Integrand size = 21, antiderivative size = 53 \[ \int \frac {\sec ^2(x) (a+b \tan (x))^2}{c+d \tan (x)} \, dx=\frac {(b c-a d)^2 \log (c+d \tan (x))}{d^3}-\frac {b (b c-a d) \tan (x)}{d^2}+\frac {(a+b \tan (x))^2}{2 d} \] Output:
(-a*d+b*c)^2*ln(c+d*tan(x))/d^3-b*(-a*d+b*c)*tan(x)/d^2+1/2*(a+b*tan(x))^2 /d
Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98 \[ \int \frac {\sec ^2(x) (a+b \tan (x))^2}{c+d \tan (x)} \, dx=\frac {2 (b c-a d)^2 \log (c+d \tan (x))+b^2 d^2 \sec ^2(x)-2 b d (b c-2 a d) \tan (x)}{2 d^3} \] Input:
Integrate[(Sec[x]^2*(a + b*Tan[x])^2)/(c + d*Tan[x]),x]
Output:
(2*(b*c - a*d)^2*Log[c + d*Tan[x]] + b^2*d^2*Sec[x]^2 - 2*b*d*(b*c - 2*a*d )*Tan[x])/(2*d^3)
Time = 0.34 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 4842, 49, 2009}
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.
\(\displaystyle \int \frac {\sec ^2(x) (a+b \tan (x))^2}{c+d \tan (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (x)^2 (a+b \tan (x))^2}{c+d \tan (x)}dx\) |
\(\Big \downarrow \) 4842 |
\(\displaystyle \int \frac {(a+b \tan (x))^2}{c+d \tan (x)}d\tan (x)\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {(a d-b c)^2}{d^2 (c+d \tan (x))}-\frac {b (b c-a d)}{d^2}+\frac {b (a+b \tan (x))}{d}\right )d\tan (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(b c-a d)^2 \log (c+d \tan (x))}{d^3}-\frac {b \tan (x) (b c-a d)}{d^2}+\frac {(a+b \tan (x))^2}{2 d}\) |
Input:
Int[(Sec[x]^2*(a + b*Tan[x])^2)/(c + d*Tan[x]),x]
Output:
((b*c - a*d)^2*Log[c + d*Tan[x]])/d^3 - (b*(b*c - a*d)*Tan[x])/d^2 + (a + b*Tan[x])^2/(2*d)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^2, x_Symbol] :> With[{d = FreeFac tors[Tan[c*(a + b*x)], x]}, Simp[d/(b*c) Subst[Int[SubstFor[1, Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d], x] /; FunctionOfQ[Tan[c*(a + b *x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Sec] | | EqQ[F, sec])
Time = 1.66 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(\frac {b \left (\frac {\tan \left (x \right )^{2} b d}{2}+2 \tan \left (x \right ) a d -\tan \left (x \right ) b c \right )}{d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (c +d \tan \left (x \right )\right )}{d^{3}}\) | \(60\) |
default | \(\frac {b \left (\frac {\tan \left (x \right )^{2} b d}{2}+2 \tan \left (x \right ) a d -\tan \left (x \right ) b c \right )}{d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (c +d \tan \left (x \right )\right )}{d^{3}}\) | \(60\) |
risch | \(\frac {2 i b \left (2 a d \,{\mathrm e}^{2 i x}-b c \,{\mathrm e}^{2 i x}-i b d \,{\mathrm e}^{2 i x}+2 a d -b c \right )}{\left (1+{\mathrm e}^{2 i x}\right )^{2} d^{2}}-\frac {\ln \left (1+{\mathrm e}^{2 i x}\right ) a^{2}}{d}+\frac {2 \ln \left (1+{\mathrm e}^{2 i x}\right ) a b c}{d^{2}}-\frac {\ln \left (1+{\mathrm e}^{2 i x}\right ) b^{2} c^{2}}{d^{3}}+\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {i d +c}{i d -c}\right ) a^{2}}{d}-\frac {2 \ln \left ({\mathrm e}^{2 i x}-\frac {i d +c}{i d -c}\right ) a b c}{d^{2}}+\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {i d +c}{i d -c}\right ) b^{2} c^{2}}{d^{3}}\) | \(206\) |
Input:
int(sec(x)^2*(a+b*tan(x))^2/(c+d*tan(x)),x,method=_RETURNVERBOSE)
Output:
b/d^2*(1/2*tan(x)^2*b*d+2*tan(x)*a*d-tan(x)*b*c)+(a^2*d^2-2*a*b*c*d+b^2*c^ 2)/d^3*ln(c+d*tan(x))
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (51) = 102\).
