Integrand size = 34, antiderivative size = 83 \[ \int \frac {\tan ^3(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {b B x}{a^2+b^2}+\frac {a B \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}-\frac {a^3 B \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right ) d}+\frac {B \tan (c+d x)}{b d} \]
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Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {21, 3647, 3707, 3698, 31, 3556} \[ \int \frac {\tan ^3(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {a B \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac {b B x}{a^2+b^2}-\frac {a^3 B \log (a+b \tan (c+d x))}{b^2 d \left (a^2+b^2\right )}+\frac {B \tan (c+d x)}{b d} \]
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Rule 21
Rule 31
Rule 3556
Rule 3647
Rule 3698
Rule 3707
Rubi steps \begin{align*} \text {integral}& = B \int \frac {\tan ^3(c+d x)}{a+b \tan (c+d x)} \, dx \\ & = \frac {B \tan (c+d x)}{b d}+\frac {B \int \frac {-a-b \tan (c+d x)-a \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b} \\ & = -\frac {b B x}{a^2+b^2}+\frac {B \tan (c+d x)}{b d}-\frac {(a B) \int \tan (c+d x) \, dx}{a^2+b^2}-\frac {\left (a^3 B\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )} \\ & = -\frac {b B x}{a^2+b^2}+\frac {a B \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}+\frac {B \tan (c+d x)}{b d}-\frac {\left (a^3 B\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^2 \left (a^2+b^2\right ) d} \\ & = -\frac {b B x}{a^2+b^2}+\frac {a B \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}-\frac {a^3 B \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right ) d}+\frac {B \tan (c+d x)}{b d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.11 \[ \int \frac {\tan ^3(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {B \left (\frac {\log (i-\tan (c+d x))}{a+i b}+\frac {\log (i+\tan (c+d x))}{a-i b}+\frac {2 a^3 \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right )}-\frac {2 \tan (c+d x)}{b}\right )}{2 d} \]
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Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {B \left (\frac {\tan \left (d x +c \right )}{b}+\frac {-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-b \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {a^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{2} \left (a^{2}+b^{2}\right )}\right )}{d}\) | \(80\) |
default | \(\frac {B \left (\frac {\tan \left (d x +c \right )}{b}+\frac {-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-b \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {a^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{2} \left (a^{2}+b^{2}\right )}\right )}{d}\) | \(80\) |
parallelrisch | \(-\frac {2 B \,b^{3} d x +B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a \,b^{2}+2 B \ln \left (a +b \tan \left (d x +c \right )\right ) a^{3}-2 B \,a^{2} b \tan \left (d x +c \right )-2 B \,b^{3} \tan \left (d x +c \right )}{2 \left (a^{2}+b^{2}\right ) b^{2} d}\) | \(86\) |
norman | \(\frac {\frac {B \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {B \,a^{2}}{d \,b^{2}}-\frac {b B a x}{a^{2}+b^{2}}-\frac {b^{2} B x \tan \left (d x +c \right )}{a^{2}+b^{2}}}{a +b \tan \left (d x +c \right )}-\frac {B a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {a^{3} B \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{2} \left (a^{2}+b^{2}\right ) d}\) | \(137\) |
risch | \(-\frac {i x B}{i b -a}-\frac {2 i a B x}{b^{2}}-\frac {2 i B a c}{b^{2} d}+\frac {2 i a^{3} B x}{b^{2} \left (a^{2}+b^{2}\right )}+\frac {2 i a^{3} B c}{d \left (a^{2}+b^{2}\right ) b^{2}}+\frac {2 i B}{d b \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B a}{b^{2} d}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{d \left (a^{2}+b^{2}\right ) b^{2}}\) | \(175\) |
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Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.43 \[ \int \frac {\tan ^3(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {2 \, B b^{3} d x + B a^{3} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (B a^{3} + B a b^{2}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (B a^{2} b + B b^{3}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} b^{2} + b^{4}\right )} d} \]
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Result contains complex when optimal does not.
Time = 1.04 (sec) , antiderivative size = 660, normalized size of antiderivative = 7.95 \[ \int \frac {\tan ^3(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 ^{2}{\left (c \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {B \left (- \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {\tan ^{2}{\left (c + d x \right )}}{2 d}\right )}{a} & \text {for}\: b = 0 \\- \frac {3 B d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {3 i B d x}{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 )} \tan {\left (c + d x \right )}}{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 )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {2 B \tan ^{2}{\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {2 i B \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {5 B}{2 b d \tan {\left (c + d x \right )} - 2 i b d} & \text {for}\: a = - i b \\- \frac {3 B d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {3 i B d x}{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 )} \tan {\left (c + d x \right )}}{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 )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {2 B \tan ^{2}{\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {2 i B \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {5 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 ^{3}{\left (c \right )}}{\left (a + b \tan {\left (c \right )}\right )^{2}} & \text {for}\: d = 0 \\- \frac {2 B a^{3} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} + \frac {2 B a^{2} b \tan {\left (c + d x \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} - \frac {B a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} - \frac {2 B b^{3} d x}{2 a^{2} b^{2} d + 2 b^{4} d} + \frac {2 B b^{3} \tan {\left (c + d x \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.07 \[ \int \frac {\tan ^3(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {2 \, B a^{3} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (d x + c\right )} B b}{a^{2} + b^{2}} + \frac {B a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, B \tan \left (d x + c\right )}{b}}{2 \, d} \]
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Time = 0.67 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.08 \[ \int \frac {\tan ^3(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {2 \, B a^{3} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (d x + c\right )} B b}{a^{2} + b^{2}} + \frac {B a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, B \tan \left (d x + c\right )}{b}}{2 \, d} \]
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Time = 7.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.18 \[ \int \frac {\tan ^3(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {B\,\mathrm {tan}\left (c+d\,x\right )}{b\,d}-\frac {B\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (a-b\,1{}\mathrm {i}\right )}-\frac {B\,a^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{b^2\,d\,\left (a^2+b^2\right )}-\frac {B\,\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-b+a\,1{}\mathrm {i}\right )} \]
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