Optimal. Leaf size=364 \[ -\frac {2 \cot (c+d x)}{5 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \cot ^3(c+d x)}{9 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot ^5(c+d x)}{13 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \tan (c+d x)}{b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {\text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}} \]
[Out]
________________________________________________________________________________________
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, 3555,
3557, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\tan ^{\frac {3}{2}}(c+d x) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\tan ^{\frac {3}{2}}(c+d x) \text {ArcTan}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \tan (c+d x)}{b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\tan ^{\frac {3}{2}}(c+d x) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {\tan ^{\frac {3}{2}}(c+d x) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot ^5(c+d x)}{13 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \cot ^3(c+d x)}{9 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot (c+d x)}{5 b^2 d \sqrt {b \tan ^3(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3555
Rule 3557
Rule 3739
Rubi steps
\begin {align*} \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{5/2}} \, dx &=\frac {\tan ^{\frac {3}{2}}(c+d x) \int \frac {1}{\tan ^{\frac {15}{2}}(c+d x)} \, dx}{b^2 \sqrt {b \tan ^3(c+d x)}}\\ &=-\frac {2 \cot ^5(c+d x)}{13 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {\tan ^{\frac {3}{2}}(c+d x) \int \frac {1}{\tan ^{\frac {11}{2}}(c+d x)} \, dx}{b^2 \sqrt {b \tan ^3(c+d x)}}\\ &=\frac {2 \cot ^3(c+d x)}{9 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot ^5(c+d x)}{13 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\tan ^{\frac {3}{2}}(c+d x) \int \frac {1}{\tan ^{\frac {7}{2}}(c+d x)} \, dx}{b^2 \sqrt {b \tan ^3(c+d x)}}\\ &=-\frac {2 \cot (c+d x)}{5 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \cot ^3(c+d x)}{9 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot ^5(c+d x)}{13 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {\tan ^{\frac {3}{2}}(c+d x) \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{b^2 \sqrt {b \tan ^3(c+d x)}}\\ &=-\frac {2 \cot (c+d x)}{5 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \cot ^3(c+d x)}{9 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot ^5(c+d x)}{13 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \tan (c+d x)}{b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\tan ^{\frac {3}{2}}(c+d x) \int \sqrt {\tan (c+d x)} \, dx}{b^2 \sqrt {b \tan ^3(c+d x)}}\\ &=-\frac {2 \cot (c+d x)}{5 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \cot ^3(c+d x)}{9 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot ^5(c+d x)}{13 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \tan (c+d x)}{b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\tan ^{\frac {3}{2}}(c+d x) \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b^2 d \sqrt {b \tan ^3(c+d x)}}\\ &=-\frac {2 \cot (c+d x)}{5 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \cot ^3(c+d x)}{9 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot ^5(c+d x)}{13 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \tan (c+d x)}{b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\left (2 \tan ^{\frac {3}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b^2 d \sqrt {b \tan ^3(c+d x)}}\\ &=-\frac {2 \cot (c+d x)}{5 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \cot ^3(c+d x)}{9 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot ^5(c+d x)}{13 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \tan (c+d x)}{b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {\tan ^{\frac {3}{2}}(c+d x) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\tan ^{\frac {3}{2}}(c+d x) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b^2 d \sqrt {b \tan ^3(c+d x)}}\\ &=-\frac {2 \cot (c+d x)}{5 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \cot ^3(c+d x)}{9 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot ^5(c+d x)}{13 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \tan (c+d x)}{b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\tan ^{\frac {3}{2}}(c+d x) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\tan ^{\frac {3}{2}}(c+d x) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\tan ^{\frac {3}{2}}(c+d x) \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} b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\tan ^{\frac {3}{2}}(c+d x) \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} b^2 d \sqrt {b \tan ^3(c+d x)}}\\ &=-\frac {2 \cot (c+d x)}{5 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \cot ^3(c+d x)}{9 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot ^5(c+d x)}{13 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \tan (c+d x)}{b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\tan ^{\frac {3}{2}}(c+d x) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {\tan ^{\frac {3}{2}}(c+d x) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}\\ &=-\frac {2 \cot (c+d x)}{5 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \cot ^3(c+d x)}{9 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot ^5(c+d x)}{13 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \tan (c+d x)}{b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.06, size = 45, normalized size = 0.12 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {13}{4},1;-\frac {9}{4};-\tan ^2(c+d x)\right ) \tan (c+d x)}{13 d \left (b \tan ^3(c+d x)\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 272, normalized size = 0.75
method | result | size |
derivativedivides | \(\frac {\tan \left (d x +c \right ) \left (585 \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}} \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}-b \tan \left (d x +c \right )-\sqrt {b^{2}}}{b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+4680 \left (b^{2}\right )^{\frac {1}{4}} \left (\tan ^{6}\left (d x +c \right )\right ) b^{6}-936 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \left (\tan ^{4}\left (d x +c \right )\right )+520 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \left (\tan ^{2}\left (d x +c \right )\right )-360 b^{6} \left (b^{2}\right )^{\frac {1}{4}}\right )}{2340 d \,b^{6} \left (b \left (\tan ^{3}\left (d x +c \right )\right )\right )^{\frac {5}{2}} \left (b^{2}\right )^{\frac {1}{4}}}\) | \(272\) |
default | \(\frac {\tan \left (d x +c \right ) \left (585 \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}} \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}-b \tan \left (d x +c \right )-\sqrt {b^{2}}}{b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+4680 \left (b^{2}\right )^{\frac {1}{4}} \left (\tan ^{6}\left (d x +c \right )\right ) b^{6}-936 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \left (\tan ^{4}\left (d x +c \right )\right )+520 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \left (\tan ^{2}\left (d x +c \right )\right )-360 b^{6} \left (b^{2}\right )^{\frac {1}{4}}\right )}{2340 d \,b^{6} \left (b \left (\tan ^{3}\left (d x +c \right )\right )\right )^{\frac {5}{2}} \left (b^{2}\right )^{\frac {1}{4}}}\) | \(272\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.52, size = 172, normalized size = 0.47 \begin {gather*} \frac {\frac {585 \, {\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 {5}{2}}} + \frac {8 \, {\left (\frac {585 \, \sqrt {b}}{\sqrt {\tan \left (d x + c\right )}} - \frac {117 \, \sqrt {b}}{\tan \left (d x + c\right )^{\frac {5}{2}}} + \frac {65 \, \sqrt {b}}{\tan \left (d x + c\right )^{\frac {9}{2}}} - \frac {45 \, \sqrt {b}}{\tan \left (d x + c\right )^{\frac {13}{2}}}\right )}}{b^{3}}}{2340 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\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.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.88, size = 305, normalized size = 0.84 \begin {gather*} \frac {1}{2340} \, b^{6} {\left (\frac {1170 \, \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^{10} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} + \frac {1170 \, \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^{10} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} - \frac {585 \, \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^{10} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} + \frac {585 \, \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^{10} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} + \frac {8 \, {\left (585 \, b^{6} \tan \left (d x + c\right )^{6} - 117 \, b^{6} \tan \left (d x + c\right )^{4} + 65 \, b^{6} \tan \left (d x + c\right )^{2} - 45 \, b^{6}\right )}}{\sqrt {b \tan \left (d x + c\right )} b^{14} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right ) \tan \left (d x + c\right )^{6}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\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.
[In]
[Out]
________________________________________________________________________________________