Optimal. Leaf size=221 \[ \frac {b \csc ^3(c+d x)}{a^4 d}-\frac {\csc ^4(c+d x)}{4 a^3 d}-\frac {2 b \left (3 a^2-5 b^2\right ) \csc (c+d x)}{a^6 d}+\frac {\left (a^2-b^2\right )^2}{2 a^5 d (a+b \sin (c+d x))^2}+\frac {\left (a^2-3 b^2\right ) \csc ^2(c+d x)}{a^5 d}+\frac {\left (a^4-12 a^2 b^2+15 b^4\right ) \log (\sin (c+d x))}{a^7 d}-\frac {\left (a^4-12 a^2 b^2+15 b^4\right ) \log (a+b \sin (c+d x))}{a^7 d}+\frac {a^4-6 a^2 b^2+5 b^4}{a^6 d (a+b \sin (c+d x))} \]
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Rubi [A] time = 0.21, antiderivative size = 221, 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 = {2721, 894} \[ \frac {-6 a^2 b^2+a^4+5 b^4}{a^6 d (a+b \sin (c+d x))}+\frac {\left (a^2-b^2\right )^2}{2 a^5 d (a+b \sin (c+d x))^2}+\frac {\left (a^2-3 b^2\right ) \csc ^2(c+d x)}{a^5 d}-\frac {2 b \left (3 a^2-5 b^2\right ) \csc (c+d x)}{a^6 d}+\frac {\left (-12 a^2 b^2+a^4+15 b^4\right ) \log (\sin (c+d x))}{a^7 d}-\frac {\left (-12 a^2 b^2+a^4+15 b^4\right ) \log (a+b \sin (c+d x))}{a^7 d}+\frac {b \csc ^3(c+d x)}{a^4 d}-\frac {\csc ^4(c+d x)}{4 a^3 d} \]
Antiderivative was successfully verified.
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Rule 894
Rule 2721
Rubi steps
\begin {align*} \int \frac {\cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{x^5 (a+x)^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {b^4}{a^3 x^5}-\frac {3 b^4}{a^4 x^4}+\frac {2 b^2 \left (-a^2+3 b^2\right )}{a^5 x^3}+\frac {2 \left (3 a^2 b^2-5 b^4\right )}{a^6 x^2}+\frac {a^4-12 a^2 b^2+15 b^4}{a^7 x}-\frac {\left (a^2-b^2\right )^2}{a^5 (a+x)^3}+\frac {-a^4+6 a^2 b^2-5 b^4}{a^6 (a+x)^2}+\frac {-a^4+12 a^2 b^2-15 b^4}{a^7 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {2 b \left (3 a^2-5 b^2\right ) \csc (c+d x)}{a^6 d}+\frac {\left (a^2-3 b^2\right ) \csc ^2(c+d x)}{a^5 d}+\frac {b \csc ^3(c+d x)}{a^4 d}-\frac {\csc ^4(c+d x)}{4 a^3 d}+\frac {\left (a^4-12 a^2 b^2+15 b^4\right ) \log (\sin (c+d x))}{a^7 d}-\frac {\left (a^4-12 a^2 b^2+15 b^4\right ) \log (a+b \sin (c+d x))}{a^7 d}+\frac {\left (a^2-b^2\right )^2}{2 a^5 d (a+b \sin (c+d x))^2}+\frac {a^4-6 a^2 b^2+5 b^4}{a^6 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 5.31, size = 195, normalized size = 0.88 \[ \frac {-a^4 \csc ^4(c+d x)+\frac {2 \left (a^3-a b^2\right )^2}{(a+b \sin (c+d x))^2}+4 a^3 b \csc ^3(c+d x)+4 a^2 \left (a^2-3 b^2\right ) \csc ^2(c+d x)-8 a b \left (3 a^2-5 b^2\right ) \csc (c+d x)+\frac {4 a \left (a^4-6 a^2 b^2+5 b^4\right )}{a+b \sin (c+d x)}+4 \left (a^4-12 a^2 b^2+15 b^4\right ) \log (\sin (c+d x))-4 \left (a^4-12 a^2 b^2+15 b^4\right ) \log (a+b \sin (c+d x))}{4 a^7 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 754, normalized size = 3.41 \[ -\frac {9 \, a^{6} - 77 \, a^{4} b^{2} + 90 \, a^{2} b^{4} + 6 \, {\left (a^{6} - 12 \, a^{4} b^{2} + 15 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{4} - {\left (16 \, a^{6} - 149 \, a^{4} b^{2} + 180 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{4} b^{2} - 12 \, a^{2} b^{4} + 15 \, b^{6}\right )} \cos \left (d x + c\right )^{6} - a^{6} + 11 \, a^{4} b^{2} - 3 \, a^{2} b^{4} - 15 \, b^{6} - {\left (a^{6} - 9 \, a^{4} b^{2} - 21 \, a^{2} b^{4} + 