Optimal. Leaf size=202 \[ \frac{a b \left (a^2-11 b^2\right )}{2 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{b \left (a^2-2 b^2\right )}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{\cos ^2(c+d x) (a \tan (c+d x)+b)}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{2 b^3 \left (5 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{a x \left (10 a^2 b^2+a^4-15 b^4\right )}{2 \left (a^2+b^2\right )^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.233042, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3506, 741, 801, 635, 203, 260} \[ \frac{a b \left (a^2-11 b^2\right )}{2 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{b \left (a^2-2 b^2\right )}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{\cos ^2(c+d x) (a \tan (c+d x)+b)}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{2 b^3 \left (5 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{a x \left (10 a^2 b^2+a^4-15 b^4\right )}{2 \left (a^2+b^2\right )^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3506
Rule 741
Rule 801
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^3 \left (1+\frac{x^2}{b^2}\right )^2} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{b \operatorname{Subst}\left (\int \frac{-4-\frac{a^2}{b^2}-\frac{3 a x}{b^2}}{(a+x)^3 \left (1+\frac{x^2}{b^2}\right )} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right ) d}\\ &=\frac{\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{b \operatorname{Subst}\left (\int \left (\frac{2 \left (a^2-2 b^2\right )}{\left (a^2+b^2\right ) (a+x)^3}+\frac{a^3-11 a b^2}{\left (a^2+b^2\right )^2 (a+x)^2}+\frac{4 b^2 \left (-5 a^2+b^2\right )}{\left (a^2+b^2\right )^3 (a+x)}+\frac{-a \left (a^4+10 a^2 b^2-15 b^4\right )+4 b^2 \left (5 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right ) d}\\ &=\frac{2 b^3 \left (5 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{b \left (a^2-2 b^2\right )}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a b \left (a^2-11 b^2\right )}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac{b \operatorname{Subst}\left (\int \frac{-a \left (a^4+10 a^2 b^2-15 b^4\right )+4 b^2 \left (5 a^2-b^2\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 d}\\ &=\frac{2 b^3 \left (5 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{b \left (a^2-2 b^2\right )}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a b \left (a^2-11 b^2\right )}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac{\left (2 b^3 \left (5 a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{\left (a^2+b^2\right )^4 d}+\frac{\left (a b \left (a^4+10 a^2 b^2-15 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 d}\\ &=\frac{a \left (a^4+10 a^2 b^2-15 b^4\right ) x}{2 \left (a^2+b^2\right )^4}+\frac{2 b^3 \left (5 a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{2 b^3 \left (5 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{b \left (a^2-2 b^2\right )}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a b \left (a^2-11 b^2\right )}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [B] time = 6.27209, size = 458, normalized size = 2.27 \[ \frac{b^3 \left (\frac{\cos ^2(c+d x) \left (a b \tan (c+d x)+b^2\right )}{2 b^4 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac{\left (2 a^2-4 b^2\right ) \left (-\frac{2 a}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{1}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac{\left (-\frac{a^3-3 a b^2}{\sqrt{-b^2}}+3 a^2-b^2\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3}+\frac{\left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}-\frac{\left (\frac{a^3-3 a b^2}{\sqrt{-b^2}}+3 a^2-b^2\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3}\right )-3 a \left (-\frac{1}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac{\left (2 a-\frac{a^2-b^2}{\sqrt{-b^2}}\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2}+\frac{2 a \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^2}-\frac{\left (\frac{a^2-b^2}{\sqrt{-b^2}}+2 a\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2}\right )}{2 b^2 \left (a^2+b^2\right )}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.122, size = 453, normalized size = 2.2 \begin{align*}{\frac{{a}^{5}\tan \left ( dx+c \right ) }{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{4} \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{{a}^{3}\tan \left ( dx+c \right ){b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4} \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{3\,a\tan \left ( dx+c \right ){b}^{4}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{4} \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }}+{\frac{3\,{a}^{4}b}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{4} \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }}+{\frac{{a}^{2}{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4} \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{{b}^{5}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{4} \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }}-5\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{2}{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){b}^{5}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+5\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-{\frac{15\,\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{4}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{5}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-{\frac{{b}^{3}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}-4\,{\frac{{b}^{3}a}{d \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}+10\,{\frac{{b}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ){a}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}}-2\,{\frac{{b}^{5}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.82646, size = 618, normalized size = 3.06 \begin{align*} \frac{\frac{{\left (a^{5} + 10 \, a^{3} b^{2} - 15 \, a b^{4}\right )}{\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{4 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac{2 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{3 \, a^{4} b - 10 \, a^{2} b^{3} - b^{5} +{\left (a^{3} b^{2} - 11 \, a b^{4}\right )} \tan \left (d x + c\right )^{3} + 2 \,{\left (a^{4} b - 6 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )^{2} +{\left (a^{5} + 3 \, a^{3} b^{2} - 10 \, a b^{4}\right )} \tan \left (d x + c\right )}{a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6} +{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{4} + 2 \,{\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )^{3} +{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} \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 )} \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 = 2.8846, size = 1102, normalized size = 5.46 \begin{align*} -\frac{3 \, a^{4} b^{3} - 16 \, a^{2} b^{5} + b^{7} - 2 \,{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{5} b^{2} + 10 \, a^{3} b^{4} - 15 \, a b^{6}\right )} d x -{\left (a^{6} b - a^{4} b^{3} - 45 \, a^{2} b^{5} - 3 \, b^{7} + 2 \,{\left (a^{7} + 9 \, a^{5} b^{2} - 25 \, a^{3} b^{4} + 15 \, a b^{6}\right )} d x\right )} \cos \left (d x + c\right )^{2} - 4 \,{\left (5 \, a^{2} b^{5} - b^{7} +{\left (5 \, a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (5 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 2 \,{\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \,{\left (a^{5} b^{2} - 3 \, a^{3} b^{4} + 6 \, a b^{6} -{\left (a^{6} b + 10 \, a^{4} b^{3} - 15 \, a^{2} b^{5}\right )} d x\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \,{\left ({\left (a^{10} + 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} - 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{2} + 2 \,{\left (a^{9} b + 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} + 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\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 = 1.26748, size = 593, normalized size = 2.94 \begin{align*} \frac{\frac{{\left (a^{5} + 10 \, a^{3} b^{2} - 15 \, a b^{4}\right )}{\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac{2 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{4 \,{\left (5 \, a^{2} b^{4} - b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} + \frac{10 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} - 2 \, b^{5} \tan \left (d x + c\right )^{2} + a^{5} \tan \left (d x + c\right ) - 2 \, a^{3} b^{2} \tan \left (d x + c\right ) - 3 \, a b^{4} \tan \left (d x + c\right ) + 3 \, a^{4} b + 12 \, a^{2} b^{3} - 3 \, b^{5}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}{\left (\tan \left (d x + c\right )^{2} + 1\right )}} - \frac{30 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} - 6 \, b^{7} \tan \left (d x + c\right )^{2} + 68 \, a^{3} b^{4} \tan \left (d x + c\right ) - 4 \, a b^{6} \tan \left (d x + c\right ) + 39 \, a^{4} b^{3} + 4 \, a^{2} b^{5} + b^{7}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, 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]