Optimal. Leaf size=138 \[ \frac{\left (a^2+b^2\right )^2}{3 a^3 b^2 d (a \cot (c+d x)+b)^3}+\frac{\frac{1}{a^3}+\frac{3 a}{b^4}}{d (a \cot (c+d x)+b)}+\frac{\frac{a}{b^3}-\frac{b}{a^3}}{d (a \cot (c+d x)+b)^2}-\frac{4 a \log (\tan (c+d x))}{b^5 d}-\frac{4 a \log (a \cot (c+d x)+b)}{b^5 d}+\frac{\tan (c+d x)}{b^4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16002, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3088, 894} \[ \frac{\left (a^2+b^2\right )^2}{3 a^3 b^2 d (a \cot (c+d x)+b)^3}+\frac{\frac{1}{a^3}+\frac{3 a}{b^4}}{d (a \cot (c+d x)+b)}+\frac{\frac{a}{b^3}-\frac{b}{a^3}}{d (a \cot (c+d x)+b)^2}-\frac{4 a \log (\tan (c+d x))}{b^5 d}-\frac{4 a \log (a \cot (c+d x)+b)}{b^5 d}+\frac{\tan (c+d x)}{b^4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3088
Rule 894
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^2 (b+a x)^4} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{b^4 x^2}-\frac{4 a}{b^5 x}+\frac{\left (a^2+b^2\right )^2}{a^2 b^2 (b+a x)^4}+\frac{2 \left (a^4-b^4\right )}{a^2 b^3 (b+a x)^3}+\frac{3 a^4+b^4}{a^2 b^4 (b+a x)^2}+\frac{4 a^2}{b^5 (b+a x)}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{\left (a^2+b^2\right )^2}{3 a^3 b^2 d (b+a \cot (c+d x))^3}+\frac{\frac{a}{b^3}-\frac{b}{a^3}}{d (b+a \cot (c+d x))^2}+\frac{\frac{1}{a^3}+\frac{3 a}{b^4}}{d (b+a \cot (c+d x))}-\frac{4 a \log (b+a \cot (c+d x))}{b^5 d}-\frac{4 a \log (\tan (c+d x))}{b^5 d}+\frac{\tan (c+d x)}{b^4 d}\\ \end{align*}
Mathematica [A] time = 2.20459, size = 133, normalized size = 0.96 \[ \frac{-4 \left (a^2+b^2\right ) \left (a^2+3 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)+b^2\right )+6 a (a+b \tan (c+d x)) \left (a^2-4 a (a+b \tan (c+d x))-2 (a+b \tan (c+d x))^2 \log (a+b \tan (c+d x))+b^2\right )+3 b^4 \sec ^4(c+d x)}{3 b^5 d (a+b \tan (c+d x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.285, size = 188, normalized size = 1.4 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{{b}^{4}d}}-4\,{\frac{a\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{{b}^{5}d}}+2\,{\frac{{a}^{3}}{{b}^{5}d \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{a}{d{b}^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}-6\,{\frac{{a}^{2}}{{b}^{5}d \left ( a+b\tan \left ( dx+c \right ) \right ) }}-2\,{\frac{1}{d{b}^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-{\frac{{a}^{4}}{3\,{b}^{5}d \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{2\,{a}^{2}}{3\,d{b}^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{3\,db \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.27017, size = 194, normalized size = 1.41 \begin{align*} -\frac{\frac{13 \, a^{4} + 2 \, a^{2} b^{2} + b^{4} + 6 \,{\left (3 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{2} + 6 \,{\left (5 \, a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )}{b^{8} \tan \left (d x + c\right )^{3} + 3 \, a b^{7} \tan \left (d x + c\right )^{2} + 3 \, a^{2} b^{6} \tan \left (d x + c\right ) + a^{3} b^{5}} + \frac{12 \, a \log \left (b \tan \left (d x + c\right ) + a\right )}{b^{5}} - \frac{3 \, \tan \left (d x + c\right )}{b^{4}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.702726, size = 1191, normalized size = 8.63 \begin{align*} \frac{3 \, a^{2} b^{4} + 3 \, b^{6} - 4 \,{\left (9 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - 2 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 6 \,{\left (5 \, a^{4} b^{2} + a^{2} b^{4} - 2 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - 6 \,{\left ({\left (a^{6} - 2 \, a^{4} b^{2} - 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (a^{4} b^{2} + a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} +{\left ({\left (3 \, a^{5} b + 2 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} +{\left (a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )\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 ) + 6 \,{\left ({\left (a^{6} - 2 \, a^{4} b^{2} - 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (a^{4} b^{2} + a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} +{\left ({\left (3 \, a^{5} b + 2 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} +{\left (a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) + 2 \,{\left (2 \,{\left (3 \, a^{5} b - 7 \, a^{3} b^{3} - 6 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} +{\left (11 \, a^{3} b^{3} + 9 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \,{\left ({\left (a^{5} b^{5} - 2 \, a^{3} b^{7} - 3 \, a b^{9}\right )} d \cos \left (d x + c\right )^{4} + 3 \,{\left (a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right )^{2} +{\left ({\left (3 \, a^{4} b^{6} + 2 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{3} +{\left (a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\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 = 1.19932, size = 186, normalized size = 1.35 \begin{align*} -\frac{\frac{12 \, a \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{5}} - \frac{3 \, \tan \left (d x + c\right )}{b^{4}} - \frac{22 \, a b^{3} \tan \left (d x + c\right )^{3} + 48 \, a^{2} b^{2} \tan \left (d x + c\right )^{2} - 6 \, b^{4} \tan \left (d x + c\right )^{2} + 36 \, a^{3} b \tan \left (d x + c\right ) - 6 \, a b^{3} \tan \left (d x + c\right ) + 9 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} b^{5}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]