Optimal. Leaf size=255 \[ \frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {b \tan ^3(c+d x)}}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right ) \sqrt {b \tan ^3(c+d x)}}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {\sqrt {b \tan ^3(c+d x)} \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\sqrt {b \tan ^3(c+d x)} \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d} \]
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Rubi [A] time = 0.11, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3658, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {b \tan ^3(c+d x)}}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right ) \sqrt {b \tan ^3(c+d x)}}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {\sqrt {b \tan ^3(c+d x)} \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\sqrt {b \tan ^3(c+d x)} \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3473
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int \sqrt {b \tan ^3(c+d x)} \, dx &=\frac {\sqrt {b \tan ^3(c+d x)} \int \tan ^{\frac {3}{2}}(c+d x) \, dx}{\tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}-\frac {\sqrt {b \tan ^3(c+d x)} \int \frac {1}{\sqrt {\tan (c+d x)}} \, dx}{\tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}-\frac {\sqrt {b \tan ^3(c+d x)} \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d \tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}-\frac {\left (2 \sqrt {b \tan ^3(c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d \tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}-\frac {\sqrt {b \tan ^3(c+d x)} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\sqrt {b \tan ^3(c+d x)} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d \tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}-\frac {\sqrt {b \tan ^3(c+d x)} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\sqrt {b \tan ^3(c+d x)} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {\sqrt {b \tan ^3(c+d x)} \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {\sqrt {b \tan ^3(c+d x)} \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {b \tan ^3(c+d x)}}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {b \tan ^3(c+d x)}}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\sqrt {b \tan ^3(c+d x)} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {\sqrt {b \tan ^3(c+d x)} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}+\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {b \tan ^3(c+d x)}}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {b \tan ^3(c+d x)}}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {b \tan ^3(c+d x)}}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {b \tan ^3(c+d x)}}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 161, normalized size = 0.63 \[ \frac {\sqrt {b \tan ^3(c+d x)} \left (2 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-2 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )+8 \sqrt {\tan (c+d x)}+\sqrt {2} \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )-\sqrt {2} \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )}{4 d \tan ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \sqrt {b \tan \left (d x + c\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 208, normalized size = 0.82 \[ -\frac {\sqrt {b \left (\tan ^{3}\left (d x +c \right )\right )}\, \left (\left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}-b \tan \left (d x +c \right )-\sqrt {b^{2}}}\right )+2 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+2 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )-8 \sqrt {b \tan \left (d x +c \right )}\right )}{4 d \tan \left (d x +c \right ) \sqrt {b \tan \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 133, normalized size = 0.52 \[ -\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} \sqrt {b} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} \sqrt {b} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} \sqrt {b} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - 8 \, \sqrt {b} \sqrt {\tan \left (d x + c\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {b\,{\mathrm {tan}\left (c+d\,x\right )}^3} \,d x \]
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 \[ \int \sqrt {b \tan ^{3}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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