Optimal. Leaf size=163 \[ -\frac {b^3}{2 a^2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))^2}+\frac {b^2 \left (3 a^2-b^2\right )}{a^2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))}+\frac {\log (1-\cos (c+d x))}{2 (a+b)^3 d}-\frac {\log (1+\cos (c+d x))}{2 (a-b)^3 d}+\frac {b \left (3 a^2+b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^3 d} \]
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Rubi [A]
time = 0.24, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3957, 2916, 12,
1643} \begin {gather*} \frac {b^2 \left (3 a^2-b^2\right )}{a^2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)}+\frac {b \left (3 a^2+b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^3}-\frac {b^3}{2 a^2 d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}+\frac {\log (1-\cos (c+d x))}{2 d (a+b)^3}-\frac {\log (\cos (c+d x)+1)}{2 d (a-b)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1643
Rule 2916
Rule 3957
Rubi steps
\begin {align*} \int \frac {\csc (c+d x)}{(a+b \sec (c+d x))^3} \, dx &=-\int \frac {\cos ^2(c+d x) \cot (c+d x)}{(-b-a \cos (c+d x))^3} \, dx\\ &=\frac {a \text {Subst}\left (\int \frac {x^3}{a^3 (-b+x)^3 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {x^3}{(-b+x)^3 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^2}{2 (a-b)^3 (a-x)}-\frac {b^3}{(a-b) (a+b) (b-x)^3}+\frac {3 a^2 b^2-b^4}{(a-b)^2 (a+b)^2 (b-x)^2}-\frac {a^2 b \left (3 a^2+b^2\right )}{(a-b)^3 (a+b)^3 (b-x)}+\frac {a^2}{2 (a+b)^3 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=-\frac {b^3}{2 a^2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))^2}+\frac {b^2 \left (3 a^2-b^2\right )}{a^2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))}+\frac {\log (1-\cos (c+d x))}{2 (a+b)^3 d}-\frac {\log (1+\cos (c+d x))}{2 (a-b)^3 d}+\frac {b \left (3 a^2+b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 203, normalized size = 1.25 \begin {gather*} \frac {(b+a \cos (c+d x)) \left (\frac {b^3}{a^2 \left (-a^2+b^2\right )}-\frac {2 b^2 \left (-3 a^2+b^2\right ) (b+a \cos (c+d x))}{a^2 (a-b)^2 (a+b)^2}+\frac {2 (b+a \cos (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{(-a+b)^3}+\frac {2 b \left (3 a^2+b^2\right ) (b+a \cos (c+d x))^2 \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^3}+\frac {2 (b+a \cos (c+d x))^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{(a+b)^3}\right ) \sec ^3(c+d x)}{2 d (a+b \sec (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 148, normalized size = 0.91
method | result | size |
derivativedivides | \(\frac {-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 \left (a -b \right )^{3}}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{2 \left (a +b \right )^{3}}-\frac {b^{3}}{2 a^{2} \left (a +b \right ) \left (a -b \right ) \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{2}+b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {b^{2} \left (3 a^{2}-b^{2}\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2} a^{2} \left (b +a \cos \left (d x +c \right )\right )}}{d}\) | \(148\) |
default | \(\frac {-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 \left (a -b \right )^{3}}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{2 \left (a +b \right )^{3}}-\frac {b^{3}}{2 a^{2} \left (a +b \right ) \left (a -b \right ) \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{2}+b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {b^{2} \left (3 a^{2}-b^{2}\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2} a^{2} \left (b +a \cos \left (d x +c \right )\right )}}{d}\) | \(148\) |
norman | \(\frac {\frac {6 b^{2} a}{d \left (a^{4}-2 b \,a^{3}+2 b^{3} a -b^{4}\right )}-\frac {2 \left (3 b^{2} a +b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{4}-2 b^{2} a^{2}+b^{4}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{3}+3 b \,a^{2}+3 b^{2} a +b^{3}\right )}+\frac {b \left (3 a^{2}+b^{2}\right ) \ln \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}\) | \(226\) |
risch | \(\frac {i x}{a^{3}-3 b \,a^{2}+3 b^{2} a -b^{3}}+\frac {i c}{d \left (a^{3}-3 b \,a^{2}+3 b^{2} a -b^{3}\right )}-\frac {i x}{a^{3}+3 b \,a^{2}+3 b^{2} a +b^{3}}-\frac {i c}{d \left (a^{3}+3 b \,a^{2}+3 b^{2} a +b^{3}\right )}-\frac {6 i b \,a^{2} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}-\frac {6 i b \,a^{2} c}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {2 i b^{3} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}-\frac {2 i b^{3} c}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {2 b^{2} \left (-3 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+b^{2} a \,{\mathrm e}^{3 i \left (d x +c \right )}-5 b \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 a^{3} {\mathrm e}^{i \left (d x +c \right )}+b^{2} a \,{\mathrm e}^{i \left (d x +c \right )}\right )}{a^{2} \left (-a^{2}+b^{2}\right )^{2} \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \left (a^{3}-3 b \,a^{2}+3 b^{2} a -b^{3}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \left (a^{3}+3 b \,a^{2}+3 b^{2} a +b^{3}\right )}+\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right ) a^{2}}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}\) | \(591\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 241, normalized size = 1.48 \begin {gather*} \frac {\frac {2 \, {\left (3 \, a^{2} b + b^{3}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {5 \, a^{2} b^{3} - b^{5} + 2 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )}{a^{6} b^{2} - 2 \, a^{4} b^{4} + a^{2} b^{6} + {\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \cos \left (d x + c\right )} - \frac {\log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {\log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 474 vs.
\(2 (157) = 314\).
time = 3.54, size = 474, normalized size = 2.91 \begin {gather*} \frac {5 \, a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7} + 2 \, {\left (3 \, a^{5} b^{2} - 4 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right ) + 2 \, {\left (3 \, a^{4} b^{3} + a^{2} b^{5} + {\left (3 \, a^{6} b + a^{4} b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) - {\left (a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5} + {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5} + {\left (a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left ({\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b - 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} - a^{3} b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b^{2} - 3 \, a^{6} b^{4} + 3 \, a^{4} b^{6} - a^{2} b^{8}\right )} d\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\csc {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 452 vs.
\(2 (157) = 314\).
time = 0.55, size = 452, normalized size = 2.77 \begin {gather*} \frac {\frac {2 \, {\left (3 \, a^{2} b + b^{3}\right )} \log \left ({\left | -a - b - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {\log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {9 \, a^{3} b + 15 \, a^{2} b^{2} + 3 \, a b^{3} - 3 \, b^{4} + \frac {18 \, a^{3} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{2} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {10 \, a b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2 \, b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, a^{3} b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {9 \, a^{2} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3 \, b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{{\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} {\left (a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.39, size = 182, normalized size = 1.12 \begin {gather*} \frac {\ln \left (\cos \left (c+d\,x\right )-1\right )}{2\,d\,{\left (a+b\right )}^3}-\frac {\frac {\cos \left (c+d\,x\right )\,\left (b^4-3\,a^2\,b^2\right )}{a\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {b\,\left (b^4-5\,a^2\,b^2\right )}{2\,a^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,{\cos \left (c+d\,x\right )}^2+2\,a\,b\,\cos \left (c+d\,x\right )+b^2\right )}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )}{2\,d\,{\left (a-b\right )}^3}-\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (\frac {1}{2\,{\left (a+b\right )}^3}-\frac {1}{2\,{\left (a-b\right )}^3}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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