Integrand size = 33, antiderivative size = 212 \[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {C \text {arctanh}(\sin (c+d x))}{b^3 d}-\frac {a \left (3 A b^4+\left (2 a^4-5 a^2 b^2+6 b^4\right ) C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^3 (a+b)^{5/2} d}+\frac {a \left (A b^2+a^2 C\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (2 A b^4-3 a^4 C+a^2 b^2 (A+6 C)\right ) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
C*arctanh(sin(d*x+c))/b^3/d-a*(3*A*b^4+(2*a^4-5*a^2*b^2+6*b^4)*C)*arctanh( (a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(5/2)/b^3/(a+b)^(5/2)/d+ 1/2*a*(A*b^2+C*a^2)*tan(d*x+c)/b^2/(a^2-b^2)/d/(a+b*sec(d*x+c))^2+1/2*(2*A *b^4-3*a^4*C+a^2*b^2*(A+6*C))*tan(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c) )
Result contains complex when optimal does not.
Time = 4.67 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.10 \[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {(b+a \cos (c+d x)) \sec (c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (-4 C (b+a \cos (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 C (b+a \cos (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {4 a \left (3 A b^4+\left (2 a^4-5 a^2 b^2+6 b^4\right ) C\right ) \arctan \left (\frac {(i \cos (c)+\sin (c)) \left (a \sin (c)+(-b+a \cos (c)) \tan \left (\frac {d x}{2}\right )\right )}{\sqrt {a^2-b^2} \sqrt {(\cos (c)-i \sin (c))^2}}\right ) (b+a \cos (c+d x))^2 (i \cos (c)+\sin (c))}{\left (a^2-b^2\right )^{5/2} \sqrt {(\cos (c)-i \sin (c))^2}}+\frac {b \left (a \sec (c) \left (\left (4 A b^5-7 a^4 b C+a^2 b^3 (5 A+16 C)\right ) \sin (d x)+a \left (a b \left (-3 A b^2+\left (a^2-4 b^2\right ) C\right ) \sin (2 c+d x)+\left (A b^4-2 a^4 C+a^2 b^2 (2 A+5 C)\right ) \sin (c+2 d x)\right )\right )+\left (a^2+2 b^2\right ) \left (-A b^4+2 a^4 C-a^2 b^2 (2 A+5 C)\right ) \tan (c)\right )}{a \left (a^2-b^2\right )^2}\right )}{2 b^3 d (A+2 C+A \cos (2 (c+d x))) (a+b \sec (c+d x))^3} \]
((b + a*Cos[c + d*x])*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*(-4*C*(b + a*Cos [c + d*x])^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*C*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (4*a*(3*A*b^4 + (2*a^4 - 5*a^2*b^2 + 6*b^4)*C)*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*C os[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])^2*(I*Cos[c] + Sin[c]))/((a^2 - b^2)^(5/2)*Sqrt[(Cos[c] - I *Sin[c])^2]) + (b*(a*Sec[c]*((4*A*b^5 - 7*a^4*b*C + a^2*b^3*(5*A + 16*C))* Sin[d*x] + a*(a*b*(-3*A*b^2 + (a^2 - 4*b^2)*C)*Sin[2*c + d*x] + (A*b^4 - 2 *a^4*C + a^2*b^2*(2*A + 5*C))*Sin[c + 2*d*x])) + (a^2 + 2*b^2)*(-(A*b^4) + 2*a^4*C - a^2*b^2*(2*A + 5*C))*Tan[c]))/(a*(a^2 - b^2)^2)))/(2*b^3*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)
Time = 1.38 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.17, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 4579, 25, 3042, 4568, 25, 25, 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.