45 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + {\left (2 \, a^{6} - 21 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + 45 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5} + {\left (a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 4 \, {\left ({\left (a^{4} b^{2} - 12 \, a^{2} b^{4} + 15 \, b^{6}\right )} \cos \left (d x + c\right )^{6} - a^{6} + 11 \, a^{4} b^{2} - 3 \, a^{2} b^{4} - 15 \, b^{6} - {\left (a^{6} - 9 \, a^{4} b^{2} - 21 \, a^{2} b^{4} + 45 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + {\left (2 \, a^{6} - 21 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + 45 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5} + {\left (a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 2 \, {\left (5 \, a^{5} b + 14 \, a^{3} b^{3} - 30 \, a b^{5} - 2 \, {\left (a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (2 \, a^{5} b + 19 \, a^{3} b^{3} - 30 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left (a^{7} b^{2} d \cos \left (d x + c\right )^{6} - {\left (a^{9} + 3 \, a^{7} b^{2}\right )} d \cos \left (d x + c\right )^{4} + {\left (2 \, a^{9} + 3 \, a^{7} b^{2}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{9} + a^{7} b^{2}\right )} d - 2 \, {\left (a^{8} b d \cos \left (d x + c\right )^{4} - 2 \, a^{8} b d \cos \left (d x + c\right )^{2} + a^{8} b 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 = 0.84, size = 327, normalized size = 1.48 \[ \frac {\frac {12 \, {\left (a^{4} - 12 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{7}} - \frac {12 \, {\left (a^{4} b - 12 \, a^{2} b^{3} + 15 \, b^{5}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{7} b} + \frac {6 \, {\left (3 \, a^{4} b^{2} \sin \left (d x + c\right )^{2} - 36 \, a^{2} b^{4} \sin \left (d x + c\right )^{2} + 45 \, b^{6} \sin \left (d x + c\right )^{2} + 8 \, a^{5} b \sin \left (d x + c\right ) - 84 \, a^{3} b^{3} \sin \left (d x + c\right ) + 100 \, a b^{5} \sin \left (d x + c\right ) + 6 \, a^{6} - 50 \, a^{4} b^{2} + 56 \, a^{2} b^{4}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2} a^{7}} - \frac {25 \, a^{4} \sin \left (d x + c\right )^{4} - 300 \, a^{2} b^{2} \sin \left (d x + c\right )^{4} + 375 \, b^{4} \sin \left (d x + c\right )^{4} + 72 \, a^{3} b \sin \left (d x + c\right )^{3} - 120 \, a b^{3} \sin \left (d x + c\right )^{3} - 12 \, a^{4} \sin \left (d x + c\right )^{2} + 36 \, a^{2} b^{2} \sin \left (d x + c\right )^{2} - 12 \, a^{3} b \sin \left (d x + c\right ) + 3 \, a^{4}}{a^{7} \sin \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 348, normalized size = 1.57 \[ -\frac {\ln \left (a +b \sin \left (d x +c \right )\right )}{a^{3} d}+\frac {12 \ln \left (a +b \sin \left (d x +c \right )\right ) b^{2}}{d \,a^{5}}-\frac {15 \ln \left (a +b \sin \left (d x +c \right )\right ) b^{4}}{d \,a^{7}}+\frac {1}{a^{2} d \left (a +b \sin \left (d x +c \right )\right )}-\frac {6 b^{2}}{d \,a^{4} \left (a +b \sin \left (d x +c \right )\right )}+\frac {5 b^{4}}{d \,a^{6} \left (a +b \sin \left (d x +c \right )\right )}+\frac {1}{2 a d \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {b^{2}}{d \,a^{3} \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {b^{4}}{2 