Optimal. Leaf size=286 \[ -\frac {2 b \sqrt {b \tan ^3(c+d x)}}{3 d}-\frac {b \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {b \tan ^3(c+d x)}}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {b \tan ^3(c+d x)}}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {b \tan ^3(c+d x)}}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}-\frac {b \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {b \tan ^3(c+d x)}}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 b \tan ^2(c+d x) \sqrt {b \tan ^3(c+d x)}}{7 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3739, 3554,
3557, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {b \text {ArcTan}\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 {b \text {ArcTan}\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 {2 b \sqrt {b \tan ^3(c+d x)}}{3 d}+\frac {2 b \tan ^2(c+d x) \sqrt {b \tan ^3(c+d x)}}{7 d}+\frac {b \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 {b \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3554
Rule 3557
Rule 3739
Rubi steps
\begin {align*} \int \left (b \tan ^3(c+d x)\right )^{3/2} \, dx &=\frac {\left (b \sqrt {b \tan ^3(c+d x)}\right ) \int \tan ^{\frac {9}{2}}(c+d x) \, dx}{\tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {2 b \tan ^2(c+d x) \sqrt {b \tan ^3(c+d x)}}{7 d}-\frac {\left (b \sqrt {b \tan ^3(c+d x)}\right ) \int \tan ^{\frac {5}{2}}(c+d x) \, dx}{\tan ^{\frac {3}{2}}(c+d x)}\\ &=-\frac {2 b \sqrt {b \tan ^3(c+d x)}}{3 d}+\frac {2 b \tan ^2(c+d x) \sqrt {b \tan ^3(c+d x)}}{7 d}+\frac {\left (b \sqrt {b \tan ^3(c+d x)}\right ) \int \sqrt {\tan (c+d x)} \, dx}{\tan ^{\frac {3}{2}}(c+d x)}\\ &=-\frac {2 b \sqrt {b \tan ^3(c+d x)}}{3 d}+\frac {2 b \tan ^2(c+d x) \sqrt {b \tan ^3(c+d x)}}{7 d}+\frac {\left (b \sqrt {b \tan ^3(c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d \tan ^{\frac {3}{2}}(c+d x)}\\ &=-\frac {2 b \sqrt {b \tan ^3(c+d x)}}{3 d}+\frac {2 b \tan ^2(c+d x) \sqrt {b \tan ^3(c+d x)}}{7 d}+\frac {\left (2 b \sqrt {b \tan ^3(c+d x)}\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d \tan ^{\frac {3}{2}}(c+d x)}\\ &=-\frac {2 b \sqrt {b \tan ^3(c+d x)}}{3 d}+\frac {2 b \tan ^2(c+d x) \sqrt {b \tan ^3(c+d x)}}{7 d}-\frac {\left (b \sqrt {b \tan ^3(c+d x)}\right ) \text {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 {\left (b \sqrt {b \tan ^3(c+d x)}\right ) \text {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 b \sqrt {b \tan ^3(c+d x)}}{3 d}+\frac {2 b \tan ^2(c+d x) \sqrt {b \tan ^3(c+d x)}}{7 d}+\frac {\left (b \sqrt {b \tan ^3(c+d x)}\right ) \text {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 {\left (b \sqrt {b \tan ^3(c+d x)}\right ) \text {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 {\left (b \sqrt {b \tan ^3(c+d x)}\right ) \text {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 {\left (b \sqrt {b \tan ^3(c+d x)}\right ) \text {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 b \sqrt {b \tan ^3(c+d x)}}{3 d}+\frac {b \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 {b \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 {2 b \tan ^2(c+d x) \sqrt {b \tan ^3(c+d x)}}{7 d}+\frac {\left (b \sqrt {b \tan ^3(c+d x)}\right ) \text {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 {\left (b \sqrt {b \tan ^3(c+d x)}\right ) \text {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 b \sqrt {b \tan ^3(c+d x)}}{3 d}-\frac {b \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 {b \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 {b \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 {b \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 {2 b \tan ^2(c+d x) \sqrt {b \tan ^3(c+d x)}}{7 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.07, size = 54, normalized size = 0.19 \begin {gather*} \frac {2 b \sqrt {b \tan ^3(c+d x)} \left (-7+7 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\tan ^2(c+d x)\right )+3 \tan ^2(c+d x)\right )}{21 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 236, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\left (b \left (\tan ^{3}\left (d x +c \right )\right )\right )^{\frac {3}{2}} \left (24 \left (b \tan \left (d x +c \right )\right )^{\frac {7}{2}} \left (b^{2}\right )^{\frac {1}{4}}+21 b^{4} \sqrt {2}\, \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}-b \tan \left (d x +c \right )-\sqrt {b^{2}}}{b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+42 b^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+42 b^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )-56 b^{2} \left (b \tan \left (d x +c \right )\right )^{\frac {3}{2}} \left (b^{2}\right )^{\frac {1}{4}}\right )}{84 d \tan \left (d x +c \right )^{3} \left (b \tan \left (d x +c \right )\right )^{\frac {3}{2}} b^{2} \left (b^{2}\right )^{\frac {1}{4}}}\) | \(236\) |
default | \(\frac {\left (b \left (\tan ^{3}\left (d x +c \right )\right )\right )^{\frac {3}{2}} \left (24 \left (b \tan \left (d x +c \right )\right )^{\frac {7}{2}} \left (b^{2}\right )^{\frac {1}{4}}+21 b^{4} \sqrt {2}\, \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}-b \tan \left (d x +c \right )-\sqrt {b^{2}}}{b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+42 b^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+42 b^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )-56 b^{2} \left (b \tan \left (d x +c \right )\right )^{\frac {3}{2}} \left (b^{2}\right )^{\frac {1}{4}}\right )}{84 d \tan \left (d x +c \right )^{3} \left (b \tan \left (d x +c \right )\right )^{\frac {3}{2}} b^{2} \left (b^{2}\right )^{\frac {1}{4}}}\) | \(236\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 140, normalized size = 0.49 \begin {gather*} \frac {24 \, b^{\frac {3}{2}} \tan \left (d x + c\right )^{\frac {7}{2}} - 56 \, b^{\frac {3}{2}} \tan \left (d x + c\right )^{\frac {3}{2}} + 21 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} b^{\frac {3}{2}}}{84 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \left (b \tan ^{3}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.53, size = 253, normalized size = 0.88 \begin {gather*} \frac {1}{84} \, b {\left (\frac {42 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b d} + \frac {42 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b d} - \frac {21 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b d} + \frac {21 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b d} + \frac {8 \, {\left (3 \, \sqrt {b \tan \left (d x + c\right )} b^{21} d^{6} \tan \left (d x + c\right )^{3} - 7 \, \sqrt {b \tan \left (d x + c\right )} b^{21} d^{6} \tan \left (d x + c\right )\right )}}{b^{21} d^{7}}\right )} \mathrm {sgn}\left (\tan \left (d x + c\right )\right ) \end {gather*}
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 \begin {gather*} \int {\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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