Integrand size = 33, antiderivative size = 153 \[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=-\frac {2 a C \text {arctanh}(\sin (c+d x))}{b^3 d}-\frac {2 \left (A b^4-2 a^4 C+3 a^2 b^2 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^3 (a+b)^{3/2} d}+\frac {C \tan (c+d x)}{b^2 d}+\frac {a \left (A b^2+a^2 C\right ) \tan (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \]
-2*a*C*arctanh(sin(d*x+c))/b^3/d-2*(A*b^4-2*C*a^4+3*C*a^2*b^2)*arctanh((a- b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(3/2)/b^3/(a+b)^(3/2)/d+C*t an(d*x+c)/b^2/d+a*(A*b^2+C*a^2)*tan(d*x+c)/b^2/(a^2-b^2)/d/(a+b*sec(d*x+c) )
Leaf count is larger than twice the leaf count of optimal. \(336\) vs. \(2(153)=306\).
Time = 3.39 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.20 \[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\frac {2 (b+a \cos (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \left (\frac {2 \left (A b^4-2 a^4 C+3 a^2 b^2 C\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))}{\left (a^2-b^2\right )^{3/2}}+2 a C (b+a \cos (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 a C (b+a \cos (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b C (b+a \cos (c+d x)) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {b C (b+a \cos (c+d x)) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {a b \left (A b^2+a^2 C\right ) \sin (c+d x)}{(a-b) (a+b)}\right )}{b^3 d (A+2 C+A \cos (2 (c+d x))) (a+b \sec (c+d x))^2} \]
(2*(b + a*Cos[c + d*x])*(A + C*Sec[c + d*x]^2)*((2*(A*b^4 - 2*a^4*C + 3*a^ 2*b^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2 - b^2)^(3/2) + 2*a*C*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x) /2] - Sin[(c + d*x)/2]] - 2*a*C*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b*C*(b + a*Cos[c + d*x])*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (b*C*(b + a*Cos[c + d*x])*Sin[(c + d*x)/2] )/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (a*b*(A*b^2 + a^2*C)*Sin[c + d*x ])/((a - b)*(a + b))))/(b^3*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^2)
Time = 1.18 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.21, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3042, 4579, 25, 3042, 4570, 3042, 4486, 3042, 4257, 4318, 3042, 3138, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4579 |
| \(\displaystyle \frac {\int -\frac {\sec (c+d x) \left (-b \left (a^2-b^2\right ) C \sec ^2(c+d x)+a \left (a^2-b^2\right ) C \sec (c+d x)+b \left (C a^2+A b^2\right )\right )}{a+b \sec (c+d x)}dx}{b^2 \left (a^2-b^2\right )}+\frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {\sec (c+d x) \left (-b \left (a^2-b^2\right ) C \sec ^2(c+d x)+a \left (a^2-b^2\right ) C \sec (c+d x)+b \left (C a^2+A b^2\right )\right )}{a+b \sec (c+d x)}dx}{b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (-b \left (a^2-b^2\right ) C \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left (a^2-b^2\right ) C \csc \left (c+d x+\frac {\pi }{2}\right )+b \left (C a^2+A b^2\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\int \frac {\sec (c+d x) \left (\left (C a^2+A b^2\right ) b^2+2 a \left (a^2-b^2\right ) C \sec (c+d x) b\right )}{a+b \sec (c+d x)}dx}{b}-\frac {C \left (a^2-b^2\right ) \tan (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\left (C a^2+A b^2\right ) b^2+2 a \left (a^2-b^2\right ) C \csc \left (c+d x+\frac {\pi }{2}\right ) b\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {C \left (a^2-b^2\right ) \tan (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4486 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {2 a C \left (a^2-b^2\right ) \int \sec (c+d x)dx+\left (-2 a^4 C+3 a^2 b^2 C+A b^4\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{b}-\frac {C \left (a^2-b^2\right ) \tan (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {2 a C \left (a^2-b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\left (-2 a^4 C+3 a^2 b^2 C+A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {C \left (a^2-b^2\right ) \tan (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (-2 a^4 C+3 a^2 b^2 C+A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a C \left (a^2-b^2\right ) \text {arctanh}(\sin (c+d x))}{d}}{b}-\frac {C \left (a^2-b^2\right ) \tan (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\frac {\left (-2 a^4 C+3 a^2 b^2 C+A b^4\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{b}+\frac {2 a C \left (a^2-b^2\right ) \text {arctanh}(\sin (c+d x))}{d}}{b}-\frac {C \left (a^2-b^2\right ) \tan (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\frac {\left (-2 a^4 C+3 a^2 b^2 