Optimal. Leaf size=226 \[ \frac {b \csc ^4(c+d x)}{2 a^3 d}-\frac {\csc ^5(c+d x)}{5 a^2 d}-\frac {b \left (a^2-b^2\right )^2}{a^6 d (a+b \sin (c+d x))}-\frac {2 b \left (a^2-b^2\right ) \csc ^2(c+d x)}{a^5 d}+\frac {\left (2 a^2-3 b^2\right ) \csc ^3(c+d x)}{3 a^4 d}-\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right ) \log (\sin (c+d x))}{a^7 d}+\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right ) \log (a+b \sin (c+d x))}{a^7 d}-\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \csc (c+d x)}{a^6 d} \]
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Rubi [A] time = 0.26, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2837, 12, 894} \[ -\frac {b \left (a^2-b^2\right )^2}{a^6 d (a+b \sin (c+d x))}+\frac {\left (2 a^2-3 b^2\right ) \csc ^3(c+d x)}{3 a^4 d}-\frac {2 b \left (a^2-b^2\right ) \csc ^2(c+d x)}{a^5 d}-\frac {\left (-6 a^2 b^2+a^4+5 b^4\right ) \csc (c+d x)}{a^6 d}-\frac {2 b \left (-4 a^2 b^2+a^4+3 b^4\right ) \log (\sin (c+d x))}{a^7 d}+\frac {2 b \left (-4 a^2 b^2+a^4+3 b^4\right ) \log (a+b \sin (c+d x))}{a^7 d}+\frac {b \csc ^4(c+d x)}{2 a^3 d}-\frac {\csc ^5(c+d x)}{5 a^2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 894
Rule 2837
Rubi steps
\begin {align*} \int \frac {\cot ^5(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^6 \left (b^2-x^2\right )^2}{x^6 (a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {b \operatorname {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{x^6 (a+x)^2} \, dx,x,b \sin (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 \sin (c+d x)\right )}{d}\\ &=-\frac {\left (a^4-6 a^2 b^2+5 b^4\right ) \csc (c+d x)}{a^6 d}-\frac {2 b \left (a^2-b^2\right ) \csc ^2(c+d x)}{a^5 d}+\frac {\left (2 a^2-3 b^2\right ) \csc ^3(c+d x)}{3 a^4 d}+\frac {b \csc ^4(c+d x)}{2 a^3 d}-\frac {\csc ^5(c+d x)}{5 a^2 d}-\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right ) \log (\sin (c+d x))}{a^7 d}+\frac {2 b \left (a^4-4 a^2 b^2+3 b^4\right ) \log (a+b \sin (c+d x))}{a^7 d}-\frac {b \left (a^2-b^2\right )^2}{a^6 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 3.17, size = 220, normalized size = 0.97 \[ \frac {-6 a^6 \csc ^6(c+d x)+9 a^5 b \csc ^5(c+d x)+\left (30 a^3 b^3-40 a^5 b\right ) \csc ^3(c+d x)+5 a^4 \left (4 a^2-3 b^2\right ) \csc ^4(c+d x)-30 a^2 \left (a^4-4 a^2 b^2+3 b^4\right ) \csc ^2(c+d x)-60 b^2 \left (a^4-4 a^2 b^2+3 b^4\right ) (\log (\sin (c+d x))-\log (a+b \sin (c+d x)))-60 a b \left (a^4-4 a^2 b^2+3 b^4\right ) \csc (c+d x) (-\log (a+b \sin (c+d x))+\log (\sin (c+d x))+1)}{30 a^7 d (a \csc (c+d x)+b)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.00, size = 696, normalized size = 3.08 \[ \frac {16 \, a^{6} - 105 \, a^{4} b^{2} + 90 \, a^{2} b^{4} + 30 \, {\left (a^{6} - 4 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{4} - 5 \, {\left (8 \, a^{6} - 45 \, a^{4} b^{2} + 36 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + 60 \, {\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 (a^{5} b - 4 \, a^{3} b^{3} + 3 \, a b^{5} + {\left (a^{5} b - 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{5} b - 4 \, a^{3} b^{3} + 3 \, 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 ) - 60 \, {\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 (a^{5} b - 4 \, a^{3} b^{3} + 3 \, a b^{5} + {\left (a^{5} b - 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{5} b - 4 \, a^{3} b^{3} + 3 \, 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 ) + {\left (91 \, a^{5} b - 270 \, a^{3} b^{3} + 180 \, a b^{5} + 60 \, {\left (a^{5} b - 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} - 10 \, {\left (16 \, a^{5} b - 51 \, a^{3} b^{3} + 36 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\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 )^{4} - 2 \, a^{8} d \cos \left (d x + c\right )^{2} + a^{8} 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 = 0.24, size = 332, normalized size = 1.47 \[ -\frac {\frac {60 \, {\left (a^{4} b - 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{7}} - \frac {60 \, {\left (a^{4} b^{2} - 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{7} b} + \frac {30 \, {\left (2 \, a^{4} b^{2} \sin \left (d x + c\right ) - 8 \, a^{2} b^{4} \sin \left (d x + c\right ) + 6 \, b^{6} \sin \left (d x + c\right ) + 3 \, a^{5} b - 10 \, a^{3} b^{3} + 7 \, a b^{5}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} a^{7}} - \frac {137 \, a^{4} b \sin \left (d x + c\right )^{5} - 