Optimal. Leaf size=205 \[ -\frac{b \left (3 a^2+10 b^2\right )}{a^6 d (a+b \tan (c+d x))}-\frac{b \left (a^2+2 b^2\right )}{a^5 d (a+b \tan (c+d x))^2}-\frac{b \left (a^2+b^2\right )}{3 a^4 d (a+b \tan (c+d x))^3}-\frac{\left (a^2+10 b^2\right ) \cot (c+d x)}{a^6 d}-\frac{4 b \left (a^2+5 b^2\right ) \log (\tan (c+d x))}{a^7 d}+\frac{4 b \left (a^2+5 b^2\right ) \log (a+b \tan (c+d x))}{a^7 d}+\frac{2 b \cot ^2(c+d x)}{a^5 d}-\frac{\cot ^3(c+d x)}{3 a^4 d} \]
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Rubi [A] time = 0.173254, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3516, 894} \[ -\frac{b \left (3 a^2+10 b^2\right )}{a^6 d (a+b \tan (c+d x))}-\frac{b \left (a^2+2 b^2\right )}{a^5 d (a+b \tan (c+d x))^2}-\frac{b \left (a^2+b^2\right )}{3 a^4 d (a+b \tan (c+d x))^3}-\frac{\left (a^2+10 b^2\right ) \cot (c+d x)}{a^6 d}-\frac{4 b \left (a^2+5 b^2\right ) \log (\tan (c+d x))}{a^7 d}+\frac{4 b \left (a^2+5 b^2\right ) \log (a+b \tan (c+d x))}{a^7 d}+\frac{2 b \cot ^2(c+d x)}{a^5 d}-\frac{\cot ^3(c+d x)}{3 a^4 d} \]
Antiderivative was successfully verified.
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Rule 3516
Rule 894
Rubi steps
\begin{align*} \int \frac{\csc ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{b^2+x^2}{x^4 (a+x)^4} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \left (\frac{b^2}{a^4 x^4}-\frac{4 b^2}{a^5 x^3}+\frac{a^2+10 b^2}{a^6 x^2}-\frac{4 \left (a^2+5 b^2\right )}{a^7 x}+\frac{a^2+b^2}{a^4 (a+x)^4}+\frac{2 \left (a^2+2 b^2\right )}{a^5 (a+x)^3}+\frac{3 a^2+10 b^2}{a^6 (a+x)^2}+\frac{4 \left (a^2+5 b^2\right )}{a^7 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac{\left (a^2+10 b^2\right ) \cot (c+d x)}{a^6 d}+\frac{2 b \cot ^2(c+d x)}{a^5 d}-\frac{\cot ^3(c+d x)}{3 a^4 d}-\frac{4 b \left (a^2+5 b^2\right ) \log (\tan (c+d x))}{a^7 d}+\frac{4 b \left (a^2+5 b^2\right ) \log (a+b \tan (c+d x))}{a^7 d}-\frac{b \left (a^2+b^2\right )}{3 a^4 d (a+b \tan (c+d x))^3}-\frac{b \left (a^2+2 b^2\right )}{a^5 d (a+b \tan (c+d x))^2}-\frac{b \left (3 a^2+10 b^2\right )}{a^6 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [B] time = 2.03439, size = 528, normalized size = 2.58 \[ \frac{\sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (-192 b \left (a^2+5 b^2\right ) \log (\sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3+192 b \left (a^2+5 b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^3 \log (a \cos (c+d x)+b \sin (c+d x))-\frac{\csc ^3(c+d x) \left (3 a^5 b^3 \sin (2 (c+d x))+84 a^5 b^3 \sin (4 (c+d x))-65 a^5 b^3 \sin (6 (c+d x))-75 a^3 b^5 \sin (2 (c+d x))+156 a^3 b^5 \sin (4 (c+d x))-79 a^3 b^5 \sin (6 (c+d x))-22 a^6 b^2 \cos (6 (c+d x))+17 a^4 b^4 \cos (6 (c+d x))+55 a^2 b^6 \cos (6 (c+d x))+3 \left (10 a^6 b^2+45 a^4 b^4+115 a^2 b^6+3 a^8+75 b^8\right ) \cos (2 (c+d x))+6 \left (-35 a^2 b^6-17 a^4 b^4+2 a^6 b^2-15 b^8\right ) \cos (4 (c+d x))-4 a^6 b^2-50 a^4 b^4-190 a^2 b^6-3 a^7 b \sin (2 (c+d x))-6 a^7 b \sin (4 (c+d x))-3 a^7 b \sin (6 (c+d x))+a^8 (-\cos (6 (c+d x)))+8 a^8-75 a b^7 \sin (2 (c+d x))+60 a b^7 \sin (4 (c+d x))-15 a b^7 \sin (6 (c+d x))+15 b^8 \cos (6 (c+d