Integrand size = 21, antiderivative size = 350 \[ \int \frac {\tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {4 a^2 b^2 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2} d}-\frac {a^2 \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2} d}-\frac {2 b^2 \left (3 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2} d}+\frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}-\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}-\frac {a^2 b \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}-\frac {3 a^3 b \cos (c+d x)}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {2 a b^3 \cos (c+d x)}{\left (a^2-b^2\right )^3 d (a+b \sin (c+d x))} \]
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Time = 0.44 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2810, 2727, 2743, 2833, 12, 2739, 632, 210} \[ \int \frac {\tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {a^2 \left (2 a^2+b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{7/2}}-\frac {4 a^2 b^2 \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{7/2}}-\frac {2 b^2 \left (3 a^2+b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{7/2}}-\frac {a^2 b \cos (c+d x)}{2 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}-\frac {2 a b^3 \cos (c+d x)}{d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac {3 a^3 b \cos (c+d x)}{2 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac {\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}-\frac {\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)} \]
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Rule 12
Rule 210
Rule 632
Rule 2727
Rule 2739
Rule 2743
Rule 2810
Rule 2833
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{2 (a+b)^3 (-1+\sin (c+d x))}+\frac {1}{2 (a-b)^3 (1+\sin (c+d x))}-\frac {a^2}{\left (a^2-b^2\right ) (a+b \sin (c+d x))^3}-\frac {2 a b^2}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac {b^2 \left (3 a^2+b^2\right )}{\left (-a^2+b^2\right )^3 (a+b \sin (c+d x))}\right ) \, dx \\ & = \frac {\int \frac {1}{1+\sin (c+d x)} \, dx}{2 (a-b)^3}-\frac {\int \frac {1}{-1+\sin (c+d x)} \, dx}{2 (a+b)^3}-\frac {\left (2 a b^2\right ) \int \frac {1}{(a+b \sin (c+d x))^2} \, dx}{\left (a^2-b^2\right )^2}-\frac {a^2 \int \frac {1}{(a+b \sin (c+d x))^3} \, dx}{a^2-b^2}-\frac {\left (b^2 \left (3 a^2+b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right )^3} \\ & = \frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}-\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}-\frac {a^2 b \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}-\frac {2 a b^3 \cos (c+d x)}{\left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {\left (2 a b^2\right ) \int \frac {a}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right )^3}+\frac {a^2 \int \frac {-2 a+b \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{2 \left (a^2-b^2\right )^2}-\frac {\left (2 b^2 \left (3 a^2+b^2\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 )}{\left (a^2-b^2\right )^3 d} \\ & = \frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}-\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}-\frac {a^2 b \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}-\frac {3 a^3 b \cos (c+d x)}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {2 a b^3 \cos (c+d x)}{\left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {a^2 \int \frac {2 a^2+b^2}{a+b \sin (c+d x)} \, dx}{2 \left (a^2-b^2\right )^3}-\frac {\left (2 a^2 b^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right )^3}+\frac {\left (4 b^2 \left (3 a^2+b^2\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 )}{\left (a^2-b^2\right )^3 d} \\ & = -\frac {2 b^2 \left (3 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2} d}+\frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}-\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}-\frac {a^2 b \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}-\frac {3 a^3 b \cos (c+d x)}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {2 a b^3 \cos (c+d x)}{\left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {\left (a^2 \left (2 a^2+b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 \left (a^2-b^2\right )^3}-\frac {\left (4 a^2 b^2\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 )}{\left (a^2-b^2\right )^3 d} \\ & = -\frac {2 b^2 \left (3 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2} d}+\frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}-\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}-\frac {a^2 b \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}-\frac {3 a^3 b \cos (c+d x)}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {2 a b^3 \cos (c+d x)}{\left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac {\left (8 a^2 b^2\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 )}{\left (a^2-b^2\right )^3 d}-\frac {\left (a^2 \left (2 a^2+b^2\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 )}{\left (a^2-b^2\right )^3 d} \\ & = -\frac {4 a^2 b^2 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2} d}-\frac {2 b^2 \left (3 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2} d}+\frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}-\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}-\frac {a^2 b \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}-\frac {3 a^3 b \cos (c+d x)}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {2 a b^3 \cos (c+d x)}{\left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac {\left (2 a^2 \left (2 a^2+b^2\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 )}{\left (a^2-b^2\right )^3 d} \\ & = -\frac {4 a^2 b^2 \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2} d}-\frac {a^2 \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2} d}-\frac {2 b^2 \left (3 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2} d}+\frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}-\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}-\frac {a^2 b \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}-\frac {3 a^3 b \cos (c+d x)}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {2 a b^3 \cos (c+d x)}{\left (a^2-b^2\right )^3 d (a+b \sin (c+d x))} \\ \end{align*}
Time = 2.25 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.61 \[ \int \frac {\tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {-\frac {2 \left (2 a^4+11 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {2}{(a+b)^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {2}{(a-b)^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )-\frac {a b \cos (c+d x) \left (4 a^3+3 a b^2+b \left (3 a^2+4 b^2\right ) \sin (c+d x)\right )}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))^2}}{2 d} \]