| \(\displaystyle \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4579 |
| \(\displaystyle \frac {\int -\frac {\sec (c+d x) \left (-2 b \left (a^2-b^2\right ) C \sec ^2(c+d x)-a \left (-C a^2+A b^2+2 b^2 C\right ) \sec (c+d x)+2 b \left (C a^2+A b^2\right )\right )}{(a+b \sec (c+d x))^2}dx}{2 b^2 \left (a^2-b^2\right )}+\frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int \frac {\sec (c+d x) \left (-2 b \left (a^2-b^2\right ) C \sec ^2(c+d x)-a \left (-C a^2+A b^2+2 b^2 C\right ) \sec (c+d x)+2 b \left (C a^2+A b^2\right )\right )}{(a+b \sec (c+d x))^2}dx}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (-2 b \left (a^2-b^2\right ) C \csc \left (c+d x+\frac {\pi }{2}\right )^2-a \left (-C a^2+A b^2+2 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 b \left (C a^2+A b^2\right )\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4568 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\int -\frac {\sec (c+d x) \left (a b^2 \left (3 A b^2-\left (a^2-4 b^2\right ) C\right )-2 b \left (a^2-b^2\right )^2 C \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\int -\frac {\sec (c+d x) \left (a \left (a^2 C-b^2 (3 A+4 C)\right ) b^2+2 \left (a^2-b^2\right )^2 C \sec (c+d x) b\right )}{a+b \sec (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\int \frac {\sec (c+d x) \left (a \left (a^2 C-b^2 (3 A+4 C)\right ) b^2+2 \left (a^2-b^2\right )^2 C \sec (c+d x) b\right )}{a+b \sec (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a \left (a^2 C-b^2 (3 A+4 C)\right ) b^2+2 \left (a^2-b^2\right )^2 C \csc \left (c+d x+\frac {\pi }{2}\right ) b\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4486 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {2 C \left (a^2-b^2\right )^2 \int \sec (c+d x)dx-a \left (C \left (2 a^4-5 a^2 b^2+6 b^4\right )+3 A b^4\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {2 C \left (a^2-b^2\right )^2 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-a \left (C \left (2 a^4-5 a^2 b^2+6 b^4\right )+3 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {2 C \left (a^2-b^2\right )^2 \text {arctanh}(\sin (c+d x))}{d}-a \left (C \left (2 a^4-5 a^2 b^2+6 b^4\right )+3 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {2 C \left (a^2-b^2\right )^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {a \left (C \left (2 a^4-5 a^2 b^2+6 b^4\right )+3 A b^4\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{b}}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {2 C \left (a^2-b^2\right )^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {a \left (C \left (2 a^4-5 a^2 b^2+6 b^4\right )+3 A b^4\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{b}}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {2 C \left (a^2-b^2\right )^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a \left (C \left (2 a^4-5 a^2 b^2+6 b^4\right )+3 A b^4\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {2 C \left (a^2-b^2\right )^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a \left (C \left (2 a^4-5 a^2 b^2+6 b^4\right )+3 A b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
(a*(A*b^2 + a^2*C)*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x]) ^2) - (-(((2*(a^2 - b^2)^2*C*ArcTanh[Sin[c + d*x]])/d - (2*a*(3*A*b^4 + (2 *a^4 - 5*a^2*b^2 + 6*b^4)*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))) - ((2*A*b^4 - 3*a^4* C + a^2*b^2*(A + 6*C))*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))) /(2*b^2*(a^2 - b^2))
3.7.93.3.1 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[(-(A*b^2 - a*b*B + a^2*C))*Cot[e + f*x]*((a + b*Csc[e + f*x] )^(m + 1)/(b*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^ 2, 0]
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 = 0.65 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.50
| method | result | size |
| derivativedivides | \(\frac {-\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{3}}+\frac {\frac {2 \left (-\frac {\left (2 A \,a^{2} b^{2}+a A \,b^{3}+2 A \,b^{4}-2 a^{4} C +a^{3} b C +6 C \,a^{2} b^{2}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b \left (2 A \,a^{2} b^{2}-a A \,b^{3}+2 A \,b^{4}-2 a^{4} C -a^{3} b C +6 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {a \left (3 A \,b^{4}+2 a^{4} C -5 C \,a^{2} b^{2}+6 C \,b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{b^{3}}+\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{3}}}{d}\) | \(317\) |
