Optimal. Leaf size=290 \[ -\frac {2 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {2 a^{10} \tan ^{15}(c+d x)}{15 d (a \sec (c+d x)+a)^{15/2}}+\frac {18 a^9 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac {62 a^8 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac {98 a^7 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac {62 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac {2 a^5 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac {2 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}+\frac {2 a^3 \tan (c+d x)}{d \sqrt {a \sec (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 290, 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} \[ \frac {2 a^{10} \tan ^{15}(c+d x)}{15 d (a \sec (c+d x)+a)^{15/2}}+\frac {18 a^9 \tan ^{13}(c+d x)}{13 d (a \sec (c+d x)+a)^{13/2}}+\frac {62 a^8 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac {98 a^7 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac {62 a^6 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac {2 a^5 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac {2 a^4 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac {2 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {2 a^3 \tan (c+d x)}{d \sqrt {a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 461
Rule 3887
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^{5/2} \tan ^6(c+d x) \, dx &=-\frac {\left (2 a^6\right ) \operatorname {Subst}\left (\int \frac {x^6 \left (2+a x^2\right )^5}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac {\left (2 a^6\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^3}-\frac {x^2}{a^2}+\frac {x^4}{a}+31 x^6+49 a x^8+31 a^2 x^{10}+9 a^3 x^{12}+a^4 x^{14}-\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^3 \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a^4 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^5 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {62 a^6 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac {98 a^7 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac {62 a^8 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac {18 a^9 \tan ^{13}(c+d x)}{13 d (a+a \sec (c+d x))^{13/2}}+\frac {2 a^{10} \tan ^{15}(c+d x)}{15 d (a+a \sec (c+d x))^{15/2}}+\frac {\left (2 a^3\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^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {2 a^3 \tan (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {2 a^4 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac {2 a^5 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac {62 a^6 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac {98 a^7 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac {62 a^8 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac {18 a^9 \tan ^{13}(c+d x)}{13 d (a+a \sec (c+d x))^{13/2}}+\frac {2 a^{10} \tan ^{15}(c+d x)}{15 d (a+a \sec (c+d x))^{15/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 10.33, size = 173, normalized size = 0.60 \[ \frac {a^2 \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^7(c+d x) \sqrt {a (\sec (c+d x)+1)} \left (604890 \sin \left (\frac {1}{2} (c+d x)\right )-87230 \sin \left (\frac {3}{2} (c+d x)\right )+450450 \sin \left (\frac {5}{2} (c+d x)\right )-137670 \sin \left (\frac {7}{2} (c+d x)\right )+210210 \sin \left (\frac {9}{2} (c+d x)\right )+75450 \sin \left (\frac {11}{2} (c+d x)\right )+90090 \sin \left (\frac {13}{2} (c+d x)\right )+16066 \sin \left (\frac {15}{2} (c+d x)\right )-2882880 \sqrt {2} \sin ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^{\frac {15}{2}}(c+d x)\right )}{2882880 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.80, size = 477, normalized size = 1.64 \[ \left [\frac {45045 \, {\left (a^{2} \cos \left (d x + c\right )^{8} + a^{2} \cos \left (d x + c\right )^{7}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (16066 \, a^{2} \cos \left (d x + c\right )^{7} + 53078 \, a^{2} \cos \left (d x + c\right )^{6} + 17286 \, a^{2} \cos \left (d x + c\right )^{5} - 30640 \, a^{2} \cos \left (d x + c\right )^{4} - 26810 \, a^{2} \cos \left (d x + c\right )^{3} + 2898 \, a^{2} \cos \left (d x + c\right )^{2} + 10164 \, a^{2} \cos \left (d x + c\right ) + 3003 \, a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right )^{8} + d \cos \left (d x + c\right )^{7}\right )}}, \frac {2 \, {\left (45045 \, {\left (a^{2} \cos \left (d x + c\right )^{8} + a^{2} \cos \left (d x + c\right )^{7}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) + {\left (16066 \, a^{2} \cos \left (d x + c\right )^{7} + 53078 \, a^{2} \cos \left (d x + c\right )^{6} + 17286 \, a^{2} \cos \left (d x + c\right )^{5} - 30640 \, a^{2} \cos \left (d x + c\right )^{4} - 26810 \, a^{2} \cos \left (d x + c\right )^{3} + 2898 \, a^{2} \cos \left (d x + c\right )^{2} + 10164 \, a^{2} \cos \left (d x + c\right ) + 3003 \, a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{45045 \, {\left (d \cos \left (d x + c\right )^{8} + d \cos \left (d x + c\right )^{7}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 10.86, size = 397, normalized size = 1.37 \[ \frac {\frac {45045 \, \sqrt {-a} a^{3} \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^{10} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (345345 \, \sqrt {2} a^{10} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (1162161 \, \sqrt {2} a^{10} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (611325 \, \sqrt {2} a^{10} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (77935 \, \sqrt {2} a^{10} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + {\left (109005 \, \sqrt {2} a^{10} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + {\left (11633 \, \sqrt {2} a^{10} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 64725 \, \sqrt {2} a^{10} \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 )^{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 )}^{7} \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 = 1.67, size = 747, normalized size = 2.58 \[ \text {result too large to display} \]
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 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^6\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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]
________________________________________________________________________________________