Optimal. Leaf size=95 \[ -\frac {3 b \log (\tan (c+d x))}{a^4 d}+\frac {3 b \log (a+b \tan (c+d x))}{a^4 d}-\frac {2 b}{a^3 d (a+b \tan (c+d x))}-\frac {\cot (c+d x)}{a^3 d}-\frac {b}{2 a^2 d (a+b \tan (c+d x))^2} \]
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Rubi [A] time = 0.08, antiderivative size = 95, normalized size of antiderivative = 1.00, 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, 44} \[ -\frac {2 b}{a^3 d (a+b \tan (c+d x))}-\frac {b}{2 a^2 d (a+b \tan (c+d x))^2}-\frac {3 b \log (\tan (c+d x))}{a^4 d}+\frac {3 b \log (a+b \tan (c+d x))}{a^4 d}-\frac {\cot (c+d x)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 44
Rule 3516
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {1}{x^2 (a+x)^3} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \left (\frac {1}{a^3 x^2}-\frac {3}{a^4 x}+\frac {1}{a^2 (a+x)^3}+\frac {2}{a^3 (a+x)^2}+\frac {3}{a^4 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac {\cot (c+d x)}{a^3 d}-\frac {3 b \log (\tan (c+d x))}{a^4 d}+\frac {3 b \log (a+b \tan (c+d x))}{a^4 d}-\frac {b}{2 a^2 d (a+b \tan (c+d x))^2}-\frac {2 b}{a^3 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [B] time = 2.69, size = 241, normalized size = 2.54 \[ \frac {b \left (a^2 \left (-b^2\right ) \sec ^2(c+d x)-2 a^2 \left (a^2+b^2\right ) (-3 \log (a \cos (c+d x)+b \sin (c+d x))+3 \log (\sin (c+d x))+2)-2 b^2 \tan ^2(c+d x) \left (3 \left (a^2+b^2\right ) \log (\sin (c+d x))-3 \left (a^2+b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))-3 a^2-2 b^2\right )+2 a b \tan (c+d x) \left (-6 \left (a^2+b^2\right ) \log (\sin (c+d x))+6 \left (a^2+b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))+2 a^2+b^2\right )\right )-2 a^3 \left (a^2+b^2\right ) \cot (c+d x)}{2 a^4 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 565, normalized size = 5.95 \[ \frac {2 \, {\left (a^{7} + 4 \, a^{5} b^{2} - 2 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (2 \, a^{5} b^{2} - 3 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \cos \left (d x + c\right ) + 3 \, {\left (2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right ) - {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7} + {\left (a^{6} b + a^{4} b^{3} - a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 3 \, {\left (2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right ) - {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7} + {\left (a^{6} b + a^{4} b^{3} - a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4}\right ) - {\left (5 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - 4 \, {\left (a^{6} b + 5 \, a^{4} b^{3} + 3 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{2 \, {\left (2 \, {\left (a^{9} b + 2 \, a^{7} b^{3} + a^{5} b^{5}\right )} d \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{9} b + 2 \, a^{7} b^{3} + a^{5} b^{5}\right )} d \cos \left (d x + c\right ) - {\left ({\left (a^{10} + a^{8} b^{2} - a^{6} b^{4} - a^{4} b^{6}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{8} b^{2} + 2 \, a^{6} b^{4} + a^{4} b^{6}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 113, normalized size = 1.19 \[ \frac {\frac {6 \, b \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4}} - \frac {6 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} + \frac {2 \, {\left (3 \, b \tan \left (d x + c\right ) - a\right )}}{a^{4} \tan \left (d x + c\right )} - \frac {9 \, b^{3} \tan \left (d x + c\right )^{2} + 22 \, a b^{2} \tan \left (d x + c\right ) + 14 \, a^{2} b}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} a^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 96, normalized size = 1.01 \[ -\frac {b}{2 a^{2} d \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {3 b \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} d}-\frac {2 b}{a^{3} d \left (a +b \tan \left (d x +c \right )\right )}-\frac {1}{d \,a^{3} \tan \left (d x +c \right )}-\frac {3 b \ln \left (\tan \left (d x +c \right )\right )}{a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 108, normalized size = 1.14 \[ -\frac {\frac {6 \, b^{2} \tan \left (d x + c\right )^{2} + 9 \, a b \tan \left (d x + c\right ) + 2 \, a^{2}}{a^{3} b^{2} \tan \left (d x + c\right )^{3} + 2 \, a^{4} b \tan \left (d x + c\right )^{2} + a^{5} \tan \left (d x + c\right )} - \frac {6 \, b \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4}} + \frac {6 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.87, size = 99, normalized size = 1.04 \[ \frac {6\,b\,\mathrm {atanh}\left (\frac {2\,b\,\mathrm {tan}\left (c+d\,x\right )}{a}+1\right )}{a^4\,d}-\frac {\frac {1}{a}+\frac {3\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2}{a^3}+\frac {9\,b\,\mathrm {tan}\left (c+d\,x\right )}{2\,a^2}}{d\,\left (a^2\,\mathrm {tan}\left (c+d\,x\right )+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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