Optimal. Leaf size=225 \[ \frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {(28 A+35 B+32 C) \sin (c+d x) \cos (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{24 d}+\frac {1}{8} a^4 x (28 A+35 B+48 C)+\frac {a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {(28 A+35 B+20 C) \sin (c+d x) \cos ^2(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{60 d}+\frac {a (4 A+5 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.58, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {4086, 4017, 3996, 3770} \[ \frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {(28 A+35 B+20 C) \sin (c+d x) \cos ^2(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{60 d}+\frac {(28 A+35 B+32 C) \sin (c+d x) \cos (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{24 d}+\frac {1}{8} a^4 x (28 A+35 B+48 C)+\frac {a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a (4 A+5 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3770
Rule 3996
Rule 4017
Rule 4086
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (a (4 A+5 B)+5 a C \sec (c+d x)) \, dx}{5 a}\\ &=\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {\int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left (a^2 (28 A+35 B+20 C)+20 a^2 C \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \left (5 a^3 (28 A+35 B+32 C)+60 a^3 C \sec (c+d x)\right ) \, dx}{60 a}\\ &=\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x)) \left (15 a^4 (28 A+35 B+40 C)+120 a^4 C \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}-\frac {\int \left (-15 a^5 (28 A+35 B+48 C)-120 a^5 C \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac {1}{8} a^4 (28 A+35 B+48 C) x+\frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\left (a^4 C\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} a^4 (28 A+35 B+48 C) x+\frac {a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac {a (4 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(28 A+35 B+20 C) \cos ^2(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(28 A+35 B+32 C) \cos (c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.63, size = 182, normalized size = 0.81 \[ \frac {a^4 \left (60 (49 A+56 B+54 C) \sin (c+d x)+120 (8 A+7 B+4 C) \sin (2 (c+d x))+290 A \sin (3 (c+d x))+60 A \sin (4 (c+d x))+6 A \sin (5 (c+d x))+1680 A d x+160 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+2100 B d x+40 C \sin (3 (c+d x))-480 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 C \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+2880 C d x\right )}{480 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 154, normalized size = 0.68 \[ \frac {15 \, {\left (28 \, A + 35 \, B + 48 \, C\right )} a^{4} d x + 60 \, C a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 60 \, C a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 30 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{3} + 8 \, {\left (34 \, A + 20 \, B + 5 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (28 \, A + 27 \, B + 16 \, C\right )} a^{4} \cos \left (d x + c\right ) + 8 \, {\left (83 \, A + 100 \, B + 100 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.35, size = 337, normalized size = 1.50 \[ \frac {120 \, C a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 120 \, C a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 15 \, {\left (28 \, A a^{4} + 35 \, B a^{4} + 48 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (420 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 525 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 600 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1960 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2450 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2720 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3584 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4480 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4720 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3160 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3950 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3680 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1500 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1395 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1080 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.66, size = 320, normalized size = 1.42 \[ \frac {a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{4}}{3 d}+\frac {20 a^{4} B \sin \left (d x +c \right )}{3 d}+\frac {7 A \,a^{4} x}{2}+\frac {35 a^{4} B x}{8}+\frac {83 A \,a^{4} \sin \left (d x +c \right )}{15 d}+\frac {A \,a^{4} \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5 d}+\frac {34 A \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{4}}{15 d}+\frac {C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{4}}{3 d}+\frac {20 a^{4} C \sin \left (d x +c \right )}{3 d}+\frac {a^{4} B \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {27 a^{4} B \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {A \,a^{4} \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{d}+\frac {7 A \,a^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {7 A \,a^{4} c}{2 d}+6 a^{4} C x +\frac {6 C \,a^{4} c}{d}+\frac {2 a^{4} C \sin \left (d x +c \right ) \cos \left (d x +c \right )}{d}+\frac {35 a^{4} B c}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 332, normalized size = 1.48 \[ \frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 480 \, {\left (d x + c\right )} B a^{4} - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 1920 \, {\left (d x + c\right )} C a^{4} + 240 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, A a^{4} \sin \left (d x + c\right ) + 1920 \, B a^{4} \sin \left (d x + c\right ) + 2880 \, C a^{4} \sin \left (d x + c\right )}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.45, size = 1151, normalized size = 5.12 \[ \text {result too large to display} \]
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]
________________________________________________________________________________________