| default | \(\frac {-\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{3}}+\frac {\frac {2 \left (-\frac {\left (2 A \,a^{2} b^{2}+a A \,b^{3}+2 A \,b^{4}-2 a^{4} C +a^{3} b C +6 C \,a^{2} b^{2}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b \left (2 A \,a^{2} b^{2}-a A \,b^{3}+2 A \,b^{4}-2 a^{4} C -a^{3} b C +6 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {a \left (3 A \,b^{4}+2 a^{4} C -5 C \,a^{2} b^{2}+6 C \,b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{b^{3}}+\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{3}}}{d}\) | \(317\) |
| risch | \(\frac {i \left (3 A \,a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-C \,a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}+4 C \,a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+2 A \,a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+5 A \,a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+2 A \,b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-2 C \,a^{6} {\mathrm e}^{2 i \left (d x +c \right )}+C \,a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+10 C \,a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+5 b^{3} A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+4 b^{5} A a \,{\mathrm e}^{i \left (d x +c \right )}-7 C \,a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}+16 C \,a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}+2 A \,a^{4} b^{2}+a^{2} A \,b^{4}-2 a^{6} C +5 a^{4} b^{2} C \right )}{a \left (-a^{2}+b^{2}\right )^{2} d \,b^{2} \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2}}+\frac {3 a 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}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {a^{5} \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}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,b^{3}}-\frac {5 a^{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}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d b}+\frac {3 a 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 ) C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {3 a 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}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {a^{5} \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}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,b^{3}}+\frac {5 a^{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}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d b}-\frac {3 a 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 ) C}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{b^{3} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{b^{3} d}\) | \(1036\) |
1/d*(-C/b^3*ln(tan(1/2*d*x+1/2*c)-1)+2/b^3*((-1/2*(2*A*a^2*b^2+A*a*b^3+2*A *b^4-2*C*a^4+C*a^3*b+6*C*a^2*b^2)*b/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2* c)^3+1/2*b*(2*A*a^2*b^2-A*a*b^3+2*A*b^4-2*C*a^4-C*a^3*b+6*C*a^2*b^2)/(a+b) /(a-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)^2-1/2*a*(3*A*b^4+2*C*a^4-5*C*a^2*b^2+6*C*b^4)/(a^4-2*a^2*b^2+b^4)/( (a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))+ C/b^3*ln(tan(1/2*d*x+1/2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (200) = 400\).
Time = 4.24 (sec) , antiderivative size = 1317, normalized size of antiderivative = 6.21 \[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
[1/4*((2*C*a^5*b^2 - 5*C*a^3*b^4 + 3*(A + 2*C)*a*b^6 + (2*C*a^7 - 5*C*a^5* b^2 + 3*(A + 2*C)*a^3*b^4)*cos(d*x + c)^2 + 2*(2*C*a^6*b - 5*C*a^4*b^3 + 3 *(A + 2*C)*a^2*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)*si n(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*b^2 - 3*C*a^4*b^4 + 3*C*a^2*b^6 - C*b^8 + (C*a^8 - 3*C*a^6*b^2 + 3*C*a^4*b^4 - C*a^2*b^6)*cos(d*x + c)^2 + 2*(C*a^7*b - 3*C*a^5*b^3 + 3* C*a^3*b^5 - C*a*b^7)*cos(d*x + c))*log(sin(d*x + c) + 1) - 2*(C*a^6*b^2 - 3*C*a^4*b^4 + 3*C*a^2*b^6 - C*b^8 + (C*a^8 - 3*C*a^6*b^2 + 3*C*a^4*b^4 - C *a^2*b^6)*cos(d*x + c)^2 + 2*(C*a^7*b - 3*C*a^5*b^3 + 3*C*a^3*b^5 - C*a*b^ 7)*cos(d*x + c))*log(-sin(d*x + c) + 1) - 2*(3*C*a^6*b^2 - (A + 9*C)*a^4*b ^4 - (A - 6*C)*a^2*b^6 + 2*A*b^8 + (2*C*a^7*b - (2*A + 7*C)*a^5*b^3 + (A + 5*C)*a^3*b^5 + A*a*b^7)*cos(d*x + c))*sin(d*x + c))/((a^8*b^3 - 3*a^6*b^5 + 3*a^4*b^7 - a^2*b^9)*d*cos(d*x + c)^2 + 2*(a^7*b^4 - 3*a^5*b^6 + 3*a^3* b^8 - a*b^10)*d*cos(d*x + c) + (a^6*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11)*d) , -1/2*((2*C*a^5*b^2 - 5*C*a^3*b^4 + 3*(A + 2*C)*a*b^6 + (2*C*a^7 - 5*C*a^ 5*b^2 + 3*(A + 2*C)*a^3*b^4)*cos(d*x + c)^2 + 2*(2*C*a^6*b - 5*C*a^4*b^3 + 3*(A + 2*C)*a^2*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*b^2 - 3*C*a^ 4*b^4 + 3*C*a^2*b^6 - C*b^8 + (C*a^8 - 3*C*a^6*b^2 + 3*C*a^4*b^4 - C*a^...
\[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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. 509 vs. \(2 (200) = 400\).
