Optimal. Leaf size=364 \[ -\frac {2 b^2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}-\frac {b^2 \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {b \tan ^3(c+d x)}}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b^2 \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {b \tan ^3(c+d x)}}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}-\frac {b^2 \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {b \tan ^3(c+d x)}}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b^2 \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {b \tan ^3(c+d x)}}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 b^2 \tan (c+d x) \sqrt {b \tan ^3(c+d x)}}{5 d}-\frac {2 b^2 \tan ^3(c+d x) \sqrt {b \tan ^3(c+d x)}}{9 d}+\frac {2 b^2 \tan ^5(c+d x) \sqrt {b \tan ^3(c+d x)}}{13 d} \]
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Rubi [A]
time = 0.11, antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3739, 3554,
3557, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {b^2 \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^2 \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^2 \tan ^3(c+d x) \sqrt {b \tan ^3(c+d x)}}{9 d}+\frac {2 b^2 \tan (c+d x) \sqrt {b \tan ^3(c+d x)}}{5 d}+\frac {2 b^2 \tan ^5(c+d x) \sqrt {b \tan ^3(c+d x)}}{13 d}-\frac {b^2 \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^2 \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 b^2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
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 )^{5/2} \, dx &=\frac {\left (b^2 \sqrt {b \tan ^3(c+d x)}\right ) \int \tan ^{\frac {15}{2}}(c+d x) \, dx}{\tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {2 b^2 \tan ^5(c+d x) \sqrt {b \tan ^3(c+d x)}}{13 d}-\frac {\left (b^2 \sqrt {b \tan ^3(c+d x)}\right ) \int \tan ^{\frac {11}{2}}(c+d x) \, dx}{\tan ^{\frac {3}{2}}(c+d x)}\\ &=-\frac {2 b^2 \tan ^3(c+d x) \sqrt {b \tan ^3(c+d x)}}{9 d}+\frac {2 b^2 \tan ^5(c+d x) \sqrt {b \tan ^3(c+d x)}}{13 d}+\frac {\left (b^2 \sqrt {b \tan ^3(c+d x)}\right ) \int \tan ^{\frac {7}{2}}(c+d x) \, dx}{\tan ^{\frac {3}{2}}(c+d x)}\\ &=\frac {2 b^2 \tan (c+d x) \sqrt {b \tan ^3(c+d x)}}{5 d}-\frac {2 b^2 \tan ^3(c+d x) \sqrt {b \tan ^3(c+d x)}}{9 d}+\frac {2 b^2 \tan ^5(c+d x) \sqrt {b \tan ^3(c+d x)}}{13 d}-\frac {\left (b^2 \sqrt {b \tan ^3(c+d x)}\right ) \int \tan ^{\frac {3}{2}}(c+d x) \, dx}{\tan ^{\frac {3}{2}}(c+d x)}\\ &=-\frac {2 b^2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}+\frac {2 b^2 \tan (c+d x) \sqrt {b \tan ^3(c+d x)}}{5 d}-\frac {2 b^2 \tan ^3(c+d x) \sqrt {b \tan ^3(c+d x)}}{9 d}+\frac {2 b^2 \tan ^5(c+d x) \sqrt {b \tan ^3(c+d x)}}{13 d}+\frac {\left (b^2 \sqrt {b \tan ^3(c+d x)}\right ) \int \frac {1}{\sqrt {\tan (c+d x)}} \, dx}{\tan ^{\frac {3}{2}}(c+d x)}\\ &=-\frac {2 b^2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}+\frac {2 b^2 \tan (c+d x) \sqrt {b \tan ^3(c+d x)}}{5 d}-\frac {2 b^2 \tan ^3(c+d x) \sqrt {b \tan ^3(c+d x)}}{9 d}+\frac {2 b^2 \tan ^5(c+d x) \sqrt {b \tan ^3(c+d x)}}{13 d}+\frac {\left (b^2 \sqrt {b \tan ^3(c+d x)}\right ) \text {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 b^2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}+\frac {2 b^2 \tan (c+d x) \sqrt {b \tan ^3(c+d x)}}{5 d}-\frac {2 b^2 \tan ^3(c+d x) \sqrt {b \tan ^3(c+d x)}}{9 d}+\frac {2 b^2 \tan ^5(c+d x) \sqrt {b \tan ^3(c+d x)}}{13 d}+\frac {\left (2 b^2 \sqrt {b \tan ^3(c+d x)}\right ) \text {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 b^2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}+\frac {2 b^2 \tan (c+d x) \sqrt {b \tan ^3(c+d x)}}{5 d}-\frac {2 b^2 \tan ^3(c+d x) \sqrt {b \tan ^3(c+d x)}}{9 d}+\frac {2 b^2 \tan ^5(c+d x) \sqrt {b \tan ^3(c+d x)}}{13 d}+\frac {\left (b^2 \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^2 \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^2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}+\frac {2 b^2 \tan (c+d x) \sqrt {b \tan ^3(c+d x)}}{5 d}-\frac {2 b^2 \tan ^3(c+d x) \sqrt {b \tan ^3(c+d x)}}{9 d}+\frac {2 b^2 \tan ^5(c+d x) \sqrt {b \tan ^3(c+d x)}}{13 d}+\frac {\left (b^2 \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^2 \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^2 \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^2 \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^2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}-\frac {b^2 \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^2 \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^2 \tan (c+d x) \sqrt {b \tan ^3(c+d x)}}{5 