C+A b^4\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{b}+\frac {2 a C \left (a^2-b^2\right ) \text {arctanh}(\sin (c+d x))}{d}}{b}-\frac {C \left (a^2-b^2\right ) \tan (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\frac {2 \left (-2 a^4 C+3 a^2 b^2 C+A b^4\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}+\frac {2 a C \left (a^2-b^2\right ) \text {arctanh}(\sin (c+d x))}{d}}{b}-\frac {C \left (a^2-b^2\right ) \tan (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\frac {2 a C \left (a^2-b^2\right ) \text {arctanh}(\sin (c+d x))}{d}+\frac {2 \left (-2 a^4 C+3 a^2 b^2 C+A b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}}{b}-\frac {C \left (a^2-b^2\right ) \tan (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
(a*(A*b^2 + a^2*C)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])) - (((2*a*(a^2 - b^2)*C*ArcTanh[Sin[c + d*x]])/d + (2*(A*b^4 - 2*a^4*C + 3* a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d))/b - ((a^2 - b^2)*C*Tan[c + d*x])/d)/(b^2*(a^2 - b^2))
3.7.85.3.1 Defintions of rubi rules used
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]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/(csc[( e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[B/b Int[Csc[e + f*x], x], x] + Simp[(A*b - a*B)/b Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x ] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e _.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_S ymbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2) )), x] + Simp[1/(b*(m + 2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[ b*A*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(cs c[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a*(A*b^2 + a^2* C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - Simp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[Csc[e + f*x]*(a + b*Csc[e + f* x])^(m + 1)*Simp[b*(m + 1)*(a^2*C + A*b^2) - a*(A*b^2*(m + 2) + C*(a^2 + b^ 2*(m + 1)))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Time = 0.51 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.52
| method | result | size |
| derivativedivides | \(\frac {\frac {2 C a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{3}}-\frac {C}{b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {C}{b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 C a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{3}}+\frac {-\frac {2 a b \left (A \,b^{2}+C \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {2 \left (A \,b^{4}-2 a^{4} C +3 C \,a^{2} b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{b^{3}}}{d}\) | \(232\) |
| default | \(\frac {\frac {2 C a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{3}}-\frac {C}{b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {C}{b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 C a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{3}}+\frac {-\frac {2 a b \left (A \,b^{2}+C \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {2 \left (A \,b^{4}-2 a^{4} C +3 C \,a^{2} b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{b^{3}}}{d}\) | \(232\) |
| risch | \(\frac {2 i \left (A \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+C \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+A a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+2 C \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-C a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+A \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+3 C \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-2 C \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+a A \,b^{2}+2 a^{3} C -C a \,b^{2}\right )}{d \,b^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left (a^{2}-b^{2}\right ) \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) a^{4} C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C \,a^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) a^{4} C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C \,a^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}-\frac {2 C a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{b^{3} d}+\frac {2 C a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{b^{3} d}\) | \(759\) |
1/d*(2*C*a/b^3*ln(tan(1/2*d*x+1/2*c)-1)-C/b^2/(tan(1/2*d*x+1/2*c)-1)-C/b^2 /(tan(1/2*d*x+1/2*c)+1)-2*C*a/b^3*ln(tan(1/2*d*x+1/2*c)+1)+2/b^3*(-a*b*(A* b^2+C*a^2)/(a^2-b^2)*tan(1/2*d*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d* x+1/2*c)^2*b-a-b)-(A*b^4-2*C*a^4+3*C*a^2*b^2)/(a+b)/(a-b)/((a+b)*(a-b))^(1 /2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (145) = 290\).
