Optimal. Leaf size=219 \[ \frac {b \cot ^4(c+d x)}{2 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^2 d}-\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right ) \log (\tan (c+d x))}{a^7 d}+\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{a^7 d}-\frac {b \left (a^2+b^2\right )^2}{a^6 d (a+b \tan (c+d x))}-\frac {\left (a^2+b^2\right ) \left (a^2+5 b^2\right ) \cot (c+d x)}{a^6 d}+\frac {2 b \left (a^2+b^2\right ) \cot ^2(c+d x)}{a^5 d}-\frac {\left (2 a^2+3 b^2\right ) \cot ^3(c+d x)}{3 a^4 d} \]
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Rubi [A] time = 0.20, antiderivative size = 219, 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, 894} \[ -\frac {b \left (a^2+b^2\right )^2}{a^6 d (a+b \tan (c+d x))}-\frac {\left (2 a^2+3 b^2\right ) \cot ^3(c+d x)}{3 a^4 d}+\frac {2 b \left (a^2+b^2\right ) \cot ^2(c+d x)}{a^5 d}-\frac {\left (a^2+b^2\right ) \left (a^2+5 b^2\right ) \cot (c+d x)}{a^6 d}-\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right ) \log (\tan (c+d x))}{a^7 d}+\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{a^7 d}+\frac {b \cot ^4(c+d x)}{2 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^2 d} \]
Antiderivative was successfully verified.
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Rule 894
Rule 3516
Rubi steps
\begin {align*} \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {\left (b^2+x^2\right )^2}{x^6 (a+x)^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \left (\frac {b^4}{a^2 x^6}-\frac {2 b^4}{a^3 x^5}+\frac {2 a^2 b^2+3 b^4}{a^4 x^4}-\frac {4 b^2 \left (a^2+b^2\right )}{a^5 x^3}+\frac {a^4+6 a^2 b^2+5 b^4}{a^6 x^2}-\frac {2 \left (a^4+4 a^2 b^2+3 b^4\right )}{a^7 x}+\frac {\left (a^2+b^2\right )^2}{a^6 (a+x)^2}+\frac {2 \left (a^4+4 a^2 b^2+3 b^4\right )}{a^7 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac {\left (a^2+b^2\right ) \left (a^2+5 b^2\right ) \cot (c+d x)}{a^6 d}+\frac {2 b \left (a^2+b^2\right ) \cot ^2(c+d x)}{a^5 d}-\frac {\left (2 a^2+3 b^2\right ) \cot ^3(c+d x)}{3 a^4 d}+\frac {b \cot ^4(c+d x)}{2 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^2 d}-\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right ) \log (\tan (c+d x))}{a^7 d}+\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{a^7 d}-\frac {b \left (a^2+b^2\right )^2}{a^6 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [B] time = 6.25, size = 589, normalized size = 2.69 \[ \frac {b \csc ^4(c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{2 a^3 d (a+b \tan (c+d x))^2}-\frac {\csc ^5(c+d x) \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{5 a^2 d (a+b \tan (c+d x))^2}+\frac {b \left (a^2+2 b^2\right ) \csc ^2(c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{a^5 d (a+b \tan (c+d x))^2}+\frac {\csc ^3(c+d x) \sec ^2(c+d x) \left (-4 a^2 \cos (c+d x)-15 b^2 \cos (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{15 a^4 d (a+b \tan (c+d x))^2}-\frac {2 \left (a^4 b+4 a^2 b^3+3 b^5\right ) \sec ^2(c+d x) \log (\sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2}{a^7 d (a+b \tan (c+d x))^2}+\frac {2 \left (a^4 b+4 a^2 b^3+3 b^5\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \log (a \cos (c+d x)+b \sin (c+d x))}{a^7 d (a+b \tan (c+d x))^2}+\frac {\sec ^2(c+d x) \left (a^4 b^2 \sin (c+d x)+2 a^2 b^4 \sin (c+d x)+b^6 \sin (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))}{a^7 d (a+b \tan (c+d x))^2}+\frac {\csc (c+d x) \sec ^2(c+d x) \left (-8 a^4 \cos (c+d x)-75 a^2 b^2 \cos (c+d x)-75 b^4 \cos (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{15 a^6 d (a+b \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 787, normalized size = 3.59 \[ \frac {4 \, {\left (4 \, a^{6} + 45 \, a^{4} b^{2} + 45 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{6} - 75 \, a^{4} b^{2} - 90 \, a^{2} b^{4} - 10 \, {\left (4 \, a^{6} + 45 \, a^{4} b^{2} + 45 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (2 \, a^{6} + 23 \, a^{4} b^{2} + 24 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left ({\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4} - 3 \, b^{6} - 3 \, {\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (a^{5} b + 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} - 2 \, {\left (a^{5} b + 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{5} b + 4 \, a^{3} b^{3} + 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 ) - 30 \, {\left ({\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4} - 3 \, b^{6} - 3 \, {\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (a^{5} b + 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} - 2 \, {\left (a^{5} b + 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{5} b + 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )\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 (4 \, {\left (4 \, a^{5} b + 