x))-150 b^8\right )}{a^2+b^2}\right )}{48 a^7 d (a+b \tan (c+d x))^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.198, size = 278, normalized size = 1.4 \begin{align*} -{\frac{1}{3\,d{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{d{a}^{4}\tan \left ( dx+c \right ) }}-10\,{\frac{{b}^{2}}{d{a}^{6}\tan \left ( dx+c \right ) }}+2\,{\frac{b}{d{a}^{5} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}-4\,{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{5}}}-20\,{\frac{{b}^{3}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{7}}}-3\,{\frac{b}{d{a}^{4} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-10\,{\frac{{b}^{3}}{d{a}^{6} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-{\frac{b}{3\,{a}^{2}d \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{b}^{3}}{3\,d{a}^{4} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{b}{{a}^{3}d \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}-2\,{\frac{{b}^{3}}{d{a}^{5} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+4\,{\frac{b\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{a}^{5}}}+20\,{\frac{{b}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{a}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12349, size = 308, normalized size = 1.5 \begin{align*} \frac{\frac{3 \, a^{4} b \tan \left (d x + c\right ) - 12 \,{\left (a^{2} b^{3} + 5 \, b^{5}\right )} \tan \left (d x + c\right )^{5} - a^{5} - 30 \,{\left (a^{3} b^{2} + 5 \, a b^{4}\right )} \tan \left (d x + c\right )^{4} - 22 \,{\left (a^{4} b + 5 \, a^{2} b^{3}\right )} \tan \left (d x + c\right )^{3} - 3 \,{\left (a^{5} + 5 \, a^{3} b^{2}\right )} \tan \left (d x + c\right )^{2}}{a^{6} b^{3} \tan \left (d x + c\right )^{6} + 3 \, a^{7} b^{2} \tan \left (d x + c\right )^{5} + 3 \, a^{8} b \tan \left (d x + c\right )^{4} + a^{9} \tan \left (d x + c\right )^{3}} + \frac{12 \,{\left (a^{2} b + 5 \, b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{7}} - \frac{12 \,{\left (a^{2} b + 5 \, b^{3}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{7}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.25469, size = 2732, normalized size = 13.33 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.33433, size = 300, normalized size = 1.46 \begin{align*} -\frac{\frac{12 \,{\left (a^{2} b + 5 \, b^{3}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{7}} - \frac{12 \,{\left (a^{2} b^{2} + 5 \, b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{7} b} + \frac{12 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 60 \, b^{5} \tan \left (d x + c\right )^{5} + 30 \, a^{3} b^{2} \tan \left (d x + c\right )^{4} + 150 \, a b^{4} \tan \left (d x + c\right )^{4} + 22 \, a^{4} b \tan \left (d x + c\right )^{3} + 110 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} + 3 \, a^{5} \tan \left (d x + c\right )^{2} + 15 \, a^{3} b^{2} \tan \left (d x + c\right )^{2} - 3 \, a^{4} b \tan \left (d x + c\right ) + a^{5}}{{\left (b \tan \left (d x + c\right )^{2} + a \tan \left (d x + c\right )\right )}^{3} a^{6}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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