Time = 0.10 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.30 \[ \int \frac {\sec ^2(x) (a+b \tan (x))^2}{c+d \tan (x)} \, dx=\frac {b^{2} d^{2} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cos \left (x\right )^{2} \log \left (2 \, c d \cos \left (x\right ) \sin \left (x\right ) + {\left (c^{2} - d^{2}\right )} \cos \left (x\right )^{2} + d^{2}\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cos \left (x\right )^{2} \log \left (\cos \left (x\right )^{2}\right ) - 2 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} \cos \left (x\right ) \sin \left (x\right )}{2 \, d^{3} \cos \left (x\right )^{2}} \] Input:
integrate(sec(x)^2*(a+b*tan(x))^2/(c+d*tan(x)),x, algorithm="fricas")
Output:
1/2*(b^2*d^2 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*cos(x)^2*log(2*c*d*cos(x)*s in(x) + (c^2 - d^2)*cos(x)^2 + d^2) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*cos( x)^2*log(cos(x)^2) - 2*(b^2*c*d - 2*a*b*d^2)*cos(x)*sin(x))/(d^3*cos(x)^2)
Time = 1.84 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.06 \[ \int \frac {\sec ^2(x) (a+b \tan (x))^2}{c+d \tan (x)} \, dx=\frac {b^{2} \tan ^{2}{\left (x \right )}}{2 d} + \frac {\left (a d - b c\right )^{2} \left (\begin {cases} \frac {\tan {\left (x \right )}}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d \tan {\left (x \right )} \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {\left (2 a b d - b^{2} c\right ) \tan {\left (x \right )}}{d^{2}} \] Input:
integrate(sec(x)**2*(a+b*tan(x))**2/(c+d*tan(x)),x)
Output:
b**2*tan(x)**2/(2*d) + (a*d - b*c)**2*Piecewise((tan(x)/c, Eq(d, 0)), (log (c + d*tan(x))/d, True))/d**2 + (2*a*b*d - b**2*c)*tan(x)/d**2
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.19 \[ \int \frac {\sec ^2(x) (a+b \tan (x))^2}{c+d \tan (x)} \, dx=\frac {b^{2} d \tan \left (x\right )^{2} - 2 \, {\left (b^{2} c - 2 \, a b d\right )} \tan \left (x\right )}{2 \, d^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d \tan \left (x\right ) + c\right )}{d^{3}} \] Input:
integrate(sec(x)^2*(a+b*tan(x))^2/(c+d*tan(x)),x, algorithm="maxima")
Output:
1/2*(b^2*d*tan(x)^2 - 2*(b^2*c - 2*a*b*d)*tan(x))/d^2 + (b^2*c^2 - 2*a*b*c *d + a^2*d^2)*log(d*tan(x) + c)/d^3
Time = 0.13 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.21 \[ \int \frac {\sec ^2(x) (a+b \tan (x))^2}{c+d \tan (x)} \, dx=\frac {b^{2} d \tan \left (x\right )^{2} - 2 \, b^{2} c \tan \left (x\right ) + 4 \, a b d \tan \left (x\right )}{2 \, d^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | d \tan \left (x\right ) + c \right |}\right )}{d^{3}} \] Input:
integrate(sec(x)^2*(a+b*tan(x))^2/(c+d*tan(x)),x, algorithm="giac")
Output:
1/2*(b^2*d*tan(x)^2 - 2*b^2*c*tan(x) + 4*a*b*d*tan(x))/d^2 + (b^2*c^2 - 2* a*b*c*d + a^2*d^2)*log(abs(d*tan(x) + c))/d^3
Time = 16.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.23 \[ \int \frac {\sec ^2(x) (a+b \tan (x))^2}{c+d \tan (x)} \, dx=\frac {\ln \left (c+d\,\mathrm {tan}\left (x\right )\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{d^3}-\mathrm {tan}\left (x\right )\,\left (\frac {b^2\,c}{d^2}-\frac {2\,a\,b}{d}\right )+\frac {b^2\,{\mathrm {tan}\left (x\right )}^2}{2\,d} \] Input:
