Optimal. Leaf size=126 \[ \frac {2 (a \sec (c+d x)+a)^{7/2}}{7 a^4 d}-\frac {6 (a \sec (c+d x)+a)^{5/2}}{5 a^3 d}+\frac {2 (a \sec (c+d x)+a)^{3/2}}{3 a^2 d}+\frac {2 \sqrt {a \sec (c+d x)+a}}{a d}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3880, 88, 50, 63, 207} \[ \frac {2 (a \sec (c+d x)+a)^{7/2}}{7 a^4 d}-\frac {6 (a \sec (c+d x)+a)^{5/2}}{5 a^3 d}+\frac {2 (a \sec (c+d x)+a)^{3/2}}{3 a^2 d}+\frac {2 \sqrt {a \sec (c+d x)+a}}{a d}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 88
Rule 207
Rule 3880
Rubi steps
\begin {align*} \int \frac {\tan ^5(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-a+a x)^2 (a+a x)^{3/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-3 a^2 (a+a x)^{3/2}+\frac {a^2 (a+a x)^{3/2}}{x}+a (a+a x)^{5/2}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=-\frac {6 (a+a \sec (c+d x))^{5/2}}{5 a^3 d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a^4 d}+\frac {\operatorname {Subst}\left (\int \frac {(a+a x)^{3/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a^2 d}-\frac {6 (a+a \sec (c+d x))^{5/2}}{5 a^3 d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a^4 d}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac {2 \sqrt {a+a \sec (c+d x)}}{a d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a^2 d}-\frac {6 (a+a \sec (c+d x))^{5/2}}{5 a^3 d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a^4 d}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {2 \sqrt {a+a \sec (c+d x)}}{a d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a^2 d}-\frac {6 (a+a \sec (c+d x))^{5/2}}{5 a^3 d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a^4 d}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{a d}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {2 \sqrt {a+a \sec (c+d x)}}{a d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a^2 d}-\frac {6 (a+a \sec (c+d x))^{5/2}}{5 a^3 d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a^4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 88, normalized size = 0.70 \[ \frac {2 \left (15 \sec ^4(c+d x)-3 \sec ^3(c+d x)-64 \sec ^2(c+d x)+46 \sec (c+d x)-105 \sqrt {\sec (c+d x)+1} \tanh ^{-1}\left (\sqrt {\sec (c+d x)+1}\right )+92\right )}{105 d \sqrt {a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 285, normalized size = 2.26 \[ \left [\frac {105 \, \sqrt {a} \cos \left (d x + c\right )^{3} \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \, {\left (92 \, \cos \left (d x + c\right )^{3} - 46 \, \cos \left (d x + c\right )^{2} - 18 \, \cos \left (d x + c\right ) + 15\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{210 \, a d \cos \left (d x + c\right )^{3}}, \frac {105 \, \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) \cos \left (d x + c\right )^{3} + 2 \, {\left (92 \, \cos \left (d x + c\right )^{3} - 46 \, \cos \left (d x + c\right )^{2} - 18 \, \cos \left (d x + c\right ) + 15\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{105 \, a d \cos \left (d x + c\right )^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 3.77, size = 190, normalized size = 1.51 \[ -\frac {\sqrt {2} {\left (\frac {105 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} + \frac {2 \, {\left (105 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} - 70 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} a - 252 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )} a^{2} - 120 \, a^{3}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.33, size = 293, normalized size = 2.33 \[ -\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (105 \left (\cos ^{3}\left (d x +c \right )\right ) \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \sqrt {2}+315 \left (\cos ^{2}\left (d x +c \right )\right ) \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \sqrt {2}+315 \cos \left (d x +c \right ) \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \sqrt {2}+105 \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}}-1472 \left (\cos ^{3}\left (d x +c \right )\right )+736 \left (\cos ^{2}\left (d x +c \right )\right )+288 \cos \left (d x +c \right )-240\right )}{840 d \cos \left (d x +c \right )^{3} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.54, size = 129, normalized size = 1.02 \[ \frac {\frac {105 \, \log \left (\frac {\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \sqrt {a}}\right )}{\sqrt {a}} + \frac {30 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {7}{2}}}{a^{4}} - \frac {126 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {5}{2}}}{a^{3}} + \frac {70 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}}}{a^{2}} + \frac {210 \, \sqrt {a + \frac {a}{\cos \left (d x + c\right )}}}{a}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^5}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{5}{\left (c + d x \right )}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________