*d + a^2*c^4*d^5 - 3*a^2*c^6*d^3 + a^2*c^7*d^2 + b^2*c^5*d^4 - b^2*c^6*d^3 - b^2
*c^7*d^2 - 2*a*b*c^9 + 2*a*b*c^8*d - 2*a*b*c^6*d^3 + 2*a*b*c^7*d^2))/(c^5*d + c^6 - c^3*d^3 - c^4*d^2) + (32*t
an(e/2 + (f*x)/2)*(a*d - b*c)*((c + d)^3*(c - d)^3)^(1/2)*(a*d^2 - 2*a*c^2 + b*c*d)*(2*c^9*d - 2*c^4*d^6 + 2*c
^5*d^5 + 4*c^6*d^4 - 4*c^7*d^3 - 2*c^8*d^2))/((c^4*d + c^5 - c^2*d^3 - c^3*d^2)*(c^8 - c^2*d^6 + 3*c^4*d^4 - 3
*c^6*d^2)))*(a*d - b*c)*((c + d)^3*(c - d)^3)^(1/2)*(a*d^2 - 2*a*c^2 + b*c*d))/(c^8 - c^2*d^6 + 3*c^4*d^4 - 3*
c^6*d^2))*(a*d^2 - 2*a*c^2 + b*c*d)*1i)/(c^8 - c^2*d^6 + 3*c^4*d^4 - 3*c^6*d^2))/((64*(a^6*d^5 - 2*a^5*b*c^5 -
a^6*c*d^4 + 2*a^6*c^4*d + 4*a^4*b^2*c^5 - 3*a^6*c^2*d^3 + 2*a^6*c^3*d^2 - 4*a^3*b^3*c^4*d - a^4*b^2*c*d^4 + a
^4*b^2*c^4*d + 2*a^5*b*c^2*d^3 + 2*a^5*b*c^3*d^2 + a^2*b^4*c^3*d^2 - a^4*b^2*c^2*d^3 + 3*a^4*b^2*c^3*d^2 - 6*a
^5*b*c^4*d))/(c^5*d + c^6 - c^3*d^3 - c^4*d^2) + ((a*d - b*c)*((c + d)^3*(c - d)^3)^(1/2)*((32*tan(e/2 + (f*x)
/2)*(a^4*c^6 + 2*a^4*d^6 - 2*a^4*c*d^5 - 2*a^4*c^5*d + 4*a^2*b^2*c^6 - 5*a^4*c^2*d^4 + 4*a^4*c^3*d^3 + 3*a^4*c
^4*d^2 + b^4*c^4*d^2 + 4*a^3*b*c^3*d^3 - 2*a^2*b^2*c^2*d^4 + 4*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d - 8*a^3*b*c^5*d
))/(c^4*d + c^5 - c^2*d^3 - c^3*d^2) + (((32*(2*a^2*c^8*d - a^2*c^9 + b^2*c^8*d + a^2*c^4*d^5 - 3*a^2*c^6*d^3
+ a^2*c^7*d^2 + b^2*c^5*d^4 - b^2*c^6*d^3 - b^2*c^7*d^2 - 2*a*b*c^9 + 2*a*b*c^8*d - 2*a*b*c^6*d^3 + 2*a*b*c^7*
d^2))/(c^5*d + c^6 - c^3*d^3 - c^4*d^2) - (32*tan(e/2 + (f*x)/2)*(a*d - b*c)*((c + d)^3*(c - d)^3)^(1/2)*(a*d^
2 - 2*a*c^2 + b*c*d)*(2*c^9*d - 2*c^4*d^6 + 2*c^5*d^5 + 4*c^6*d^4 - 4*c^7*d^3 - 2*c^8*d^2))/((c^4*d + c^5 - c^
2*d^3 - c^3*d^2)*(c^8 - c^2*d^6 + 3*c^4*d^4 - 3*c^6*d^2)))*(a*d - b*c)*((c + d)^3*(c - d)^3)^(1/2)*(a*d^2 - 2*
a*c^2 + b*c*d))/(c^8 - c^2*d^6 + 3*c^4*d^4 - 3*c^6*d^2))*(a*d^2 - 2*a*c^2 + b*c*d))/(c^8 - c^2*d^6 + 3*c^4*d^4
- 3*c^6*d^2) - ((a*d - b*c)*((c + d)^3*(c - d)^3)^(1/2)*((32*tan(e/2 + (f*x)/2)*(a^4*c^6 + 2*a^4*d^6 - 2*a^4*
c*d^5 - 2*a^4*c^5*d + 4*a^2*b^2*c^6 - 5*a^4*c^2*d^4 + 4*a^4*c^3*d^3 + 3*a^4*c^4*d^2 + b^4*c^4*d^2 + 4*a^3*b*c^
3*d^3 - 2*a^2*b^2*c^2*d^4 + 4*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d - 8*a^3*b*c^5*d))/(c^4*d + c^5 - c^2*d^3 - c^3*d
^2) - (((32*(2*a^2*c^8*d - a^2*c^9 + b^2*c^8*d + a^2*c^4*d^5 - 3*a^2*c^6*d^3 + a^2*c^7*d^2 + b^2*c^5*d^4 - b^2
*c^6*d^3 - b^2*c^7*d^2 - 2*a*b*c^9 + 2*a*b*c^8*d - 2*a*b*c^6*d^3 + 2*a*b*c^7*d^2))/(c^5*d + c^6 - c^3*d^3 - c^
4*d^2) + (32*tan(e/2 + (f*x)/2)*(a*d - b*c)*((c + d)^3*(c - d)^3)^(1/2)*(a*d^2 - 2*a*c^2 + b*c*d)*(2*c^9*d - 2
*c^4*d^6 + 2*c^5*d^5 + 4*c^6*d^4 - 4*c^7*d^3 - 2*c^8*d^2))/((c^4*d + c^5 - c^2*d^3 - c^3*d^2)*(c^8 - c^2*d^6 +
3*c^4*d^4 - 3*c^6*d^2)))*(a*d - b*c)*((c + d)^3*(c - d)^3)^(1/2)*(a*d^2 - 2*a*c^2 + b*c*d))/(c^8 - c^2*d^6 +
3*c^4*d^4 - 3*c^6*d^2))*(a*d^2 - 2*a*c^2 + b*c*d))/(c^8 - c^2*d^6 + 3*c^4*d^4 - 3*c^6*d^2)))*(a*d - b*c)*((c +
d)^3*(c - d)^3)^(1/2)*(a*d^2 - 2*a*c^2 + b*c*d)*2i)/(f*(c^8 - c^2*d^6 + 3*c^4*d^4 - 3*c^6*d^2)) - (2*tan(e/2
+ (f*x)/2)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(f*(c + d)*(c*d - c^2)*(c + d - tan(e/2 + (f*x)/2)^2*(c - d)))