Optimal. Leaf size=258 \[ -\frac {2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {2 a^8 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac {14 a^7 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac {34 a^6 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac {30 a^5 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac {2 a^4 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac {2 a^3 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac {2 a^2 \tan (c+d x)}{d \sqrt {a \sec (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3887, 461, 203} \[ \frac {2 a^8 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac {14 a^7 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac {34 a^6 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac {30 a^5 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac {2 a^4 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac {2 a^3 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac {2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {2 a^2 \tan (c+d x)}{d \sqrt {a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 461
Rule 3887
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^{3/2} \tan ^6(c+d x) \, dx &=-\frac {\left (2 a^5\right ) \operatorname {Subst}\left (\int \frac {x^6 \left (2+a x^2\right )^4}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac {\left (2 a^5\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^3}-\frac {x^2}{a^2}+\frac {x^4}{a}+15 x^6+17 a x^8+7 a^2 x^{10}+a^3 x^{12}-\frac {1}{a^3 \left (1+a x^2\right )}\right ) \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}\\ &=\frac {2 a^2 \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a^3 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^4 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {30 a^5 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac {34 a^6 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac {14 a^7 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac {2 a^8 \tan ^{13}(c+d x)}{13 d (a+a \sec (c+d x))^{13/2}}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac {2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {2 a^2 \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a^3 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^4 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {30 a^5 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac {34 a^6 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac {14 a^7 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac {2 a^8 \tan ^{13}(c+d x)}{13 d (a+a \sec (c+d x))^{13/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 8.62, size = 147, normalized size = 0.57 \[ \frac {a \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^6(c+d x) \sqrt {a (\sec (c+d x)+1)} \left (164736 \sin \left (\frac {1}{2} (c+d x)\right )+81081 \sin \left (\frac {3}{2} (c+d x)\right )+134849 \sin \left (\frac {5}{2} (c+d x)\right )+98176 \sin \left (\frac {9}{2} (c+d x)\right )+45045 \sin \left (\frac {11}{2} (c+d x)\right )+32429 \sin \left (\frac {13}{2} (c+d x)\right )-1441440 \sqrt {2} \sin ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^{\frac {13}{2}}(c+d x)\right )}{1441440 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 415, normalized size = 1.61 \[ \left [\frac {45045 \, {\left (a \cos \left (d x + c\right )^{7} + a \cos \left (d x + c\right )^{6}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (32429 \, a \cos \left (d x + c\right )^{6} + 38737 \, a \cos \left (d x + c\right )^{5} - 4731 \, a \cos \left (d x + c\right )^{4} - 26465 \, a \cos \left (d x + c\right )^{3} - 6265 \, a \cos \left (d x + c\right )^{2} + 7875 \, a \cos \left (d x + c\right ) + 3465 \, a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}}, \frac {2 \, {\left (45045 \, {\left (a \cos \left (d x + c\right )^{7} + a \cos \left (d x + c\right )^{6}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) + {\left (32429 \, a \cos \left (d x + c\right )^{6} + 38737 \, a \cos \left (d x + c\right )^{5} - 4731 \, a \cos \left (d x + c\right )^{4} - 26465 \, a \cos \left (d x + c\right )^{3} - 6265 \, a \cos \left (d x + c\right )^{2} + 7875 \, a \cos \left (d x + c\right ) + 3465 \, a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{45045 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 6.47, size = 369, normalized size = 1.43 \[ \frac {\frac {45045 \, \sqrt {-a} a^{2} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right ) \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{{\left | a \right |}} + \frac {2 \, {\left (45045 \, \sqrt {2} a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (300300 \, \sqrt {2} a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (861861 \, \sqrt {2} a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (573144 \, \sqrt {2} a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (236951 \, \sqrt {2} a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + {\left (4751 \, \sqrt {2} a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 53404 \, \sqrt {2} a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{6} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}}{45045 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.36, size = 656, normalized size = 2.54 \[ \frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (45045 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {13}{2}} \sqrt {2}+270270 \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {13}{2}} \sqrt {2}+675675 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {13}{2}} \sqrt {2}+900900 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {13}{2}} \sqrt {2}+675675 \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {13}{2}} \sqrt {2}+270270 \sin \left (d x +c \right ) \cos \left (d x +c \right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {13}{2}} \sqrt {2}+45045 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {13}{2}} \sin \left (d x +c \right )-4150912 \left (\cos ^{7}\left (d x +c \right )\right )-807424 \left (\cos ^{6}\left (d x +c \right )\right )+5563904 \left (\cos ^{5}\left (d x +c \right )\right )+2781952 \left (\cos ^{4}\left (d x +c \right )\right )-2585600 \left (\cos ^{3}\left (d x +c \right )\right )-1809920 \left (\cos ^{2}\left (d x +c \right )\right )+564480 \cos \left (d x +c \right )+443520\right ) a}{2882880 d \cos \left (d x +c \right )^{6} \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^6\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \tan ^{6}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________