d}-\frac {2 b^2 \tan ^3(c+d x) \sqrt {b \tan ^3(c+d x)}}{9 d}+\frac {2 b^2 \tan ^5(c+d x) \sqrt {b \tan ^3(c+d x)}}{13 d}+\frac {\left (b^2 \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^2 \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^2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}-\frac {b^2 \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^2 \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^2 \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^2 \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^2 \tan (c+d x) \sqrt {b \tan ^3(c+d x)}}{5 d}-\frac {2 b^2 \tan ^3(c+d x) \sqrt {b \tan ^3(c+d x)}}{9 d}+\frac {2 b^2 \tan ^5(c+d x) \sqrt {b \tan ^3(c+d x)}}{13 d}\\ \end {align*}
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Mathematica [A]
time = 0.81, size = 199, normalized size = 0.55 \begin {gather*} \frac {b \left (b \tan ^3(c+d x)\right )^{3/2} \left (-1170 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )+1170 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )-585 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+585 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-4680 \sqrt {\tan (c+d x)}+936 \tan ^{\frac {5}{2}}(c+d x)-520 \tan ^{\frac {9}{2}}(c+d x)+360 \tan ^{\frac {13}{2}}(c+d x)\right )}{2340 d \tan ^{\frac {9}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 266, normalized size = 0.73
method | result | size |
derivativedivides | \(\frac {\left (b \left (\tan ^{3}\left (d x +c \right )\right )\right )^{\frac {5}{2}} \left (360 \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}}-520 b^{2} \left (b \tan \left (d x +c \right )\right )^{\frac {9}{2}}+936 b^{4} \left (b \tan \left (d x +c \right )\right )^{\frac {5}{2}}+585 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}-b \tan \left (d x +c \right )-\sqrt {b^{2}}}\right )+1170 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+1170 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )-4680 b^{6} \sqrt {b \tan \left (d x +c \right )}\right )}{2340 d \tan \left (d x +c \right )^{5} \left (b \tan \left (d x +c \right )\right )^{\frac {5}{2}} b^{4}}\) | \(266\) |
default | \(\frac {\left (b \left (\tan ^{3}\left (d x +c \right )\right )\right )^{\frac {5}{2}} \left (360 \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}}-520 b^{2} \left (b \tan \left (d x +c \right )\right )^{\frac {9}{2}}+936 b^{4} \left (b \tan \left (d x +c \right )\right )^{\frac {5}{2}}+585 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}-b \tan \left (d x +c \right )-\sqrt {b^{2}}}\right )+1170 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+1170 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )-4680 b^{6} \sqrt {b \tan \left (d x +c \right )}\right )}{2340 d \tan \left (d x +c \right )^{5} \left (b \tan \left (d x +c \right )\right )^{\frac {5}{2}} b^{4}}\) | \(266\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 178, normalized size = 0.49 \begin {gather*} \frac {360 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{\frac {13}{2}} - 520 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{\frac {9}{2}} + 936 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{\frac {5}{2}} + 585 \, {\left (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 )\right )} b^{2} - 4680 \, b^{\frac {5}{2}} \sqrt {\tan \left (d x + c\right )}}{2340 \, 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 {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 291, normalized size = 0.80 \begin {gather*} \frac {1}{2340} \, {\left (\frac {1170 \, \sqrt {2} b \sqrt {{\left | b \right |}} \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 )}{d} + \frac {1170 \, \sqrt {2} b \sqrt {{\left | b \right |}} \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 )}{d} + \frac {585 \, \sqrt {2} b \sqrt {{\left | b \right |}} \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 )}{d} - \frac {585 \, \sqrt {2} b \sqrt {{\left | b \right |}} \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 )}{d} + \frac {8 \, {\left (45 \, \sqrt {b \tan \left (d x + c\right )} b^{66} d^{12} \tan \left (d x + c\right )^{6} - 65 \, \sqrt {b \tan \left (d x + c\right )} b^{66} d^{12} \tan \left (d x + c\right )^{4} + 117 \, \sqrt {b \tan \left (d x + c\right )} b^{66} d^{12} \tan \left (d x + c\right )^{2} - 585 \, \sqrt {b \tan \left (d x + c\right )} b^{66} d^{12}\right )}}{b^{65} d^{13}}\right )} b \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 )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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