Optimal. Leaf size=128 \[ -\frac {3}{128 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {3 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}-\frac {a^2}{40 d (a \cos (c+d x)+a)^5}+\frac {3 a}{64 d (a \cos (c+d x)+a)^4}-\frac {1}{128 a d (a-a \cos (c+d x))^2}-\frac {1}{64 a d (a \cos (c+d x)+a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3872, 2836, 12, 88, 206} \[ -\frac {a^2}{40 d (a \cos (c+d x)+a)^5}-\frac {3}{128 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {3 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}+\frac {3 a}{64 d (a \cos (c+d x)+a)^4}-\frac {1}{128 a d (a-a \cos (c+d x))^2}-\frac {1}{64 a d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 88
Rule 206
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac {\cot ^3(c+d x) \csc ^2(c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=\frac {a^5 \operatorname {Subst}\left (\int \frac {x^3}{a^3 (-a-x)^3 (-a+x)^6} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {x^3}{(-a-x)^3 (-a+x)^6} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \left (-\frac {1}{8 (a-x)^6}+\frac {3}{16 a (a-x)^5}-\frac {1}{32 a^3 (a-x)^3}-\frac {3}{128 a^4 (a-x)^2}+\frac {1}{64 a^3 (a+x)^3}-\frac {3}{128 a^4 \left (a^2-x^2\right )}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {1}{128 a d (a-a \cos (c+d x))^2}-\frac {a^2}{40 d (a+a \cos (c+d x))^5}+\frac {3 a}{64 d (a+a \cos (c+d x))^4}-\frac {1}{64 a d (a+a \cos (c+d x))^2}-\frac {3}{128 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,-a \cos (c+d x)\right )}{128 a^2 d}\\ &=\frac {3 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}-\frac {1}{128 a d (a-a \cos (c+d x))^2}-\frac {a^2}{40 d (a+a \cos (c+d x))^5}+\frac {3 a}{64 d (a+a \cos (c+d x))^4}-\frac {1}{64 a d (a+a \cos (c+d x))^2}-\frac {3}{128 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 5.29, size = 137, normalized size = 1.07 \[ -\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (60 \cos ^8\left (\frac {1}{2} (c+d x)\right )-15 \cos ^2\left (\frac {1}{2} (c+d x)\right )+10 \cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (\cot ^4\left (\frac {1}{2} (c+d x)\right )+2\right )-120 \cos ^{10}\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+4\right )}{640 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.85, size = 317, normalized size = 2.48 \[ -\frac {30 \, \cos \left (d x + c\right )^{6} + 90 \, \cos \left (d x + c\right )^{5} + 40 \, \cos \left (d x + c\right )^{4} - 120 \, \cos \left (d x + c\right )^{3} + 122 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 126 \, \cos \left (d x + c\right ) + 32}{1280 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.07, size = 232, normalized size = 1.81 \[ \frac {\frac {10 \, {\left (\frac {2 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {60 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3}} + \frac {\frac {60 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {30 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {20 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {5 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{15}}}{5120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.80, size = 126, normalized size = 0.98 \[ -\frac {1}{128 d \,a^{3} \left (-1+\cos \left (d x +c \right )\right )^{2}}-\frac {3 \ln \left (-1+\cos \left (d x +c \right )\right )}{256 d \,a^{3}}-\frac {1}{40 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )^{5}}+\frac {3}{64 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )^{4}}-\frac {1}{64 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {3}{128 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {3 \ln \left (1+\cos \left (d x +c \right )\right )}{256 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.38, size = 188, normalized size = 1.47 \[ -\frac {\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{6} + 45 \, \cos \left (d x + c\right )^{5} + 20 \, \cos \left (d x + c\right )^{4} - 60 \, \cos \left (d x + c\right )^{3} + 61 \, \cos \left (d x + c\right )^{2} + 63 \, \cos \left (d x + c\right ) + 16\right )}}{a^{3} \cos \left (d x + c\right )^{7} + 3 \, a^{3} \cos \left (d x + c\right )^{6} + a^{3} \cos \left (d x + c\right )^{5} - 5 \, a^{3} \cos \left (d x + c\right )^{4} - 5 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}} - \frac {15 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {15 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3}}}{1280 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.09, size = 173, normalized size = 1.35 \[ \frac {3\,\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )}{128\,a^3\,d}-\frac {\frac {3\,{\cos \left (c+d\,x\right )}^6}{128}+\frac {9\,{\cos \left (c+d\,x\right )}^5}{128}+\frac {{\cos \left (c+d\,x\right )}^4}{32}-\frac {3\,{\cos \left (c+d\,x\right )}^3}{32}+\frac {61\,{\cos \left (c+d\,x\right )}^2}{640}+\frac {63\,\cos \left (c+d\,x\right )}{640}+\frac {1}{40}}{d\,\left (a^3\,{\cos \left (c+d\,x\right )}^7+3\,a^3\,{\cos \left (c+d\,x\right )}^6+a^3\,{\cos \left (c+d\,x\right )}^5-5\,a^3\,{\cos \left (c+d\,x\right )}^4-5\,a^3\,{\cos \left (c+d\,x\right )}^3+a^3\,{\cos \left (c+d\,x\right )}^2+3\,a^3\,\cos \left (c+d\,x\right )+a^3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\csc ^{5}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________