Time = 1.89 (sec) , antiderivative size = 852, normalized size of antiderivative = 5.57 \[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\left [\frac {{\left ({\left (2 \, C a^{5} - 3 \, C a^{3} b^{2} - A a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, C a^{4} b - 3 \, C a^{2} b^{3} - A b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - 2 \, {\left ({\left (C a^{6} - 2 \, C a^{4} b^{2} + C a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (C a^{5} b - 2 \, C a^{3} b^{3} + C a b^{5}\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left ({\left (C a^{6} - 2 \, C a^{4} b^{2} + C a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (C a^{5} b - 2 \, C a^{3} b^{3} + C a b^{5}\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (C a^{4} b^{2} - 2 \, C a^{2} b^{4} + C b^{6} + {\left (2 \, C a^{5} b + {\left (A - 3 \, C\right )} a^{3} b^{3} - {\left (A - C\right )} a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right )\right )}}, \frac {{\left ({\left (2 \, C a^{5} - 3 \, C a^{3} b^{2} - A a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, C a^{4} b - 3 \, C a^{2} b^{3} - A b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left ({\left (C a^{6} - 2 \, C a^{4} b^{2} + C a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (C a^{5} b - 2 \, C a^{3} b^{3} + C a b^{5}\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (C a^{6} - 2 \, C a^{4} b^{2} + C a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (C a^{5} b - 2 \, C a^{3} b^{3} + C a b^{5}\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (C a^{4} b^{2} - 2 \, C a^{2} b^{4} + C b^{6} + {\left (2 \, C a^{5} b + {\left (A - 3 \, C\right )} a^{3} b^{3} - {\left (A - C\right )} a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{{\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right )}\right ] \]
[1/2*(((2*C*a^5 - 3*C*a^3*b^2 - A*a*b^4)*cos(d*x + c)^2 + (2*C*a^4*b - 3*C *a^2*b^3 - A*b^5)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 + 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin( d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - 2*((C*a^6 - 2*C*a^4*b^2 + C*a^2*b^4)*cos(d*x + c)^2 + (C*a^5*b - 2*C*a^3* b^3 + C*a*b^5)*cos(d*x + c))*log(sin(d*x + c) + 1) + 2*((C*a^6 - 2*C*a^4*b ^2 + C*a^2*b^4)*cos(d*x + c)^2 + (C*a^5*b - 2*C*a^3*b^3 + C*a*b^5)*cos(d*x + c))*log(-sin(d*x + c) + 1) + 2*(C*a^4*b^2 - 2*C*a^2*b^4 + C*b^6 + (2*C* a^5*b + (A - 3*C)*a^3*b^3 - (A - C)*a*b^5)*cos(d*x + c))*sin(d*x + c))/((a ^5*b^3 - 2*a^3*b^5 + a*b^7)*d*cos(d*x + c)^2 + (a^4*b^4 - 2*a^2*b^6 + b^8) *d*cos(d*x + c)), (((2*C*a^5 - 3*C*a^3*b^2 - A*a*b^4)*cos(d*x + c)^2 + (2* C*a^4*b - 3*C*a^2*b^3 - A*b^5)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt (-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - ((C*a^6 - 2*C*a^4*b^2 + C*a^2*b^4)*cos(d*x + c)^2 + (C*a^5*b - 2*C*a^3*b^3 + C*a*b^5 )*cos(d*x + c))*log(sin(d*x + c) + 1) + ((C*a^6 - 2*C*a^4*b^2 + C*a^2*b^4) *cos(d*x + c)^2 + (C*a^5*b - 2*C*a^3*b^3 + C*a*b^5)*cos(d*x + c))*log(-sin (d*x + c) + 1) + (C*a^4*b^2 - 2*C*a^2*b^4 + C*b^6 + (2*C*a^5*b + (A - 3*C) *a^3*b^3 - (A - C)*a*b^5)*cos(d*x + c))*sin(d*x + c))/((a^5*b^3 - 2*a^3*b^ 5 + a*b^7)*d*cos(d*x + c)^2 + (a^4*b^4 - 2*a^2*b^6 + b^8)*d*cos(d*x + c))]
\[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (145) = 290\).
