Optimal. Leaf size=88 \[ \frac {22 \cot (c+d x)}{15 d \left (a^3 \csc (c+d x)+a^3\right )}+\frac {x}{a^3}+\frac {7 \cot (c+d x)}{15 a d (a \csc (c+d x)+a)^2}+\frac {\cot (c+d x)}{5 d (a \csc (c+d x)+a)^3} \]
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Rubi [A] time = 0.11, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3777, 3922, 3919, 3794} \[ \frac {22 \cot (c+d x)}{15 d \left (a^3 \csc (c+d x)+a^3\right )}+\frac {x}{a^3}+\frac {7 \cot (c+d x)}{15 a d (a \csc (c+d x)+a)^2}+\frac {\cot (c+d x)}{5 d (a \csc (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3777
Rule 3794
Rule 3919
Rule 3922
Rubi steps
\begin {align*} \int \frac {1}{(a+a \csc (c+d x))^3} \, dx &=\frac {\cot (c+d x)}{5 d (a+a \csc (c+d x))^3}-\frac {\int \frac {-5 a+2 a \csc (c+d x)}{(a+a \csc (c+d x))^2} \, dx}{5 a^2}\\ &=\frac {\cot (c+d x)}{5 d (a+a \csc (c+d x))^3}+\frac {7 \cot (c+d x)}{15 a d (a+a \csc (c+d x))^2}+\frac {\int \frac {15 a^2-7 a^2 \csc (c+d x)}{a+a \csc (c+d x)} \, dx}{15 a^4}\\ &=\frac {x}{a^3}+\frac {\cot (c+d x)}{5 d (a+a \csc (c+d x))^3}+\frac {7 \cot (c+d x)}{15 a d (a+a \csc (c+d x))^2}-\frac {22 \int \frac {\csc (c+d x)}{a+a \csc (c+d x)} \, dx}{15 a^2}\\ &=\frac {x}{a^3}+\frac {\cot (c+d x)}{5 d (a+a \csc (c+d x))^3}+\frac {7 \cot (c+d x)}{15 a d (a+a \csc (c+d x))^2}+\frac {22 \cot (c+d x)}{15 d \left (a^3+a^3 \csc (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 1.04, size = 123, normalized size = 1.40 \[ \frac {\frac {2 \sin \left (\frac {1}{2} (c+d x)\right ) (-51 \sin (c+d x)+16 \cos (2 (c+d x))-38)}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5}-\frac {13}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {3}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4}+15 c+15 d x}{15 a^3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 181, normalized size = 2.06 \[ \frac {{\left (15 \, d x + 32\right )} \cos \left (d x + c\right )^{3} + {\left (45 \, d x - 19\right )} \cos \left (d x + c\right )^{2} - 60 \, d x - 6 \, {\left (5 \, d x + 9\right )} \cos \left (d x + c\right ) + {\left ({\left (15 \, d x - 32\right )} \cos \left (d x + c\right )^{2} - 60 \, d x - 3 \, {\left (10 \, d x + 17\right )} \cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 3}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 86, normalized size = 0.98 \[ \frac {\frac {15 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 75 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 145 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 95 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 22\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.79, size = 125, normalized size = 1.42 \[ -\frac {4}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {8}{5 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {4}{3 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {2}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 228, normalized size = 2.59 \[ \frac {2 \, {\left (\frac {\frac {95 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {145 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {75 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 22}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} + \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.18, size = 78, normalized size = 0.89 \[ \frac {x}{a^3}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {58\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {38\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {44}{15}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5} \]
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 \[ \frac {\int \frac {1}{\csc ^{3}{\left (c + d x \right )} + 3 \csc ^{2}{\left (c + d x \right )} + 3 \csc {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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