d \,a^{5} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {1}{4 d \,a^{3} \sin \left (d x +c \right )^{4}}+\frac {1}{a^{3} d \sin \left (d x +c \right )^{2}}-\frac {3 b^{2}}{d \,a^{5} \sin \left (d x +c \right )^{2}}+\frac {\ln \left (\sin \left (d x +c \right )\right )}{a^{3} d}-\frac {12 \ln \left (\sin \left (d x +c \right )\right ) b^{2}}{d \,a^{5}}+\frac {15 \ln \left (\sin \left (d x +c \right )\right ) b^{4}}{d \,a^{7}}+\frac {b}{d \,a^{4} \sin \left (d x +c \right )^{3}}-\frac {6 b}{d \,a^{4} \sin \left (d x +c \right )}+\frac {10 b^{3}}{d \,a^{6} \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 236, normalized size = 1.07 \[ \frac {\frac {2 \, a^{4} b \sin \left (d x + c\right ) + 4 \, {\left (a^{4} b - 12 \, a^{2} b^{3} + 15 \, b^{5}\right )} \sin \left (d x + c\right )^{5} - a^{5} + 6 \, {\left (a^{5} - 12 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \sin \left (d x + c\right )^{4} - 4 \, {\left (4 \, a^{4} b - 5 \, a^{2} b^{3}\right )} \sin \left (d x + c\right )^{3} + {\left (4 \, a^{5} - 5 \, a^{3} b^{2}\right )} \sin \left (d x + c\right )^{2}}{a^{6} b^{2} \sin \left (d x + c\right )^{6} + 2 \, a^{7} b \sin \left (d x + c\right )^{5} + a^{8} \sin \left (d x + c\right )^{4}} - \frac {4 \, {\left (a^{4} - 12 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{7}} + \frac {4 \, {\left (a^{4} - 12 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{7}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.26, size = 563, normalized size = 2.55 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {23\,a^5}{4}-172\,a^3\,b^2+272\,a\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (27\,a^4\,b-40\,a^2\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (-134\,a^4\,b+200\,a^2\,b^3+128\,b^5\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (106\,a^4\,b-336\,a^2\,b^3+192\,b^5\right )-\frac {a^5}{4}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {5\,a^5}{2}-5\,a^3\,b^2\right )+a^4\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (3\,a^6-276\,a^4\,b^2+768\,a^2\,b^4-352\,b^6\right )}{a}}{d\,\left (16\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+16\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (32\,a^8+64\,a^6\,b^2\right )+64\,a^7\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+64\,a^7\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a^3\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {3\,\left (a^2+4\,b^2\right )}{32\,a^5}+\frac {3}{32\,a^3}-\frac {9\,b^2}{8\,a^5}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {6\,b\,\left (\frac {3\,\left (a^2+4\,b^2\right )}{16\,a^5}+\frac {3}{16\,a^3}-\frac {9\,b^2}{4\,a^5}\right )}{a}-\frac {192\,a^2\,b+128\,b^3}{256\,a^6}+\frac {9\,b\,\left (a^2+4\,b^2\right )}{8\,a^6}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^4-12\,a^2\,b^2+15\,b^4\right )}{a^7\,d}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,a^4\,d}-\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (a^4-12\,a^2\,b^2+15\,b^4\right )}{a^7\,d} \]
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 \[ \int \frac {\cot ^{5}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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