Integrand size = 21, antiderivative size = 289 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\left (2 a^4-19 a^2 b^2+20 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}-\frac {b \left (9 a^2-20 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))} \]
[Out]
Time = 0.73 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2803, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {b \left (9 a^2-20 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\left (2 a^4-19 a^2 b^2+20 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d \sqrt {a^2-b^2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2} \]
[In]
[Out]
Rule 210
Rule 632
Rule 2739
Rule 2803
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^3(c+d x) \left (2 \left (3 a^2-10 b^2\right )-2 a b \sin (c+d x)-3 \left (a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{6 a^2 b} \\ & = \frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (12 \left (a^4-6 a^2 b^2+5 b^4\right )-5 a b \left (a^2-b^2\right ) \sin (c+d x)-2 \left (3 a^4-23 a^2 b^2+20 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 a^3 b \left (a^2-b^2\right )} \\ & = -\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (-2 b \left (17 a^4-77 a^2 b^2+60 b^4\right )+20 a b^2 \left (a^2-b^2\right ) \sin (c+d x)+12 b \left (a^4-6 a^2 b^2+5 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12 a^4 b \left (a^2-b^2\right )} \\ & = \frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (6 b^2 \left (9 a^4-29 a^2 b^2+20 b^4\right )+12 a b \left (a^4-6 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12 a^5 b \left (a^2-b^2\right )} \\ & = \frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\left (b \left (9 a^4-29 a^2 b^2+20 b^4\right )\right ) \int \csc (c+d x) \, dx}{2 a^6 \left (a^2-b^2\right )}+\frac {\left (2 a^6-21 a^4 b^2+39 a^2 b^4-20 b^6\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^6 \left (a^2-b^2\right )} \\ & = -\frac {b \left (9 a^2-20 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\left (2 a^4-19 a^2 b^2+20 b^4\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^6 d} \\ & = -\frac {b \left (9 a^2-20 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}-\frac {\left (2 \left (2 a^4-19 a^2 b^2+20 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^6 d} \\ & = \frac {\left (2 a^4-19 a^2 b^2+20 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}-\frac {b \left (9 a^2-20 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))} \\ \end{align*}
Time = 6.14 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.59 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\left (2 a^4-19 a^2 b^2+20 b^4\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}+\frac {\left (2 a^2 \cos \left (\frac {1}{2} (c+d x)\right )-9 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{3 a^5 d}+\frac {3 b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^3 d}+\frac {\left (-9 a^2 b+20 b^3\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {\left (9 a^2 b-20 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}-\frac {3 b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (-2 a^2 \sin \left (\frac {1}{2} (c+d x)\right )+9 b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^5 d}+\frac {a^2 b \cos (c+d x)-b^3 \cos (c+d x)}{2 a^4 d (a+b \sin (c+d x))^2}+\frac {3 a^2 b \cos (c+d x)-8 b^3 \cos (c+d x)}{2 a^5 d (a+b \sin (c+d x))}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 a^3 d} \]
[In]
[Out]
Time = 0.92 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.25
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{8 a^{5}}-\frac {1}{24 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{2}+24 b^{2}}{8 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (9 a^{2}-20 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{6}}+\frac {\frac {2 \left (\left (\frac {5}{2} a^{3} b^{2}-5 a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (4 a^{4}-a^{2} b^{2}-18 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (11 a^{2}-26 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (4 a^{2}-9 b^{2}\right )}{2}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (2 a^{4}-19 a^{2} b^{2}+20 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{6}}}{d}\) | \(360\) |
default | \(\frac {\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{3}-3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{8 a^{5}}-\frac {1}{24 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{2}+24 b^{2}}{8 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 b}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (9 a^{2}-20 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{6}}+\frac {\frac {2 \left (\left (\frac {5}{2} a^{3} b^{2}-5 a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (4 a^{4}-a^{2} b^{2}-18 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (11 a^{2}-26 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (4 a^{2}-9 b^{2}\right )}{2}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (2 a^{4}-19 a^{2} b^{2}+20 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{6}}}{d}\) | \(360\) |
risch | \(\frac {-60 i b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-50 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+252 i b^{2} a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+6 a^{3} b \,{\mathrm e}^{9 i \left (d x +c \right )}-30 a \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-360 i b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+92 i b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-298 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )} a^{2}-60 a^{3} b \,{\mathrm e}^{7 i \left (d x +c \right )}+300 b^{3} a \,{\mathrm e}^{7 i \left (d x +c \right )}-60 i b^{4}+17 i a^{2} b^{2}-63 i a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+204 a^{3} {\mathrm e}^{5 i \left (d x +c \right )} b -720 b^{3} a \,{\mathrm e}^{5 i \left (d x +c \right )}+240 i b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+240 i b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-102 i a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-212 b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+660 b^{3} a \,{\mathrm e}^{3 i \left (d x +c \right )}+102 i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+18 i a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+62 b \,a^{3} {\mathrm e}^{i \left (d x +c \right )}-210 \,{\mathrm e}^{i \left (d x +c \right )} b^{3} a}{3 \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 )^{2} a^{5} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,a^{2}}+\frac {19 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2}}{2 \sqrt {-a^{2}+b^{2}}\, d \,a^{4}}-\frac {10 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{4}}{\sqrt {-a^{2}+b^{2}}\, d \,a^{6}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,a^{2}}-\frac {19 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2}}{2 \sqrt {-a^{2}+b^{2}}\, d \,a^{4}}+\frac {10 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{4}}{\sqrt {-a^{2}+b^{2}}\, d \,a^{6}}-\frac {9 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 a^{4} d}+\frac {10 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{6} d}+\frac {9 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 a^{4} d}-\frac {10 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{6} d}\) | \(924\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 972 vs. \(2 (274) = 548\).
Time = 0.58 (sec) , antiderivative size = 2027, normalized size of antiderivative = 7.01 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.38 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.56 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {12 \, {\left (9 \, a^{2} b - 20 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{6}} + \frac {24 \, {\left (2 \, a^{4} - 19 \, a^{2} b^{2} + 20 \, 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^{6}} + \frac {24 \, {\left (5 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 10 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 18 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 11 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 26 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, a^{4} b - 9 \, a^{2} 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 )}^{2} a^{6}} + \frac {a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{9}} - \frac {198 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 440 \, 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} + 72 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
[In]
[Out]
Time = 11.30 (sec) , antiderivative size = 1261, normalized size of antiderivative = 4.36 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
[In]
[Out]