Optimal. Leaf size=189 \[ -\frac{a^2 (b B-a C)}{2 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{a \left (a^3 (-C)-3 a b^2 C+2 b^3 B\right )}{b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{\left (3 a^2 b B+a^3 (-C)+3 a b^2 C-b^3 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}-\frac{x \left (3 a^2 b C+a^3 B-3 a b^2 B-b^3 C\right )}{\left (a^2+b^2\right )^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.427431, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {3632, 3604, 3628, 3531, 3530} \[ -\frac{a^2 (b B-a C)}{2 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{a \left (a^3 (-C)-3 a b^2 C+2 b^3 B\right )}{b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{\left (3 a^2 b B+a^3 (-C)+3 a b^2 C-b^3 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}-\frac{x \left (3 a^2 b C+a^3 B-3 a b^2 B-b^3 C\right )}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3632
Rule 3604
Rule 3628
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{\tan (c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx &=\int \frac{\tan ^2(c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\\ &=-\frac{a^2 (b B-a C)}{2 b^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\int \frac{-a (b B-a C)+b (b B-a C) \tan (c+d x)+\left (a^2+b^2\right ) C \tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac{a^2 (b B-a C)}{2 b^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a \left (2 b^3 B-a^3 C-3 a b^2 C\right )}{b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\int \frac{-b \left (a^2 B-b^2 B+2 a b C\right )+b \left (2 a b B-a^2 C+b^2 C\right ) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )^2}\\ &=-\frac{\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}-\frac{a^2 (b B-a C)}{2 b^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a \left (2 b^3 B-a^3 C-3 a b^2 C\right )}{b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^3}\\ &=-\frac{\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}-\frac{\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac{a^2 (b B-a C)}{2 b^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a \left (2 b^3 B-a^3 C-3 a b^2 C\right )}{b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 4.6474, size = 288, normalized size = 1.52 \[ \frac{(b B-a C) \left (\frac{b \left (\frac{\left (a^2+b^2\right ) \left (5 a^2+4 a b \tan (c+d x)+b^2\right )}{(a+b \tan (c+d x))^2}+\left (2 b^2-6 a^2\right ) \log (a+b \tan (c+d x))\right )}{\left (a^2+b^2\right )^3}+\frac{i \log (-\tan (c+d x)+i)}{(a+i b)^3}-\frac{\log (\tan (c+d x)+i)}{(b+i a)^3}\right )+C \left (\frac{2 b \left (\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))\right )}{\left (a^2+b^2\right )^2}+\frac{i \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac{i \log (\tan (c+d x)+i)}{(a-i b)^2}\right )-\frac{a C+b B}{b (a+b \tan (c+d x))^2}-\frac{2 C \tan (c+d x)}{(a+b \tan (c+d x))^2}}{2 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.051, size = 495, normalized size = 2.6 \begin{align*}{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) B{a}^{2}b}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) B{b}^{3}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) C{a}^{3}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) Ca{b}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+3\,{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-3\,{\frac{C\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}b}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{C\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{{a}^{2}B}{2\,bd \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+{\frac{C{a}^{3}}{2\,{b}^{2}d \left ({a}^{2}+{b}^{2} \right ) \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 ) B}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ) B{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{{a}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) C}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-3\,{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ) Ca{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+2\,{\frac{Bab}{d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-{\frac{C{a}^{4}}{{b}^{2}d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-3\,{\frac{C{a}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.81723, size = 450, normalized size = 2.38 \begin{align*} -\frac{\frac{2 \,{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{2 \,{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{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 (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{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{C a^{5} + B a^{4} b + 5 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3} + 2 \,{\left (C a^{4} b + 3 \, C a^{2} b^{3} - 2 \, B a b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6} +{\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\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 = 1.1764, size = 1038, normalized size = 5.49 \begin{align*} \frac{C a^{5} - 3 \, B a^{4} b - 5 \, C a^{3} b^{2} + 3 \, B a^{2} b^{3} - 2 \,{\left (B a^{5} + 3 \, C a^{4} b - 3 \, B a^{3} b^{2} - C a^{2} b^{3}\right )} d x +{\left (C a^{5} + B a^{4} b + 7 \, C a^{3} b^{2} - 5 \, B a^{2} b^{3} - 2 \,{\left (B a^{3} b^{2} + 3 \, C a^{2} b^{3} - 3 \, B a b^{4} - C b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} +{\left (C a^{5} - 3 \, B a^{4} b - 3 \, C a^{3} b^{2} + B a^{2} b^{3} +{\left (C a^{3} b^{2} - 3 \, B a^{2} b^{3} - 3 \, C a b^{4} + B b^{5}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (C a^{4} b - 3 \, B a^{3} b^{2} - 3 \, C a^{2} b^{3} + B 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 (B a^{5} + 3 \, C a^{4} b - 3 \, B a^{3} b^{2} - 3 \, C a^{2} b^{3} + 2 \, B a b^{4} - 2 \,{\left (B a^{4} b + 3 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3} - C a b^{4}\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 = 1.62453, size = 554, normalized size = 2.93 \begin{align*} -\frac{\frac{2 \,{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{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 (C a^{3} b - 3 \, B a^{2} b^{2} - 3 \, C a b^{3} + B 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{3 \, C a^{3} b^{4} \tan \left (d x + c\right )^{2} - 9 \, B a^{2} b^{5} \tan \left (d x + c\right )^{2} - 9 \, C a b^{6} \tan \left (d x + c\right )^{2} + 3 \, B b^{7} \tan \left (d x + c\right )^{2} + 2 \, C a^{6} b \tan \left (d x + c\right ) + 14 \, C a^{4} b^{3} \tan \left (d x + c\right ) - 22 \, B a^{3} b^{4} \tan \left (d x + c\right ) - 12 \, C a^{2} b^{5} \tan \left (d x + c\right ) + 2 \, B a b^{6} \tan \left (d x + c\right ) + C a^{7} + B a^{6} b + 9 \, C a^{5} b^{2} - 11 \, B a^{4} b^{3} - 4 \, C a^{3} b^{4}}{{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\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]