Optimal. Leaf size=103 \[ \frac{b^2 (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left (a^2+b^2\right )}-\frac{x (a B+b C)}{a^2+b^2}-\frac{(b B-a C) \log (\sin (c+d x))}{a^2 d}-\frac{B \cot (c+d x)}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.342237, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3632, 3609, 3651, 3530, 3475} \[ \frac{b^2 (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left (a^2+b^2\right )}-\frac{x (a B+b C)}{a^2+b^2}-\frac{(b B-a C) \log (\sin (c+d x))}{a^2 d}-\frac{B \cot (c+d x)}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3632
Rule 3609
Rule 3651
Rule 3530
Rule 3475
Rubi steps
\begin{align*} \int \frac{\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx &=\int \frac{\cot ^2(c+d x) (B+C \tan (c+d x))}{a+b \tan (c+d x)} \, dx\\ &=-\frac{B \cot (c+d x)}{a d}-\frac{\int \frac{\cot (c+d x) \left (b B-a C+a B \tan (c+d x)+b B \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{a}\\ &=-\frac{(a B+b C) x}{a^2+b^2}-\frac{B \cot (c+d x)}{a d}-\frac{(b B-a C) \int \cot (c+d x) \, dx}{a^2}+\frac{\left (b^2 (b B-a C)\right ) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )}\\ &=-\frac{(a B+b C) x}{a^2+b^2}-\frac{B \cot (c+d x)}{a d}-\frac{(b B-a C) \log (\sin (c+d x))}{a^2 d}+\frac{b^2 (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [C] time = 0.83286, size = 138, normalized size = 1.34 \[ \frac{\frac{2 b^2 (b B-a C) \log (a+b \tan (c+d x))}{a^2 \left (a^2+b^2\right )}+\frac{2 (a C-b B) \log (\tan (c+d x))}{a^2}+\frac{i (B+i C) \log (-\tan (c+d x)+i)}{a+i b}-\frac{(C+i B) \log (\tan (c+d x)+i)}{a-i b}-\frac{2 B \cot (c+d x)}{a}}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.119, size = 214, normalized size = 2.1 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) Bb}{2\,d \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) Ca}{2\,d \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ) a}{d \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{C\arctan \left ( \tan \left ( dx+c \right ) \right ) b}{d \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{B}{ad\tan \left ( dx+c \right ) }}-{\frac{\ln \left ( \tan \left ( dx+c \right ) \right ) Bb}{{a}^{2}d}}+{\frac{\ln \left ( \tan \left ( dx+c \right ) \right ) C}{ad}}+{\frac{{b}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) B}{d \left ({a}^{2}+{b}^{2} \right ){a}^{2}}}-{\frac{{b}^{2}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) C}{d \left ({a}^{2}+{b}^{2} \right ) a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.78672, size = 177, normalized size = 1.72 \begin{align*} -\frac{\frac{2 \,{\left (B a + C b\right )}{\left (d x + c\right )}}{a^{2} + b^{2}} + \frac{2 \,{\left (C a b^{2} - B b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + a^{2} b^{2}} + \frac{{\left (C a - B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac{2 \,{\left (C a - B b\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{2}} + \frac{2 \, B}{a \tan \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.23669, size = 404, normalized size = 3.92 \begin{align*} -\frac{2 \, B a^{3} + 2 \, B a b^{2} + 2 \,{\left (B a^{3} + C a^{2} b\right )} d x \tan \left (d x + c\right ) -{\left (C a^{3} - B a^{2} b + C a b^{2} - B b^{3}\right )} \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) +{\left (C a b^{2} - B b^{3}\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 ) \tan \left (d x + c\right )}{2 \,{\left (a^{4} + a^{2} b^{2}\right )} d \tan \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.69312, size = 212, normalized size = 2.06 \begin{align*} -\frac{\frac{2 \,{\left (B a + C b\right )}{\left (d x + c\right )}}{a^{2} + b^{2}} + \frac{{\left (C a - B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac{2 \,{\left (C a b^{3} - B b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + a^{2} b^{3}} - \frac{2 \,{\left (C a - B b\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{2}} + \frac{2 \,{\left (C a \tan \left (d x + c\right ) - B b \tan \left (d x + c\right ) + B a\right )}}{a^{2} \tan \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]