Optimal. Leaf size=148 \[ -\frac {2 (80 A-3 B-4 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac {(55 A-6 B-8 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac {A x}{a^4}-\frac {(10 A-3 B-4 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A-B+C) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.28, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4052, 3922, 3919, 3794} \[ -\frac {2 (80 A-3 B-4 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac {(55 A-6 B-8 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac {A x}{a^4}-\frac {(10 A-3 B-4 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A-B+C) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3794
Rule 3919
Rule 3922
Rule 4052
Rubi steps
\begin {align*} \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx &=-\frac {(A-B+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {\int \frac {-7 a A+a (3 A-3 B-4 C) \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {(A-B+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(10 A-3 B-4 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {35 a^2 A-2 a^2 (10 A-3 B-4 C) \sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {(55 A-6 B-8 C) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(10 A-3 B-4 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {\int \frac {-105 a^3 A+a^3 (55 A-6 B-8 C) \sec (c+d x)}{a+a \sec (c+d x)} \, dx}{105 a^6}\\ &=\frac {A x}{a^4}-\frac {(55 A-6 B-8 C) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(10 A-3 B-4 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(2 (80 A-3 B-4 C)) \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{105 a^3}\\ &=\frac {A x}{a^4}-\frac {(55 A-6 B-8 C) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(10 A-3 B-4 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {2 (80 A-3 B-4 C) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 1.40, size = 405, normalized size = 2.74 \[ \frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (8260 A \sin \left (c+\frac {d x}{2}\right )-7140 A \sin \left (c+\frac {3 d x}{2}\right )+3780 A \sin \left (2 c+\frac {3 d x}{2}\right )-2800 A \sin \left (2 c+\frac {5 d x}{2}\right )+840 A \sin \left (3 c+\frac {5 d x}{2}\right )-520 A \sin \left (3 c+\frac {7 d x}{2}\right )+3675 A d x \cos \left (c+\frac {d x}{2}\right )+2205 A d x \cos \left (c+\frac {3 d x}{2}\right )+2205 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+735 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+735 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+105 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+105 A d x \cos \left (4 c+\frac {7 d x}{2}\right )-9940 A \sin \left (\frac {d x}{2}\right )+3675 A d x \cos \left (\frac {d x}{2}\right )-1260 B \sin \left (c+\frac {d x}{2}\right )+882 B \sin \left (c+\frac {3 d x}{2}\right )-630 B \sin \left (2 c+\frac {3 d x}{2}\right )+294 B \sin \left (2 c+\frac {5 d x}{2}\right )-210 B \sin \left (3 c+\frac {5 d x}{2}\right )+72 B \sin \left (3 c+\frac {7 d x}{2}\right )+1260 B \sin \left (\frac {d x}{2}\right )-350 C \sin \left (c+\frac {d x}{2}\right )+336 C \sin \left (c+\frac {3 d x}{2}\right )-210 C \sin \left (2 c+\frac {3 d x}{2}\right )+182 C \sin \left (2 c+\frac {5 d x}{2}\right )+26 C \sin \left (3 c+\frac {7 d x}{2}\right )+560 C \sin \left (\frac {d x}{2}\right )\right )}{13440 a^4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 192, normalized size = 1.30 \[ \frac {105 \, A d x \cos \left (d x + c\right )^{4} + 420 \, A d x \cos \left (d x + c\right )^{3} + 630 \, A d x \cos \left (d x + c\right )^{2} + 420 \, A d x \cos \left (d x + c\right ) + 105 \, A d x - {\left ({\left (260 \, A - 36 \, B - 13 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (620 \, A - 39 \, B - 52 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (535 \, A - 24 \, B - 32 \, C\right )} \cos \left (d x + c\right ) + 160 \, A - 6 \, B - 8 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.26, size = 220, normalized size = 1.49 \[ \frac {\frac {840 \, {\left (d x + c\right )} A}{a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 63 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1575 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.92, size = 255, normalized size = 1.72 \[ \frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{56 d \,a^{4}}-\frac {B \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}+\frac {C \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}-\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}+\frac {3 B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{40 d \,a^{4}}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{24 d \,a^{4}}-\frac {B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{24 d \,a^{4}}-\frac {15 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {2 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.47, size = 286, normalized size = 1.93 \[ -\frac {5 \, A {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - \frac {C {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} - \frac {3 \, B {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.69, size = 229, normalized size = 1.55 \[ \frac {A\,x}{a^4}+\frac {{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}-\frac {15\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}\right )-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {B\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}-\frac {11\,A\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {C\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}\right )-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {A\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8}-\frac {3\,B\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40}+\frac {C\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40}\right )+\frac {A\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56}-\frac {B\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56}+\frac {C\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {A}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec {\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________