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Time = 2.12 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {-\frac {1}{\left (a -b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{\left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2 \left (\frac {\left (\frac {5}{2} a^{3} b^{2}+a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (4 a^{4}+11 a^{2} b^{2}+6 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (11 a^{2}+10 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (4 a^{2}+3 b^{2}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (2 a^{4}+11 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}}{d}\) | \(258\) |
default | \(\frac {-\frac {1}{\left (a -b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{\left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2 \left (\frac {\left (\frac {5}{2} a^{3} b^{2}+a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (4 a^{4}+11 a^{2} b^{2}+6 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (11 a^{2}+10 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (4 a^{2}+3 b^{2}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (2 a^{4}+11 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}}{d}\) | \(258\) |
risch | \(\frac {i \left (4 i b^{5} {\mathrm e}^{3 i \left (d x +c \right )}-2 i b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+29 i a^{2} b^{3} {\mathrm e}^{i \left (d x +c \right )}+24 i a^{4} b \,{\mathrm e}^{3 i \left (d x +c \right )}+18 i a^{4} b \,{\mathrm e}^{i \left (d x +c \right )}-2 i b^{5} {\mathrm e}^{i \left (d x +c \right )}+6 a^{5} {\mathrm e}^{4 i \left (d x +c \right )}+33 a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+6 a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-2 i a^{4} b \,{\mathrm e}^{5 i \left (d x +c \right )}+2 i a^{2} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-11 i a^{2} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+14 a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+12 a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+4 a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-5 a^{3} b^{2}-10 a \,b^{4}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} d \left (a^{2}-b^{2}\right )^{3}}+\frac {a^{4} \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}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}+\frac {11 a^{2} \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}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} 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 ) b^{4}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}-\frac {a^{4} \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}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}-\frac {11 a^{2} \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}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} 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 ) b^{4}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}\) | \(808\) |
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Time = 0.32 (sec) , antiderivative size = 934, normalized size of antiderivative = 2.67 \[ \int \frac {\tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\left [\frac {4 \, a^{6} b - 12 \, a^{4} b^{3} + 12 \, a^{2} b^{5} - 4 \, b^{7} + 2 \, {\left (8 \, a^{6} b + a^{4} b^{3} - 11 \, a^{2} b^{5} + 2 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left ({\left (2 \, a^{4} b^{2} + 11 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (2 \, a^{5} b + 11 \, a^{3} b^{3} + 2 \, a b^{5}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (2 \, a^{6} + 13 \, a^{4} b^{2} + 13 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left (2 \, a^{7} - 6 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - 2 \, a b^{6} - 5 \, {\left (a^{5} b^{2} + a^{3} b^{4} - 2 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a^{10} - 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} + 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )\right )}}, \frac {2 \, a^{6} b - 6 \, a^{4} b^{3} + 6 \, a^{2} b^{5} - 2 \, b^{7} + {\left (8 \, a^{6} b + a^{4} b^{3} - 11 \, a^{2} b^{5} + 2 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left ({\left (2 \, a^{4} b^{2} + 11 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (2 \, a^{5} b + 11 \, a^{3} b^{3} + 2 \, a b^{5}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (2 \, a^{6} + 13 \, a^{4} b^{2} + 13 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - {\left (2 \, a^{7} - 6 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - 2 \, a b^{6} - 5 \, {\left (a^{5} b^{2} + a^{3} b^{4} - 2 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a^{10} - 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} + 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )\right )}}\right ] \]
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\[ \int \frac {\tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
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Exception generated. \[ \int \frac {\tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.41 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.10 \[ \int \frac {\tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\frac {{\left (2 \, a^{4} + 11 \, a^{2} b^{2} + 2 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2} b - b^{3}\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}} + \frac {5 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 11 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 11 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 10 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, a^{4} b + 3 \, a^{2} b^{3}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2}}}{d} \]
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Time = 17.80 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.79 \[ \int \frac {\tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {5\,\left (2\,a^4\,b+a^2\,b^3\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,a^5+a^3\,b^2+12\,a\,b^4\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a^4\,b+6\,a^2\,b^3+7\,b^5\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (2\,a^4\,b+11\,a^2\,b^3+2\,b^5\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-2\,a^4+29\,a^2\,b^2+18\,b^4\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (2\,a^4+11\,a^2\,b^2+2\,b^4\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-a^2-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a^2+4\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^2+4\,b^2\right )+4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {\mathrm {atan}\left (\frac {\frac {\left (2\,a^4+11\,a^2\,b^2+2\,b^4\right )\,\left (2\,a^6\,b-6\,a^4\,b^3+6\,a^2\,b^5-2\,b^7\right )}{2\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^4+11\,a^2\,b^2+2\,b^4\right )\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}{{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}}}{2\,a^4+11\,a^2\,b^2+2\,b^4}\right )\,\left (2\,a^4+11\,a^2\,b^2+2\,b^4\right )}{d\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}} \]
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