\(\int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{(a+b \sec (e+f x))^2} \, dx\) [261]

Optimal result

Integrand size = 31, antiderivative size = 198 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{(a+b \sec (e+f x))^2} \, dx=\frac {d^2 \text {arctanh}(\sin (e+f x))}{b^2 f}+\frac {2 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a (a-b)^{3/2} (a+b)^{3/2} f}+\frac {2 \left (b^2 c^2-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} b^2 \sqrt {a+b} f}-\frac {(b c-a d)^2 \sin (e+f x)}{b \left (a^2-b^2\right ) f (b+a \cos (e+f x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4073, 3031, 2743, 12, 2738, 214, 3855} \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{(a+b \sec (e+f x))^2} \, dx=\frac {2 \left (b^2 c^2-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a b^2 f \sqrt {a-b} \sqrt {a+b}}-\frac {(b c-a d)^2 \sin (e+f x)}{b f \left (a^2-b^2\right ) (a \cos (e+f x)+b)}+\frac {2 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a f (a-b)^{3/2} (a+b)^{3/2}}+\frac {d^2 \text {arctanh}(\sin (e+f x))}{b^2 f} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2743

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((a + b*Sin[c + d*x])^(n
+ 1)/(d*(n + 1)*(a^2 - b^2))), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n +
1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ
erQ[2*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 3855

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

Rule 4073

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]
*(d_.) + (c_))^(n_), x_Symbol] :> Dist[1/g^(m + n), Int[(g*Csc[e + f*x])^(m + n + p)*(b + a*Sin[e + f*x])^m*(d
+ c*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && Inte
gerQ[n]

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

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.91 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{(a+b \sec (e+f x))^2} \, dx=\frac {\frac {2 \left (2 b^3 c d+a^3 d^2-a b^2 \left (c^2+2 d^2\right )\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-d^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+d^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+\frac {b (b c-a d)^2 \sin (e+f x)}{(-a+b) (a+b) (b+a \cos (e+f x))}}{b^2 f} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {-\frac {d^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{b^{2}}+\frac {d^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{b^{2}}+\frac {\frac {2 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} b -a -b \right )}-\frac {2 \left (a^{3} d^{2}-b^{2} c^{2} a -2 b^{2} d^{2} a +2 c d \,b^{3}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{b^{2}}}{f}\)	\(215\)
default	\(\frac {-\frac {d^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{b^{2}}+\frac {d^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{b^{2}}+\frac {\frac {2 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} b -a -b \right )}-\frac {2 \left (a^{3} d^{2}-b^{2} c^{2} a -2 b^{2} d^{2} a +2 c d \,b^{3}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{b^{2}}}{f}\)	\(215\)
risch	\(-\frac {2 i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left ({\mathrm e}^{i \left (f x +e \right )} b +a \right )}{\left (a^{2}-b^{2}\right ) f b a \left ({\mathrm e}^{2 i \left (f x +e \right )} a +2 \,{\mathrm e}^{i \left (f x +e \right )} b +a \right )}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a^{3} d^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) f \,b^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) c^{2} a}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}-\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) d^{2} a}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}+\frac {2 b \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) c d}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a^{3} d^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) f \,b^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) c^{2} a}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}+\frac {2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) d^{2} a}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}-\frac {2 b \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) c d}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) f}+\frac {d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{b^{2} f}-\frac {d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{b^{2} f}\)	\(819\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (180) = 360\).

Time = 5.70 (sec) , antiderivative size = 798, normalized size of antiderivative = 4.03 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{(a+b \sec (e+f x))^2} \, dx=\left [-\frac {{\left (a b^{3} c^{2} - 2 \, b^{4} c d - {\left (a^{3} b - 2 \, a b^{3}\right )} d^{2} + {\left (a^{2} b^{2} c^{2} - 2 \, a b^{3} c d - {\left (a^{4} - 2 \, a^{2} b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (f x + e\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (f x + e\right ) + a\right )} \sin \left (f x + e\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (f x + e\right )^{2} + 2 \, a b \cos \left (f x + e\right ) + b^{2}}\right ) - {\left ({\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d^{2} \cos \left (f x + e\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) + {\left ({\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d^{2} \cos \left (f x + e\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d^{2}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left ({\left (a^{2} b^{3} - b^{5}\right )} c^{2} - 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} c d + {\left (a^{4} b - a^{2} b^{3}\right )} d^{2}\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} f \cos \left (f x + e\right ) + {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} f\right )}}, \frac {2 \, {\left (a b^{3} c^{2} - 2 \, b^{4} c d - {\left (a^{3} b - 2 \, a b^{3}\right )} d^{2} + {\left (a^{2} b^{2} c^{2} - 2 \, a b^{3} c d - {\left (a^{4} - 2 \, a^{2} b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (f x + e\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (f x + e\right )}\right ) + {\left ({\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d^{2} \cos \left (f x + e\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left ({\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d^{2} \cos \left (f x + e\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d^{2}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left ({\left (a^{2} b^{3} - b^{5}\right )} c^{2} - 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} c d + {\left (a^{4} b - a^{2} b^{3}\right )} d^{2}\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} f \cos \left (f x + e\right ) + {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} f\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.35 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{(a+b \sec (e+f x))^2} \, dx=\frac {\frac {d^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{b^{2}} - \frac {d^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{b^{2}} - \frac {2 \, {\left (a b^{2} c^{2} - 2 \, b^{3} c d - a^{3} d^{2} + 2 \, a b^{2} d^{2}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{2} b^{2} - b^{4}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, {\left (b^{2} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a b c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (a^{2} b - b^{3}\right )} {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a - b\right )}}}{f} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 21.29 (sec) , antiderivative size = 4926, normalized size of antiderivative = 24.88 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{(a+b \sec (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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