Optimal. Leaf size=194 \[ \frac {2 a^4 (8 A+7 B) \tan ^3(c+d x)}{15 d}+\frac {4 a^4 (8 A+7 B) \tan (c+d x)}{5 d}+\frac {7 a^4 (8 A+7 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a^4 (8 A+7 B) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac {27 a^4 (8 A+7 B) \tan (c+d x) \sec (c+d x)}{80 d}+\frac {(6 A-B) \tan (c+d x) (a \sec (c+d x)+a)^4}{30 d}+\frac {B \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.32, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4010, 4001, 3791, 3770, 3767, 8, 3768} \[ \frac {2 a^4 (8 A+7 B) \tan ^3(c+d x)}{15 d}+\frac {4 a^4 (8 A+7 B) \tan (c+d x)}{5 d}+\frac {7 a^4 (8 A+7 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a^4 (8 A+7 B) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac {27 a^4 (8 A+7 B) \tan (c+d x) \sec (c+d x)}{80 d}+\frac {(6 A-B) \tan (c+d x) (a \sec (c+d x)+a)^4}{30 d}+\frac {B \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3767
Rule 3768
Rule 3770
Rule 3791
Rule 4001
Rule 4010
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx &=\frac {B (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^4 (5 a B+a (6 A-B) \sec (c+d x)) \, dx}{6 a}\\ &=\frac {(6 A-B) (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {B (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {1}{10} (8 A+7 B) \int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx\\ &=\frac {(6 A-B) (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {B (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {1}{10} (8 A+7 B) \int \left (a^4 \sec (c+d x)+4 a^4 \sec ^2(c+d x)+6 a^4 \sec ^3(c+d x)+4 a^4 \sec ^4(c+d x)+a^4 \sec ^5(c+d x)\right ) \, dx\\ &=\frac {(6 A-B) (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {B (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {1}{10} \left (a^4 (8 A+7 B)\right ) \int \sec (c+d x) \, dx+\frac {1}{10} \left (a^4 (8 A+7 B)\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{5} \left (2 a^4 (8 A+7 B)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{5} \left (2 a^4 (8 A+7 B)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{5} \left (3 a^4 (8 A+7 B)\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac {a^4 (8 A+7 B) \tanh ^{-1}(\sin (c+d x))}{10 d}+\frac {3 a^4 (8 A+7 B) \sec (c+d x) \tan (c+d x)}{10 d}+\frac {a^4 (8 A+7 B) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {(6 A-B) (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {B (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {1}{40} \left (3 a^4 (8 A+7 B)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{10} \left (3 a^4 (8 A+7 B)\right ) \int \sec (c+d x) \, dx-\frac {\left (2 a^4 (8 A+7 B)\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{5 d}-\frac {\left (2 a^4 (8 A+7 B)\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac {2 a^4 (8 A+7 B) \tanh ^{-1}(\sin (c+d x))}{5 d}+\frac {4 a^4 (8 A+7 B) \tan (c+d x)}{5 d}+\frac {27 a^4 (8 A+7 B) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {a^4 (8 A+7 B) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {(6 A-B) (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {B (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {2 a^4 (8 A+7 B) \tan ^3(c+d x)}{15 d}+\frac {1}{80} \left (3 a^4 (8 A+7 B)\right ) \int \sec (c+d x) \, dx\\ &=\frac {7 a^4 (8 A+7 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {4 a^4 (8 A+7 B) \tan (c+d x)}{5 d}+\frac {27 a^4 (8 A+7 B) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {a^4 (8 A+7 B) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {(6 A-B) (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {B (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {2 a^4 (8 A+7 B) \tan ^3(c+d x)}{15 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.38, size = 358, normalized size = 1.85 \[ -\frac {a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \sec ^6(c+d x) \left (3360 (8 A+7 B) \cos ^6(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )-\sec (c) (-160 (83 A+72 B) \sin (c)+30 (88 A+125 B) \sin (d x)+2640 A \sin (2 c+d x)+15840 A \sin (c+2 d x)-4080 A \sin (3 c+2 d x)+3480 A \sin (2 c+3 d x)+3480 A \sin (4 c+3 d x)+7728 A \sin (3 c+4 d x)-240 A \sin (5 c+4 d x)+840 A \sin (4 c+5 d x)+840 A \sin (6 c+5 d x)+1328 A \sin (5 c+6 d x)+3750 B \sin (2 c+d x)+15360 B \sin (c+2 d x)-1920 B \sin (3 c+2 d x)+3845 B \sin (2 c+3 d x)+3845 B \sin (4 c+3 d x)+6912 B \sin (3 c+4 d x)+735 B \sin (4 c+5 d x)+735 B \sin (6 c+5 d x)+1152 B \sin (5 c+6 d x))\right )}{122880 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 185, normalized size = 0.95 \[ \frac {105 \, {\left (8 \, A + 7 \, B\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (8 \, A + 7 \, B\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (83 \, A + 72 \, B\right )} a^{4} \cos \left (d x + c\right )^{5} + 105 \, {\left (8 \, A + 7 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} + 32 \, {\left (17 \, A + 18 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 10 \, {\left (24 \, A + 41 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 48 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right ) + 40 \, B a^{4}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.78, size = 280, normalized size = 1.44 \[ \frac {105 \, {\left (8 \, A a^{4} + 7 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (8 \, A a^{4} + 7 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (840 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 735 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 4760 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 4165 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 11088 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9702 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 13488 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 11802 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9320 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7355 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3000 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3105 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.80, size = 280, normalized size = 1.44 \[ \frac {83 A \,a^{4} \tan \left (d x +c \right )}{15 d}+\frac {49 a^{4} B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{16 d}+\frac {49 a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16 d}+\frac {7 A \,a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {7 A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {24 a^{4} B \tan \left (d x +c \right )}{5 d}+\frac {12 a^{4} B \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{5 d}+\frac {34 A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{15 d}+\frac {41 a^{4} B \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{24 d}+\frac {A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{d}+\frac {4 a^{4} B \tan \left (d x +c \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{5 d}+\frac {A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{5 d}+\frac {a^{4} B \tan \left (d x +c \right ) \left (\sec ^{5}\left (d x +c \right )\right )}{6 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.35, size = 464, normalized size = 2.39 \[ \frac {32 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{4} + 960 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 128 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a^{4} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} - 5 \, B a^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, B a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 480 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, B a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, A a^{4} \tan \left (d x + c\right )}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.61, size = 262, normalized size = 1.35 \[ \frac {\left (-7\,A\,a^4-\frac {49\,B\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {119\,A\,a^4}{3}+\frac {833\,B\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {462\,A\,a^4}{5}-\frac {1617\,B\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {562\,A\,a^4}{5}+\frac {1967\,B\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {233\,A\,a^4}{3}-\frac {1471\,B\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (25\,A\,a^4+\frac {207\,B\,a^4}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {7\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (8\,A+7\,B\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{4} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 6 A \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{5}{\left (c + d x \right )}\, dx + \int A \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{4}{\left (c + d x \right )}\, dx + \int 6 B \sec ^{5}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{7}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________