45 \, a^{3} b^{3} + 45 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} - 10 \, {\left (a^{5} b + 33 \, a^{3} b^{3} + 36 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (a^{5} b - 10 \, a^{3} b^{3} - 12 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{7} b d \cos \left (d x + c\right )^{6} - 3 \, a^{7} b d \cos \left (d x + c\right )^{4} + 3 \, a^{7} b d \cos \left (d x + c\right )^{2} - a^{7} b d - {\left (a^{8} d \cos \left (d x + c\right )^{5} - 2 \, a^{8} d \cos \left (d x + c\right )^{3} + a^{8} d \cos \left (d x + c\right )\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 = 1.01, size = 332, normalized size = 1.52 \[ -\frac {\frac {60 \, {\left (a^{4} b + 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{7}} - \frac {60 \, {\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{7} b} + \frac {30 \, {\left (2 \, a^{4} b^{2} \tan \left (d x + c\right ) + 8 \, a^{2} b^{4} \tan \left (d x + c\right ) + 6 \, b^{6} \tan \left (d x + c\right ) + 3 \, a^{5} b + 10 \, a^{3} b^{3} + 7 \, a b^{5}\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )} a^{7}} - \frac {137 \, a^{4} b \tan \left (d x + c\right )^{5} + 548 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 411 \, b^{5} \tan \left (d x + c\right )^{5} - 30 \, a^{5} \tan \left (d x + c\right )^{4} - 180 \, a^{3} b^{2} \tan \left (d x + c\right )^{4} - 150 \, a b^{4} \tan \left (d x + c\right )^{4} + 60 \, a^{4} b \tan \left (d x + c\right )^{3} + 60 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} - 20 \, a^{5} \tan \left (d x + c\right )^{2} - 30 \, a^{3} b^{2} \tan \left (d x + c\right )^{2} + 15 \, a^{4} b \tan \left (d x + c\right ) - 6 \, a^{5}}{a^{7} \tan \left (d x + c\right )^{5}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 343, normalized size = 1.57 \[ -\frac {b}{a^{2} d \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 b^{3}}{d \,a^{4} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{5}}{d \,a^{6} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{3} d}+\frac {8 b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,a^{5}}+\frac {6 b^{5} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,a^{7}}-\frac {1}{5 d \,a^{2} \tan \left (d x +c \right )^{5}}-\frac {2}{3 d \,a^{2} \tan \left (d x +c \right )^{3}}-\frac {b^{2}}{d \,a^{4} \tan \left (d x +c \right )^{3}}-\frac {1}{d \,a^{2} \tan \left (d x +c \right )}-\frac {6 b^{2}}{d \,a^{4} \tan \left (d x +c \right )}-\frac {5 b^{4}}{d \,a^{6} \tan \left (d x +c \right )}+\frac {b}{2 d \,a^{3} \tan \left (d x +c \right )^{4}}+\frac {2 b}{d \,a^{3} \tan \left (d x +c \right )^{2}}+\frac {2 b^{3}}{d \,a^{5} \tan \left (d x +c \right )^{2}}-\frac {2 b \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}-\frac {8 b^{3} \ln \left (\tan \left (d x +c \right )\right )}{d \,a^{5}}-\frac {6 b^{5} \ln \left (\tan \left (d x +c \right )\right )}{d \,a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 225, normalized size = 1.03 \[ \frac {\frac {9 \, a^{4} b \tan \left (d x + c\right ) - 60 \, {\left (a^{4} b + 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \tan \left (d x + c\right )^{5} - 6 \, a^{5} - 30 \, {\left (a^{5} + 4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \tan \left (d x + c\right )^{4} + 10 \, {\left (4 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \tan \left (d x + c\right )^{3} - 5 \, {\left (4 \, a^{5} + 3 \, a^{3} b^{2}\right )} \tan \left (d x + c\right )^{2}}{a^{6} b \tan \left (d x + c\right )^{6} + a^{7} \tan \left (d x + c\right )^{5}} + \frac {60 \, {\left (a^{4} b + 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{7}} - \frac {60 \, {\left (a^{4} b + 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{7}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.01, size = 237, normalized size = 1.08 \[ \frac {4\,b\,\mathrm {atanh}\left (\frac {2\,b\,\left (a^2+3\,b^2\right )\,\left (a^2+b^2\right )\,\left (a+2\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}{a\,\left (2\,a^4\,b+8\,a^2\,b^3+6\,b^5\right )}\right )\,\left (a^2+3\,b^2\right )\,\left (a^2+b^2\right )}{a^7\,d}-\frac {\frac {1}{5\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^4+4\,a^2\,b^2+3\,b^4\right )}{a^5}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (4\,a^2+3\,b^2\right )}{6\,a^3}-\frac {3\,b\,\mathrm {tan}\left (c+d\,x\right )}{10\,a^2}+\frac {2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (a^4+4\,a^2\,b^2+3\,b^4\right )}{a^6}-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (4\,a^2+3\,b^2\right )}{3\,a^4}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^6+a\,{\mathrm {tan}\left (c+d\,x\right )}^5\right )} \]
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 \[ \int \frac {\csc ^{6}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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