Time = 0.34 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.50 \[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\frac {2 \, {\left (\frac {{\left (2 \, C a^{4} - 3 \, C a^{2} b^{2} - A b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{2} b^{3} - b^{5}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {C a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{3}} + \frac {C a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{3}} - \frac {2 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )} {\left (a^{2} b^{2} - b^{4}\right )}}\right )}}{d} \]
2*((2*C*a^4 - 3*C*a^2*b^2 - A*b^4)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(- 2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqr t(-a^2 + b^2)))/((a^2*b^3 - b^5)*sqrt(-a^2 + b^2)) - C*a*log(abs(tan(1/2*d *x + 1/2*c) + 1))/b^3 + C*a*log(abs(tan(1/2*d*x + 1/2*c) - 1))/b^3 - (2*C* a^3*tan(1/2*d*x + 1/2*c)^3 - C*a^2*b*tan(1/2*d*x + 1/2*c)^3 + A*a*b^2*tan( 1/2*d*x + 1/2*c)^3 - C*a*b^2*tan(1/2*d*x + 1/2*c)^3 + C*b^3*tan(1/2*d*x + 1/2*c)^3 - 2*C*a^3*tan(1/2*d*x + 1/2*c) - C*a^2*b*tan(1/2*d*x + 1/2*c) - A *a*b^2*tan(1/2*d*x + 1/2*c) + C*a*b^2*tan(1/2*d*x + 1/2*c) + C*b^3*tan(1/2 *d*x + 1/2*c))/((a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a *tan(1/2*d*x + 1/2*c)^2 + a + b)*(a^2*b^2 - b^4)))/d
Time = 25.39 (sec) , antiderivative size = 4105, normalized size of antiderivative = 26.83 \[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
(C*a*atan(((C*a*((32*tan(c/2 + (d*x)/2)*(A^2*b^8 + 8*C^2*a^8 - 8*C^2*a^7*b + 4*C^2*a^2*b^6 - 8*C^2*a^3*b^5 + 5*C^2*a^4*b^4 + 16*C^2*a^5*b^3 - 16*C^2 *a^6*b^2 + 6*A*C*a^2*b^6 - 4*A*C*a^4*b^4))/(a*b^6 + b^7 - a^2*b^5 - a^3*b^ 4) - (2*C*a*((32*(A*b^12 - A*a^2*b^10 + A*a^3*b^9 + 3*C*a^2*b^10 + 3*C*a^3 *b^9 - 5*C*a^4*b^8 - C*a^5*b^7 + 2*C*a^6*b^6 - A*a*b^11 - 2*C*a*b^11))/(a* b^8 + b^9 - a^2*b^7 - a^3*b^6) - (64*C*a*tan(c/2 + (d*x)/2)*(2*a*b^11 - 2* a^2*b^10 - 4*a^3*b^9 + 4*a^4*b^8 + 2*a^5*b^7 - 2*a^6*b^6))/(b^3*(a*b^6 + b ^7 - a^2*b^5 - a^3*b^4))))/b^3)*2i)/b^3 + (C*a*((32*tan(c/2 + (d*x)/2)*(A^ 2*b^8 + 8*C^2*a^8 - 8*C^2*a^7*b + 4*C^2*a^2*b^6 - 8*C^2*a^3*b^5 + 5*C^2*a^ 4*b^4 + 16*C^2*a^5*b^3 - 16*C^2*a^6*b^2 + 6*A*C*a^2*b^6 - 4*A*C*a^4*b^4))/ (a*b^6 + b^7 - a^2*b^5 - a^3*b^4) + (2*C*a*((32*(A*b^12 - A*a^2*b^10 + A*a ^3*b^9 + 3*C*a^2*b^10 + 3*C*a^3*b^9 - 5*C*a^4*b^8 - C*a^5*b^7 + 2*C*a^6*b^ 6 - A*a*b^11 - 2*C*a*b^11))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (64*C*a*ta n(c/2 + (d*x)/2)*(2*a*b^11 - 2*a^2*b^10 - 4*a^3*b^9 + 4*a^4*b^8 + 2*a^5*b^ 7 - 2*a^6*b^6))/(b^3*(a*b^6 + b^7 - a^2*b^5 - a^3*b^4))))/b^3)*2i)/b^3)/(( 64*(8*C^3*a^8 - 4*C^3*a^7*b + 12*C^3*a^4*b^4 + 6*C^3*a^5*b^3 - 20*C^3*a^6* b^2 + 2*A^2*C*a*b^7 + 4*A*C^2*a^2*b^6 + 8*A*C^2*a^3*b^5 - 4*A*C^2*a^4*b^4 - 4*A*C^2*a^5*b^3))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (2*C*a*((32*tan(c/ 2 + (d*x)/2)*(A^2*b^8 + 8*C^2*a^8 - 8*C^2*a^7*b + 4*C^2*a^2*b^6 - 8*C^2*a^ 3*b^5 + 5*C^2*a^4*b^4 + 16*C^2*a^5*b^3 - 16*C^2*a^6*b^2 + 6*A*C*a^2*b^6...