Integrand size = 21, antiderivative size = 277 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=\frac {77 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}-\frac {77 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}-\frac {77 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}+\frac {77 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}+\frac {77 d (d \tan (a+b x))^{3/2}}{48 b}-\frac {11 \cos ^2(a+b x) (d \tan (a+b x))^{7/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{11/2}}{4 b d^3} \]
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Time = 0.24 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2671, 294, 327, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=\frac {77 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}-\frac {77 d^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}+1\right )}{32 \sqrt {2} b}-\frac {77 d^{5/2} \log \left (\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}+\sqrt {d}\right )}{64 \sqrt {2} b}+\frac {77 d^{5/2} \log \left (\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}+\sqrt {d}\right )}{64 \sqrt {2} b}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{11/2}}{4 b d^3}+\frac {77 d (d \tan (a+b x))^{3/2}}{48 b}-\frac {11 \cos ^2(a+b x) (d \tan (a+b x))^{7/2}}{16 b d} \]
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Rule 210
Rule 294
Rule 303
Rule 327
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2671
Rubi steps \begin{align*} \text {integral}& = \frac {d \text {Subst}\left (\int \frac {x^{13/2}}{\left (d^2+x^2\right )^3} \, dx,x,d \tan (a+b x)\right )}{b} \\ & = -\frac {\cos ^4(a+b x) (d \tan (a+b x))^{11/2}}{4 b d^3}+\frac {(11 d) \text {Subst}\left (\int \frac {x^{9/2}}{\left (d^2+x^2\right )^2} \, dx,x,d \tan (a+b x)\right )}{8 b} \\ & = -\frac {11 \cos ^2(a+b x) (d \tan (a+b x))^{7/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{11/2}}{4 b d^3}+\frac {(77 d) \text {Subst}\left (\int \frac {x^{5/2}}{d^2+x^2} \, dx,x,d \tan (a+b x)\right )}{32 b} \\ & = \frac {77 d (d \tan (a+b x))^{3/2}}{48 b}-\frac {11 \cos ^2(a+b x) (d \tan (a+b x))^{7/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{11/2}}{4 b d^3}-\frac {\left (77 d^3\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{d^2+x^2} \, dx,x,d \tan (a+b x)\right )}{32 b} \\ & = \frac {77 d (d \tan (a+b x))^{3/2}}{48 b}-\frac {11 \cos ^2(a+b x) (d \tan (a+b x))^{7/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{11/2}}{4 b d^3}-\frac {\left (77 d^3\right ) \text {Subst}\left (\int \frac {x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{16 b} \\ & = \frac {77 d (d \tan (a+b x))^{3/2}}{48 b}-\frac {11 \cos ^2(a+b x) (d \tan (a+b x))^{7/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{11/2}}{4 b d^3}+\frac {\left (77 d^3\right ) \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{32 b}-\frac {\left (77 d^3\right ) \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{32 b} \\ & = \frac {77 d (d \tan (a+b x))^{3/2}}{48 b}-\frac {11 \cos ^2(a+b x) (d \tan (a+b x))^{7/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{11/2}}{4 b d^3}-\frac {\left (77 d^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}-\frac {\left (77 d^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}-\frac {\left (77 d^3\right ) \text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{64 b}-\frac {\left (77 d^3\right ) \text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{64 b} \\ & = -\frac {77 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}+\frac {77 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}+\frac {77 d (d \tan (a+b x))^{3/2}}{48 b}-\frac {11 \cos ^2(a+b x) (d \tan (a+b x))^{7/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{11/2}}{4 b d^3}-\frac {\left (77 d^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}+\frac {\left (77 d^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b} \\ & = \frac {77 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}-\frac {77 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}-\frac {77 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}+\frac {77 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}+\frac {77 d (d \tan (a+b x))^{3/2}}{48 b}-\frac {11 \cos ^2(a+b x) (d \tan (a+b x))^{7/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{11/2}}{4 b d^3} \\ \end{align*}
Time = 0.83 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.51 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=\frac {d \left (128+204 \cos ^2(a+b x)+231 \arcsin (\cos (a+b x)-\sin (a+b x)) \cot (a+b x) \csc (a+b x) \sqrt {\sin (2 (a+b x))}+231 \cot (a+b x) \csc (a+b x) \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right ) \sqrt {\sin (2 (a+b x))}-6 \cot (a+b x) \sin (4 (a+b x))\right ) (d \tan (a+b x))^{3/2}}{192 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(525\) vs. \(2(213)=426\).
Time = 48.45 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.90
method | result | size |
default | \(-\frac {\tan \left (b x +a \right ) \sqrt {d \tan \left (b x +a \right )}\, \left (48 \left (\cos ^{5}\left (b x +a \right )\right ) \sqrt {2}-48 \sqrt {2}\, \left (\cos ^{4}\left (b x +a \right )\right )-228 \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}+228 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}-462 \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \arctan \left (\frac {-\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+\cos \left (b x +a \right )-1}{-1+\cos \left (b x +a \right )}\right ) \cos \left (b x +a \right )+462 \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+\cos \left (b x +a \right )-1}{-1+\cos \left (b x +a \right )}\right ) \cos \left (b x +a \right )-231 \cos \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \ln \left (2 \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {2}\, \csc \left (b x +a \right )+2 \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {2}\, \cot \left (b x +a \right )-2 \cot \left (b x +a \right )+2\right )+231 \cos \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \ln \left (-2 \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {2}\, \csc \left (b x +a \right )-2 \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {2}\, \cot \left (b x +a \right )-2 \cot \left (b x +a \right )+2\right )-128 \sqrt {2}\, \cos \left (b x +a \right )+128 \sqrt {2}\right ) d^{2} \sqrt {2}}{384 b \left (-1+\cos \left (b x +a \right )\right )}\) | \(526\) |
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Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 1048, normalized size of antiderivative = 3.78 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.87 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {231 \, d^{8} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - 256 \, \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} d^{6} - \frac {24 \, {\left (19 \, \left (d \tan \left (b x + a\right )\right )^{\frac {7}{2}} d^{8} + 15 \, \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} d^{10}\right )}}{d^{4} \tan \left (b x + a\right )^{4} + 2 \, d^{4} \tan \left (b x + a\right )^{2} + d^{4}}}{384 \, b d^{5}} \]
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Time = 0.35 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {1}{384} \, d^{2} {\left (\frac {462 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b d} + \frac {462 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b d} - \frac {231 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b d} + \frac {231 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b d} - \frac {256 \, \sqrt {d \tan \left (b x + a\right )} \tan \left (b x + a\right )}{b} - \frac {24 \, {\left (19 \, \sqrt {d \tan \left (b x + a\right )} d^{4} \tan \left (b x + a\right )^{3} + 15 \, \sqrt {d \tan \left (b x + a\right )} d^{4} \tan \left (b x + a\right )\right )}}{{\left (d^{2} \tan \left (b x + a\right )^{2} + d^{2}\right )}^{2} b}\right )} \]
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Timed out. \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=\int {\sin \left (a+b\,x\right )}^4\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{5/2} \,d x \]
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