Time = 0.38 (sec) , antiderivative size = 509, normalized size of antiderivative = 2.40 \[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=-\frac {\frac {{\left (2 \, C a^{5} - 5 \, C a^{3} b^{2} + 3 \, A a b^{4} + 6 \, C 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^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{3}} + \frac {C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{3}} - \frac {2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{2}}}{d} \]
-((2*C*a^5 - 5*C*a^3*b^2 + 3*A*a*b^4 + 6*C*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))/sqrt(-a^2 + b^2)))/((a^4*b^3 - 2*a^2*b^5 + b^7)*sqrt(-a^2 + b^ 2)) - C*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^3 + C*log(abs(tan(1/2*d*x + 1 /2*c) - 1))/b^3 - (2*C*a^5*tan(1/2*d*x + 1/2*c)^3 - 3*C*a^4*b*tan(1/2*d*x + 1/2*c)^3 - 2*A*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 5*C*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 + A*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - A*a*b^4*tan(1/2*d*x + 1/2*c)^3 + 2*A*b^5*tan(1/2*d*x + 1/2*c)^3 - 2*C*a^5*tan(1/2*d*x + 1/2*c) - 3*C*a^4*b*tan(1/2*d*x + 1/2*c) + 2*A*a^3 *b^2*tan(1/2*d*x + 1/2*c) + 5*C*a^3*b^2*tan(1/2*d*x + 1/2*c) + A*a^2*b^3*t an(1/2*d*x + 1/2*c) + 6*C*a^2*b^3*tan(1/2*d*x + 1/2*c) + A*a*b^4*tan(1/2*d *x + 1/2*c) + 2*A*b^5*tan(1/2*d*x + 1/2*c))/((a^4*b^2 - 2*a^2*b^4 + b^6)*( a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^2))/d
Time = 26.21 (sec) , antiderivative size = 6565, normalized size of antiderivative = 30.97 \[ \int \frac {\sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
(C*atan(((C*((C*((8*(4*C*b^15 + 6*A*a^2*b^13 + 12*A*a^3*b^12 - 12*A*a^4*b^ 11 - 6*A*a^5*b^10 + 6*A*a^6*b^9 - 8*C*a^2*b^13 + 34*C*a^3*b^12 + 6*C*a^4*b ^11 - 36*C*a^5*b^10 - 4*C*a^6*b^9 + 18*C*a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b ^6 - 6*A*a*b^14 - 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) - (8*C*tan(c/2 + (d*x)/2)*(8*a *b^15 - 8*a^2*b^14 - 32*a^3*b^13 + 32*a^4*b^12 + 48*a^5*b^11 - 48*a^6*b^10 - 32*a^7*b^9 + 32*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6))/(b^3*(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4))))/b^ 3 - (8*tan(c/2 + (d*x)/2)*(8*C^2*a^10 + 4*C^2*b^10 - 8*C^2*a*b^9 - 8*C^2*a ^9*b + 9*A^2*a^2*b^8 + 24*C^2*a^2*b^8 + 32*C^2*a^3*b^7 - 52*C^2*a^4*b^6 - 48*C^2*a^5*b^5 + 57*C^2*a^6*b^4 + 32*C^2*a^7*b^3 - 32*C^2*a^8*b^2 + 36*A*C *a^2*b^8 - 30*A*C*a^4*b^6 + 12*A*C*a^6*b^4))/(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4))*1i)/b^3 - (C*((C*( (8*(4*C*b^15 + 6*A*a^2*b^13 + 12*A*a^3*b^12 - 12*A*a^4*b^11 - 6*A*a^5*b^10 + 6*A*a^6*b^9 - 8*C*a^2*b^13 + 34*C*a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5*b^ 10 - 4*C*a^6*b^9 + 18*C*a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b^6 - 6*A*a*b^14 - 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^ 5*b^8 - a^6*b^7 - a^7*b^6) + (8*C*tan(c/2 + (d*x)/2)*(8*a*b^15 - 8*a^2*b^1 4 - 32*a^3*b^13 + 32*a^4*b^12 + 48*a^5*b^11 - 48*a^6*b^10 - 32*a^7*b^9 + 3 2*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6))/(b^3*(a*b^10 + b^11 - 3*a^2*b^9 - ...