Integrand size = 21, antiderivative size = 257 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{3/2}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{3/2}}+\frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{3/2}}-\frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{3/2}}+\frac {3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3} \]
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Time = 0.21 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2671, 294, 296, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{3/2}}+\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}+1\right )}{32 \sqrt {2} b d^{3/2}}+\frac {3 \log \left (\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}+\sqrt {d}\right )}{64 \sqrt {2} b d^{3/2}}-\frac {3 \log \left (\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}+\sqrt {d}\right )}{64 \sqrt {2} b d^{3/2}}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}+\frac {3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3} \]
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Rule 210
Rule 294
Rule 296
Rule 303
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^{5/2}}{\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))^{3/2}}{4 b d^3}+\frac {(3 d) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (d^2+x^2\right )^2} \, dx,x,d \tan (a+b x)\right )}{8 b} \\ & = \frac {3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {x}}{d^2+x^2} \, dx,x,d \tan (a+b x)\right )}{32 b d} \\ & = \frac {3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}+\frac {3 \text {Subst}\left (\int \frac {x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{16 b d} \\ & = \frac {3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}-\frac {3 \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{32 b d}+\frac {3 \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{32 b d} \\ & = \frac {3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}+\frac {3 \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 d^{3/2}}+\frac {3 \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 d^{3/2}}+\frac {3 \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 d}+\frac {3 \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 d} \\ & = \frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{3/2}}-\frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{3/2}}+\frac {3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}+\frac {3 \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 d^{3/2}}-\frac {3 \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 d^{3/2}} \\ & = -\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{3/2}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{3/2}}+\frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{3/2}}-\frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{3/2}}+\frac {3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.48 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\csc (a+b x) \left (\cos (a+b x)-2 \cos (3 (a+b x))+\cos (5 (a+b x))-3 \arcsin (\cos (a+b x)-\sin (a+b x)) \sqrt {\sin (2 (a+b x))}-3 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right ) \sqrt {\sin (2 (a+b x))}\right ) \sqrt {d \tan (a+b x)}}{64 b d^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(620\) vs. \(2(197)=394\).
Time = 10.06 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.42
method | result | size |
default | \(\frac {\csc \left (b x +a \right ) \left (-1+\cos \left (b x +a \right )\right ) \left (16 \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}}}\, \left (\cos ^{3}\left (b x +a \right )\right ) \sin \left (b x +a \right )+16 \left (\cos ^{2}\left (b x +a \right )\right ) \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}}}-12 \cos \left (b x +a \right ) \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}}}-12 \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}}}-6 \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 )-6 \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 )-3 \ln \left (-\frac {\cot \left (b x +a \right ) \cos \left (b x +a \right )-2 \cot \left (b x +a \right )-2 \sin \left (b x +a \right ) \sqrt {-\left (\cot ^{3}\left (b x +a \right )\right )+3 \left (\cot ^{2}\left (b x +a \right )\right ) \csc \left (b x +a \right )-3 \cot \left (b x +a \right ) \left (\csc ^{2}\left (b x +a \right )\right )+\csc ^{3}\left (b x +a \right )+\cot \left (b x +a \right )-\csc \left (b x +a \right )}-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{-1+\cos \left (b x +a \right )}\right )+3 \ln \left (-\frac {\cot \left (b x +a \right ) \cos \left (b x +a \right )-2 \cot \left (b x +a \right )+2 \sin \left (b x +a \right ) \sqrt {-\left (\cot ^{3}\left (b x +a \right )\right )+3 \left (\cot ^{2}\left (b x +a \right )\right ) \csc \left (b x +a \right )-3 \cot \left (b x +a \right ) \left (\csc ^{2}\left (b x +a \right )\right )+\csc ^{3}\left (b x +a \right )+\cot \left (b x +a \right )-\csc \left (b x +a \right )}-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{-1+\cos \left (b x +a \right )}\right )\right ) \sqrt {2}}{128 b \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 {d \tan \left (b x +a \right )}\, d}\) | \(621\) |
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Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 986, normalized size of antiderivative = 3.84 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {\sin ^{4}{\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.37 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.88 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {3 \, d^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} + \frac {8 \, {\left (3 \, \left (d \tan \left (b x + a\right )\right )^{\frac {7}{2}} d^{4} - \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} d^{6}\right )}}{d^{4} \tan \left (b x + a\right )^{4} + 2 \, d^{4} \tan \left (b x + a\right )^{2} + d^{4}}}{128 \, b d^{5}} \]
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Time = 0.43 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\frac {6 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b d^{2}} + \frac {6 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b d^{2}} - \frac {3 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b d^{2}} + \frac {3 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b d^{2}} + \frac {8 \, {\left (3 \, \sqrt {d \tan \left (b x + a\right )} d^{3} \tan \left (b x + a\right )^{3} - \sqrt {d \tan \left (b x + a\right )} d^{3} \tan \left (b x + a\right )\right )}}{{\left (d^{2} \tan \left (b x + a\right )^{2} + d^{2}\right )}^{2} b}}{128 \, d} \]
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Timed out. \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^4}{{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \]
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