548 \, a^{2} b^{3} \sin \left (d x + c\right )^{5} + 411 \, b^{5} \sin \left (d x + c\right )^{5} - 30 \, a^{5} \sin \left (d x + c\right )^{4} + 180 \, a^{3} b^{2} \sin \left (d x + c\right )^{4} - 150 \, a b^{4} \sin \left (d x + c\right )^{4} - 60 \, a^{4} b \sin \left (d x + c\right )^{3} + 60 \, a^{2} b^{3} \sin \left (d x + c\right )^{3} + 20 \, a^{5} \sin \left (d x + c\right )^{2} - 30 \, a^{3} b^{2} \sin \left (d x + c\right )^{2} + 15 \, a^{4} b \sin \left (d x + c\right ) - 6 \, a^{5}}{a^{7} \sin \left (d x + c\right )^{5}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.90, size = 343, normalized size = 1.52 \[ -\frac {b}{d \,a^{2} \left (a +b \sin \left (d x +c \right )\right )}+\frac {2 b^{3}}{d \,a^{4} \left (a +b \sin \left (d x +c \right )\right )}-\frac {b^{5}}{d \,a^{6} \left (a +b \sin \left (d x +c \right )\right )}+\frac {2 b \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,a^{3}}-\frac {8 b^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,a^{5}}+\frac {6 b^{5} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,a^{7}}-\frac {1}{5 d \,a^{2} \sin \left (d x +c \right )^{5}}+\frac {2}{3 d \,a^{2} \sin \left (d x +c \right )^{3}}-\frac {b^{2}}{d \,a^{4} \sin \left (d x +c \right )^{3}}-\frac {1}{d \,a^{2} \sin \left (d x +c \right )}+\frac {6 b^{2}}{d \,a^{4} \sin \left (d x +c \right )}-\frac {5 b^{4}}{d \,a^{6} \sin \left (d x +c \right )}+\frac {b}{2 d \,a^{3} \sin \left (d x +c \right )^{4}}-\frac {2 b}{d \,a^{3} \sin \left (d x +c \right )^{2}}+\frac {2 b^{3}}{d \,a^{5} \sin \left (d x +c \right )^{2}}-\frac {2 b \ln \left (\sin \left (d x +c \right )\right )}{a^{3} d}+\frac {8 b^{3} \ln \left (\sin \left (d x +c \right )\right )}{d \,a^{5}}-\frac {6 b^{5} \ln \left (\sin \left (d x +c \right )\right )}{d \,a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 225, normalized size = 1.00 \[ \frac {\frac {9 \, a^{4} b \sin \left (d x + c\right ) - 60 \, {\left (a^{4} b - 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \sin \left (d x + c\right )^{5} - 6 \, a^{5} - 30 \, {\left (a^{5} - 4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \sin \left (d x + c\right )^{4} - 10 \, {\left (4 \, a^{4} b - 3 \, a^{2} b^{3}\right )} \sin \left (d x + c\right )^{3} + 5 \, {\left (4 \, a^{5} - 3 \, a^{3} b^{2}\right )} \sin \left (d x + c\right )^{2}}{a^{6} b \sin \left (d x + c\right )^{6} + a^{7} \sin \left (d x + c\right )^{5}} + \frac {60 \, {\left (a^{4} b - 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{7}} - \frac {60 \, {\left (a^{4} b - 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{7}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.90, size = 628, normalized size = 2.78 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {64\,a^2+128\,b^2}{3072\,a^4}+\frac {1}{32\,a^2}-\frac {b^2}{6\,a^4}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {b^2}{2\,a^4}-\frac {4\,b\,\left (\frac {b\,\left (64\,a^2+128\,b^2\right )}{256\,a^5}-\frac {b}{8\,a^3}+\frac {4\,b\,\left (\frac {64\,a^2+128\,b^2}{1024\,a^4}+\frac {3}{32\,a^2}-\frac {b^2}{2\,a^4}\right )}{a}\right )}{a}+\frac {\left (64\,a^2+128\,b^2\right )\,\left (\frac {64\,a^2+128\,b^2}{1024\,a^4}+\frac {3}{32\,a^2}-\frac {b^2}{2\,a^4}\right )}{32\,a^2}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {23\,a^4\,b}{3}-8\,a^2\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {25\,a^5}{3}-56\,a^3\,b^2+48\,a\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (32\,a^4\,b-184\,a^2\,b^3+160\,b^5\right )+\frac {a^5}{5}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {22\,a^5}{15}-2\,a^3\,b^2\right )-\frac {3\,a^4\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (5\,a^6-74\,a^4\,b^2+104\,a^2\,b^4-32\,b^6\right )}{a}}{d\,\left (32\,a^7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,a^7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+64\,b\,a^6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {b\,\left (64\,a^2+128\,b^2\right )}{512\,a^5}-\frac {b}{16\,a^3}+\frac {2\,b\,\left (\frac {64\,a^2+128\,b^2}{1024\,a^4}+\frac {3}{32\,a^2}-\frac {b^2}{2\,a^4}\right )}{a}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a^4\,b-8\,a^2\,b^3+6\,b^5\right )}{a^7\,d}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,a^3\,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 (2\,a^4\,b-8\,a^2\,b^3+6\,b^5\right )}{a^7\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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