Integrand size = 23, antiderivative size = 126 \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)^{7/2} d}+\frac {\left (a^2-3 a b+3 b^2\right ) \sin (c+d x)}{(a-b)^3 d}-\frac {(2 a-3 b) \sin ^3(c+d x)}{3 (a-b)^2 d}+\frac {\sin ^5(c+d x)}{5 (a-b) d} \]
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Time = 0.18 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3757, 398, 214} \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=\frac {\left (a^2-3 a b+3 b^2\right ) \sin (c+d x)}{d (a-b)^3}-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a-b)^{7/2}}+\frac {\sin ^5(c+d x)}{5 d (a-b)}-\frac {(2 a-3 b) \sin ^3(c+d x)}{3 d (a-b)^2} \]
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Rule 214
Rule 398
Rule 3757
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{a-(a-b) x^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^2-3 a b+3 b^2}{(a-b)^3}-\frac {(2 a-3 b) x^2}{(a-b)^2}+\frac {x^4}{a-b}-\frac {b^3}{(a-b)^3 \left (a-(a-b) x^2\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\left (a^2-3 a b+3 b^2\right ) \sin (c+d x)}{(a-b)^3 d}-\frac {(2 a-3 b) \sin ^3(c+d x)}{3 (a-b)^2 d}+\frac {\sin ^5(c+d x)}{5 (a-b) d}-\frac {b^3 \text {Subst}\left (\int \frac {1}{a-(a-b) x^2} \, dx,x,\sin (c+d x)\right )}{(a-b)^3 d} \\ & = -\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)^{7/2} d}+\frac {\left (a^2-3 a b+3 b^2\right ) \sin (c+d x)}{(a-b)^3 d}-\frac {(2 a-3 b) \sin ^3(c+d x)}{3 (a-b)^2 d}+\frac {\sin ^5(c+d x)}{5 (a-b) d} \\ \end{align*}
Time = 1.90 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.17 \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=\frac {\frac {120 b^3 \left (\log \left (\sqrt {a}-\sqrt {a-b} \sin (c+d x)\right )-\log \left (\sqrt {a}+\sqrt {a-b} \sin (c+d x)\right )\right )}{\sqrt {a} (a-b)^{7/2}}+\frac {30 \left (5 a^2-16 a b+19 b^2\right ) \sin (c+d x)}{(a-b)^3}+\frac {5 (5 a-9 b) \sin (3 (c+d x))}{(a-b)^2}+\frac {3 \sin (5 (c+d x))}{a-b}}{240 d} \]
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Time = 10.57 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.31
method | result | size |
derivativedivides | \(\frac {\frac {\frac {a^{2} \sin \left (d x +c \right )^{5}}{5}-\frac {2 a b \sin \left (d x +c \right )^{5}}{5}+\frac {b^{2} \sin \left (d x +c \right )^{5}}{5}-\frac {2 a^{2} \sin \left (d x +c \right )^{3}}{3}+\frac {5 a b \sin \left (d x +c \right )^{3}}{3}-b^{2} \sin \left (d x +c \right )^{3}+a^{2} \sin \left (d x +c \right )-3 a b \sin \left (d x +c \right )+3 b^{2} \sin \left (d x +c \right )}{\left (a -b \right )^{3}}-\frac {b^{3} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{\left (a -b \right )^{3} \sqrt {a \left (a -b \right )}}}{d}\) | \(165\) |
default | \(\frac {\frac {\frac {a^{2} \sin \left (d x +c \right )^{5}}{5}-\frac {2 a b \sin \left (d x +c \right )^{5}}{5}+\frac {b^{2} \sin \left (d x +c \right )^{5}}{5}-\frac {2 a^{2} \sin \left (d x +c \right )^{3}}{3}+\frac {5 a b \sin \left (d x +c \right )^{3}}{3}-b^{2} \sin \left (d x +c \right )^{3}+a^{2} \sin \left (d x +c \right )-3 a b \sin \left (d x +c \right )+3 b^{2} \sin \left (d x +c \right )}{\left (a -b \right )^{3}}-\frac {b^{3} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{\left (a -b \right )^{3} \sqrt {a \left (a -b \right )}}}{d}\) | \(165\) |
