Optimal. Leaf size=122 \[ -\frac{2 a b}{d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{b}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{b \left (3 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac{a x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.155545, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3483, 3529, 3531, 3530} \[ -\frac{2 a b}{d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{b}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{b \left (3 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac{a x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3483
Rule 3529
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{1}{(a+b \tan (c+d x))^3} \, dx &=-\frac{b}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\int \frac{a-b \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx}{a^2+b^2}\\ &=-\frac{b}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{2 a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\int \frac{a^2-b^2-2 a b \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac{a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac{b}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{2 a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\left (b \left (3 a^2-b^2\right )\right ) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^3}\\ &=\frac{a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac{b \left (3 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac{b}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{2 a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 1.68205, size = 127, normalized size = 1.04 \[ \frac{\frac{b \left (\left (6 a^2-2 b^2\right ) \log (a+b \tan (c+d x))-\frac{\left (a^2+b^2\right ) \left (5 a^2+4 a b \tan (c+d x)+b^2\right )}{(a+b \tan (c+d x))^2}\right )}{\left (a^2+b^2\right )^3}+\frac{\log (-\tan (c+d x)+i)}{(b-i a)^3}+\frac{\log (\tan (c+d x)+i)}{(b+i a)^3}}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.028, size = 219, normalized size = 1.8 \begin{align*} -{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) b{a}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){b}^{3}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-3\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{b}{ \left ( 2\,{a}^{2}+2\,{b}^{2} \right ) d \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{b{a}^{2}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{{b}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-2\,{\frac{ab}{ \left ({a}^{2}+{b}^{2} \right ) ^{2}d \left ( a+b\tan \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.51543, size = 335, normalized size = 2.75 \begin{align*} \frac{\frac{2 \,{\left (a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left (3 \, a^{2} b - b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{4 \, a b^{2} \tan \left (d x + c\right ) + 5 \, a^{2} b + b^{3}}{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4} +{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.80712, size = 689, normalized size = 5.65 \begin{align*} -\frac{7 \, a^{2} b^{3} + b^{5} - 2 \,{\left (a^{5} - 3 \, a^{3} b^{2}\right )} d x -{\left (5 \, a^{2} b^{3} - b^{5} + 2 \,{\left (a^{3} b^{2} - 3 \, a b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{2} -{\left (3 \, a^{4} b - a^{2} b^{3} +{\left (3 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (3 \, a^{3} b^{2} - a b^{4}\right )} \tan \left (d x + c\right )\right )} \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 ) - 2 \,{\left (3 \, a^{3} b^{2} - 3 \, a b^{4} + 2 \,{\left (a^{4} b - 3 \, a^{2} b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{2 \,{\left ({\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} d \tan \left (d x + c\right ) +{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.24899, size = 358, normalized size = 2.93 \begin{align*} \frac{\frac{2 \,{\left (a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left (3 \, a^{2} b^{2} - b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac{9 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} - 3 \, b^{5} \tan \left (d x + c\right )^{2} + 22 \, a^{3} b^{2} \tan \left (d x + c\right ) - 2 \, a b^{4} \tan \left (d x + c\right ) + 14 \, a^{4} b + 3 \, a^{2} b^{3} + b^{5}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]