Optimal. Leaf size=92 \[ -\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}+\frac {3 \tan ^{-1}\left (\frac {(1-\cot (x)) \csc ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{\sqrt {2}}\right )}{4 \sqrt {2}}+\frac {3 \log \left (\sin (x)+\cos (x)-\sqrt {2} \cot (x) \csc (x) \sqrt {\sin ^4(x) \tan (x)}\right )}{4 \sqrt {2}} \]
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Rubi [B] time = 0.24, antiderivative size = 204, normalized size of antiderivative = 2.22, number of steps used = 13, number of rules used = 9, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {6719, 288, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{4 \sqrt {2} \tan ^{\frac {5}{2}}(x)}+\frac {3 \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (x)}+1\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{4 \sqrt {2} \tan ^{\frac {5}{2}}(x)}+\frac {3 \sec ^2(x) \log \left (\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin ^4(x) \tan (x)}}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)}-\frac {3 \sec ^2(x) \log \left (\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1\right ) \sqrt {\sin ^4(x) \tan (x)}}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 204
Rule 288
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6719
Rubi steps
\begin {align*} \int \sqrt {\sin ^4(x) \tan (x)} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {\frac {x^5}{\left (1+x^2\right )^2}}}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\frac {\left (\sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \operatorname {Subst}\left (\int \frac {x^{5/2}}{\left (1+x^2\right )^2} \, dx,x,\tan (x)\right )}{\tan ^{\frac {5}{2}}(x)}\\ &=-\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (x)\right )}{4 \tan ^{\frac {5}{2}}(x)}\\ &=-\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{2 \tan ^{\frac {5}{2}}(x)}\\ &=-\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}-\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{4 \tan ^{\frac {5}{2}}(x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )}{4 \tan ^{\frac {5}{2}}(x)}\\ &=-\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{8 \tan ^{\frac {5}{2}}(x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{8 \tan ^{\frac {5}{2}}(x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)}\\ &=-\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}+\frac {3 \log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)}-\frac {3 \log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)}+\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (x)}\right )}{4 \sqrt {2} \tan ^{\frac {5}{2}}(x)}-\frac {\left (3 \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (x)}\right )}{4 \sqrt {2} \tan ^{\frac {5}{2}}(x)}\\ &=-\frac {1}{2} \cot (x) \sqrt {\sin ^4(x) \tan (x)}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{4 \sqrt {2} \tan ^{\frac {5}{2}}(x)}+\frac {3 \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (x)}\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{4 \sqrt {2} \tan ^{\frac {5}{2}}(x)}+\frac {3 \log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)}-\frac {3 \log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right ) \sec ^2(x) \sqrt {\sin ^4(x) \tan (x)}}{8 \sqrt {2} \tan ^{\frac {5}{2}}(x)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 66, normalized size = 0.72 \[ -\frac {1}{8} \sqrt {\sin (2 x)} \csc ^3(x) \sqrt {\sin ^4(x) \tan (x)} \left (2 \sin (x) \sqrt {\sin (2 x)}+3 \sin ^{-1}(\cos (x)-\sin (x))+3 \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sin ^4(x) \tan (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 114.30, size = 1006, normalized size = 10.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {\sin \relax (x)^{5}}{\cos \relax (x)}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.51, size = 318, normalized size = 3.46
method | result | size |
default | \(-\frac {\sqrt {32}\, \left (-1+\cos \relax (x )\right ) \left (-3 i \EllipticPi \left (\sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sqrt {\frac {-1+\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}+3 i \EllipticPi \left (\sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sqrt {\frac {-1+\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}-3 \EllipticPi \left (\sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sqrt {\frac {-1+\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}-3 \EllipticPi \left (\sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sqrt {\frac {-1+\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {-\frac {-1+\cos \relax (x )-\sin \relax (x )}{\sin \relax (x )}}+2 \left (\cos ^{2}\relax (x )\right ) \sqrt {2}-2 \cos \relax (x ) \sqrt {2}\right ) \left (1+\cos \relax (x )\right )^{2} \sqrt {\frac {\sin ^{5}\relax (x )}{\cos \relax (x )}}}{32 \sin \relax (x )^{5}}\) | \(318\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {\sin \relax (x)^{5}}{\cos \relax (x)}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {\frac {{\sin \relax (x)}^5}{\cos \relax (x)}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {\sin ^{5}{\relax (x )}}{\cos {\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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