int((a + b*tan(x))^2/(cos(x)^2*(c + d*tan(x))),x)
Output:
(log(c + d*tan(x))*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/d^3 - tan(x)*((b^2*c)/ d^2 - (2*a*b)/d) + (b^2*tan(x)^2)/(2*d)
Time = 0.16 (sec) , antiderivative size = 427, normalized size of antiderivative = 8.06 \[ \int \frac {\sec ^2(x) (a+b \tan (x))^2}{c+d \tan (x)} \, dx=\frac {-4 \cos \left (x \right ) \sin \left (x \right ) a b \,d^{2}+2 \cos \left (x \right ) \sin \left (x \right ) b^{2} c d -2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{2} a^{2} d^{2}+4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{2} a b c d -2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{2} b^{2} c^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) a^{2} d^{2}-4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) a b c d +2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) b^{2} c^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{2} a^{2} d^{2}+4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{2} a b c d -2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{2} b^{2} c^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) a^{2} d^{2}-4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) a b c d +2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) b^{2} c^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} c -2 \tan \left (\frac {x}{2}\right ) d -c \right ) \sin \left (x \right )^{2} a^{2} d^{2}-4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} c -2 \tan \left (\frac {x}{2}\right ) d -c \right ) \sin \left (x \right )^{2} a b c d +2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} c -2 \tan \left (\frac {x}{2}\right ) d -c \right ) \sin \left (x \right )^{2} b^{2} c^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} c -2 \tan \left (\frac {x}{2}\right ) d -c \right ) a^{2} d^{2}+4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} c -2 \tan \left (\frac {x}{2}\right ) d -c \right ) a b c d -2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} c -2 \tan \left (\frac {x}{2}\right ) d -c \right ) b^{2} c^{2}+\sin \left (x \right )^{2} b^{2} d^{2}-2 b^{2} d^{2}}{2 d^{3} \left (\sin \left (x \right )^{2}-1\right )} \] Input:
int(sec(x)^2*(a+b*tan(x))^2/(c+d*tan(x)),x)
Output:
( - 4*cos(x)*sin(x)*a*b*d**2 + 2*cos(x)*sin(x)*b**2*c*d - 2*log(tan(x/2) - 1)*sin(x)**2*a**2*d**2 + 4*log(tan(x/2) - 1)*sin(x)**2*a*b*c*d - 2*log(ta n(x/2) - 1)*sin(x)**2*b**2*c**2 + 2*log(tan(x/2) - 1)*a**2*d**2 - 4*log(ta n(x/2) - 1)*a*b*c*d + 2*log(tan(x/2) - 1)*b**2*c**2 - 2*log(tan(x/2) + 1)* sin(x)**2*a**2*d**2 + 4*log(tan(x/2) + 1)*sin(x)**2*a*b*c*d - 2*log(tan(x/ 2) + 1)*sin(x)**2*b**2*c**2 + 2*log(tan(x/2) + 1)*a**2*d**2 - 4*log(tan(x/ 2) + 1)*a*b*c*d + 2*log(tan(x/2) + 1)*b**2*c**2 + 2*log(tan(x/2)**2*c - 2* tan(x/2)*d - c)*sin(x)**2*a**2*d**2 - 4*log(tan(x/2)**2*c - 2*tan(x/2)*d - c)*sin(x)**2*a*b*c*d + 2*log(tan(x/2)**2*c - 2*tan(x/2)*d - c)*sin(x)**2* b**2*c**2 - 2*log(tan(x/2)**2*c - 2*tan(x/2)*d - c)*a**2*d**2 + 4*log(tan( x/2)**2*c - 2*tan(x/2)*d - c)*a*b*c*d - 2*log(tan(x/2)**2*c - 2*tan(x/2)*d - c)*b**2*c**2 + sin(x)**2*b**2*d**2 - 2*b**2*d**2)/(2*d**3*(sin(x)**2 - 1))