risch | \(-\frac {5 i {\mathrm e}^{i \left (d x +c \right )} a^{2}}{16 \left (a -b \right )^{3} d}+\frac {i {\mathrm e}^{i \left (d x +c \right )} a b}{\left (a -b \right )^{3} d}-\frac {19 i {\mathrm e}^{i \left (d x +c \right )} b^{2}}{16 \left (a -b \right )^{3} d}+\frac {5 i {\mathrm e}^{-i \left (d x +c \right )} a^{2}}{16 \left (a -b \right ) \left (a^{2}-2 a b +b^{2}\right ) d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} a b}{\left (a -b \right ) \left (a^{2}-2 a b +b^{2}\right ) d}+\frac {19 i {\mathrm e}^{-i \left (d x +c \right )} b^{2}}{16 \left (a -b \right ) \left (a^{2}-2 a b +b^{2}\right ) d}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{\sqrt {a^{2}-a b}}-1\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right )^{3} d}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{\sqrt {a^{2}-a b}}-1\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right )^{3} d}+\frac {\sin \left (5 d x +5 c \right )}{80 \left (a -b \right ) d}+\frac {5 \sin \left (3 d x +3 c \right ) a}{48 \left (a -b \right )^{2} d}-\frac {3 \sin \left (3 d x +3 c \right ) b}{16 \left (a -b \right )^{2} d}\) | \(374\) |
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Time = 0.33 (sec) , antiderivative size = 395, normalized size of antiderivative = 3.13 \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=\left [-\frac {15 \, \sqrt {a^{2} - a b} b^{3} \log \left (-\frac {{\left (a - b\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - a b} \sin \left (d x + c\right ) - 2 \, a + b}{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + b}\right ) - 2 \, {\left (3 \, {\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} \cos \left (d x + c\right )^{4} + 8 \, a^{4} - 34 \, a^{3} b + 59 \, a^{2} b^{2} - 33 \, a b^{3} + {\left (4 \, a^{4} - 17 \, a^{3} b + 22 \, a^{2} b^{2} - 9 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right )} d}, \frac {15 \, \sqrt {-a^{2} + a b} b^{3} \arctan \left (\frac {\sqrt {-a^{2} + a b} \sin \left (d x + c\right )}{a}\right ) + {\left (3 \, {\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} \cos \left (d x + c\right )^{4} + 8 \, a^{4} - 34 \, a^{3} b + 59 \, a^{2} b^{2} - 33 \, a b^{3} + {\left (4 \, a^{4} - 17 \, a^{3} b + 22 \, a^{2} b^{2} - 9 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{15 \, {\left (a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right )} d}\right ] \]
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Timed out. \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (114) = 228\).
Time = 0.51 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.53 \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=-\frac {\frac {15 \, b^{3} \arctan \left (-\frac {a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{\sqrt {-a^{2} + a b}}\right )}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \sqrt {-a^{2} + a b}} - \frac {3 \, a^{4} \sin \left (d x + c\right )^{5} - 12 \, a^{3} b \sin \left (d x + c\right )^{5} + 18 \, a^{2} b^{2} \sin \left (d x + c\right )^{5} - 12 \, a b^{3} \sin \left (d x + c\right )^{5} + 3 \, b^{4} \sin \left (d x + c\right )^{5} - 10 \, a^{4} \sin \left (d x + c\right )^{3} + 45 \, a^{3} b \sin \left (d x + c\right )^{3} - 75 \, a^{2} b^{2} \sin \left (d x + c\right )^{3} + 55 \, a b^{3} \sin \left (d x + c\right )^{3} - 15 \, b^{4} \sin \left (d x + c\right )^{3} + 15 \, a^{4} \sin \left (d x + c\right ) - 75 \, a^{3} b \sin \left (d x + c\right ) + 150 \, a^{2} b^{2} \sin \left (d x + c\right ) - 135 \, a b^{3} \sin \left (d x + c\right ) + 45 \, b^{4} \sin \left (d x + c\right )}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}}}{15 \, d} \]
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Time = 15.39 (sec) , antiderivative size = 1493, normalized size of antiderivative = 11.85 \[ \int \frac {\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx=\text {Too large to display} \]
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