Integrand size = 34, antiderivative size = 81 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {a B x}{b^2}+\frac {a^3 B x}{b^2 \left (a^2+b^2\right )}-\frac {B \log (\cos (c+d x))}{b d}+\frac {a^2 B \log (a \cos (c+d x)+b \sin (c+d x))}{b \left (a^2+b^2\right ) d} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {21, 3622, 3556, 3565, 3611} \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {a^2 B \log (a \cos (c+d x)+b \sin (c+d x))}{b d \left (a^2+b^2\right )}+\frac {a^3 B x}{b^2 \left (a^2+b^2\right )}-\frac {a B x}{b^2}-\frac {B \log (\cos (c+d x))}{b d} \]
[In]
[Out]
Rule 21
Rule 3556
Rule 3565
Rule 3611
Rule 3622
Rubi steps \begin{align*} \text {integral}& = B \int \frac {\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx \\ & = -\frac {a B x}{b^2}+\frac {\left (a^2 B\right ) \int \frac {1}{a+b \tan (c+d x)} \, dx}{b^2}+\frac {B \int \tan (c+d x) \, dx}{b} \\ & = -\frac {a B x}{b^2}+\frac {a^3 B x}{b^2 \left (a^2+b^2\right )}-\frac {B \log (\cos (c+d x))}{b d}+\frac {\left (a^2 B\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )} \\ & = -\frac {a B x}{b^2}+\frac {a^3 B x}{b^2 \left (a^2+b^2\right )}-\frac {B \log (\cos (c+d x))}{b d}+\frac {a^2 B \log (a \cos (c+d x)+b \sin (c+d x))}{b \left (a^2+b^2\right ) d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {B \left (b (i a+b) \log (i-\tan (c+d x))+b (-i a+b) \log (i+\tan (c+d x))+2 a^2 \log (a+b \tan (c+d x))\right )}{2 b \left (a^2+b^2\right ) d} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.73
method | result | size |
parallelrisch | \(\frac {-2 B a b d x +B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{2}+2 B \ln \left (a +b \tan \left (d x +c \right )\right ) a^{2}}{2 b d \left (a^{2}+b^{2}\right )}\) | \(59\) |
derivativedivides | \(\frac {B \left (\frac {\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-a \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}+\frac {a^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) b}\right )}{d}\) | \(69\) |
default | \(\frac {B \left (\frac {\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-a \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}+\frac {a^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) b}\right )}{d}\) | \(69\) |
norman | \(\frac {-\frac {B \,a^{2} x}{a^{2}+b^{2}}-\frac {b B a x \tan \left (d x +c \right )}{a^{2}+b^{2}}}{a +b \tan \left (d x +c \right )}+\frac {B \,a^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{b d \left (a^{2}+b^{2}\right )}+\frac {B b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}\) | \(111\) |
risch | \(\frac {x B}{i b -a}+\frac {2 i B x}{b}+\frac {2 i B c}{b d}-\frac {2 i a^{2} B x}{b \left (a^{2}+b^{2}\right )}-\frac {2 i a^{2} B c}{b d \left (a^{2}+b^{2}\right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B}{b d}+\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{b d \left (a^{2}+b^{2}\right )}\) | \(147\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.17 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {2 \, B a b d x - B a^{2} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (B a^{2} + B b^{2}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, {\left (a^{2} b + b^{3}\right )} d} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.74 (sec) , antiderivative size = 442, normalized size of antiderivative = 5.46 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\begin {cases} \tilde {\infty } B x \tan {\left (c \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {B \left (- x + \frac {\tan {\left (c + d x \right )}}{d}\right )}{a} & \text {for}\: b = 0 \\\frac {i B d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {B d x}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {B \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {i B \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {i B}{2 b d \tan {\left (c + d x \right )} - 2 i b d} & \text {for}\: a = - i b \\- \frac {i B d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {B d x}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {B \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {i B \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {i B}{2 b d \tan {\left (c + d x \right )} + 2 i b d} & \text {for}\: a = i b \\\frac {x \left (B a + B b \tan {\left (c \right )}\right ) \tan ^{2}{\left (c \right )}}{\left (a + b \tan {\left (c \right )}\right )^{2}} & \text {for}\: d = 0 \\\frac {2 B a^{2} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{2} b d + 2 b^{3} d} - \frac {2 B a b d x}{2 a^{2} b d + 2 b^{3} d} + \frac {B b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{2} b d + 2 b^{3} d} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {2 \, B a^{2} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{2} b + b^{3}} - \frac {2 \, {\left (d x + c\right )} B a}{a^{2} + b^{2}} + \frac {B b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}}}{2 \, d} \]
[In]
[Out]
none
Time = 0.51 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.94 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {2 \, B a^{2} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2} b + b^{3}} - \frac {2 \, {\left (d x + c\right )} B a}{a^{2} + b^{2}} + \frac {B b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}}}{2 \, d} \]
[In]
[Out]
Time = 7.36 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {B\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (b+a\,1{}\mathrm {i}\right )}+\frac {B\,a^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{b\,d\,\left (a^2+b^2\right )}+\frac {B\,\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a+b\,1{}\mathrm {i}\right )} \]
[In]
[Out]