Integrand size = 21, antiderivative size = 238 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 \left (a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2} d}-\frac {b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))} \]
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Time = 0.47 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2803, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {2 \left (a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 d \sqrt {a^2-b^2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))} \]
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Rule 210
Rule 632
Rule 2739
Rule 2803
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (6 \left (a^2-2 b^2\right )-a b \sin (c+d x)-\left (3 a^2-8 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{3 a^2 b} \\ & = -\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (-2 b \left (7 a^2-12 b^2\right )+4 a b^2 \sin (c+d x)+6 b \left (a^2-2 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 a^3 b} \\ & = \frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (6 b^2 \left (3 a^2-4 b^2\right )+6 a b \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 a^4 b} \\ & = \frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}+\frac {\left (b \left (3 a^2-4 b^2\right )\right ) \int \csc (c+d x) \, dx}{a^5}+\frac {\left (a^4-5 a^2 b^2+4 b^4\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^5} \\ & = -\frac {b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}+\frac {\left (2 \left (a^4-5 a^2 b^2+4 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d} \\ & = -\frac {b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}-\frac {\left (4 \left (a^4-5 a^2 b^2+4 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d} \\ & = \frac {2 \left (a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2} d}-\frac {b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {\left (7 a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))} \\ \end{align*}
Time = 6.15 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.69 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 \left (a^4-5 a^2 b^2+4 b^4\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2} d}+\frac {\left (4 a^2 \cos \left (\frac {1}{2} (c+d x)\right )-9 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{6 a^4 d}+\frac {b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{4 a^3 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}+\frac {\left (-3 a^2 b+4 b^3\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}+\frac {\left (3 a^2 b-4 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}-\frac {b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{4 a^3 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (-4 a^2 \sin \left (\frac {1}{2} (c+d x)\right )+9 b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}+\frac {a^2 b \cos (c+d x)-b^3 \cos (c+d x)}{a^4 d (a+b \sin (c+d x))}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 a^2 d} \]
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Time = 0.71 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.21
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{8 a^{4}}-\frac {1}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{2}+12 b^{2}}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (3 a^{2}-4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}+\frac {\frac {2 \left (b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+b a \left (a^{2}-b^{2}\right )\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {2 \left (a^{4}-5 a^{2} b^{2}+4 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{5}}}{d}\) | \(287\) |
default | \(\frac {\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{8 a^{4}}-\frac {1}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{2}+12 b^{2}}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (3 a^{2}-4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}+\frac {\frac {2 \left (b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+b a \left (a^{2}-b^{2}\right )\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {2 \left (a^{4}-5 a^{2} b^{2}+4 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{5}}}{d}\) | \(287\) |
risch | \(\frac {-14 a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-24 i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+\frac {50 i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}}{3}+14 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+8 i b^{3}-4 a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+20 a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-28 a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+24 i b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+2 i a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-14 i a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+12 a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-8 i b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-\frac {14 i a^{2} b}{3}-\frac {22 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{3}+2 a^{3} {\mathrm e}^{7 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right ) a^{4} d}+\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{3}}-\frac {4 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) b^{2}}{d \,a^{5}}-\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{3}}+\frac {4 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {-i a +\sqrt {-a^{2}+b^{2}}}{b}\right ) b^{2}}{d \,a^{5}}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{3} d}+\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{5} d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{3} d}-\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{5} d}\) | \(571\) |
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Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (227) = 454\).
Time = 0.41 (sec) , antiderivative size = 1149, normalized size of antiderivative = 4.83 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.49 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.50 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {24 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} + \frac {48 \, {\left (a^{4} - 5 \, a^{2} b^{2} + 4 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{5}} + \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} + \frac {48 \, {\left (a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3} b - a b^{3}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{5}} - \frac {132 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 176 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
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Time = 10.67 (sec) , antiderivative size = 973, normalized size of antiderivative = 4.09 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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