Optimal. Leaf size=273 \[ \frac {b \left (6 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \tan (c+d x)}{15 d}+\frac {a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{120 d}+\frac {b \left (3 a^2 (4 A+5 C)+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac {a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \tan (c+d x) \sec (c+d x)}{16 d}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac {A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{10 d} \]
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Rubi [A] time = 0.79, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3048, 3047, 3031, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac {b \left (6 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \tan (c+d x)}{15 d}+\frac {a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{120 d}+\frac {b \left (3 a^2 (4 A+5 C)+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac {a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \tan (c+d x) \sec (c+d x)}{16 d}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac {A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{10 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 3021
Rule 3031
Rule 3047
Rule 3048
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx &=\frac {A (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{6} \int (a+b \cos (c+d x))^2 \left (3 A b+a (5 A+6 C) \cos (c+d x)+2 b (A+3 C) \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx\\ &=\frac {A b (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{30} \int (a+b \cos (c+d x)) \left (6 A b^2+5 a^2 (5 A+6 C)+a b (47 A+60 C) \cos (c+d x)+2 b^2 (8 A+15 C) \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac {a \left (6 A b^2+5 a^2 (5 A+6 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {A b (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {1}{120} \int \left (-24 b \left (A b^2+3 a^2 (4 A+5 C)\right )-15 a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \cos (c+d x)-8 b^3 (8 A+15 C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac {b \left (A b^2+3 a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{15 d}+\frac {a \left (6 A b^2+5 a^2 (5 A+6 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {A b (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {1}{360} \int \left (-45 a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right )-24 b \left (5 b^2 (2 A+3 C)+6 a^2 (4 A+5 C)\right ) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac {b \left (A b^2+3 a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{15 d}+\frac {a \left (6 A b^2+5 a^2 (5 A+6 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {A b (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{15} \left (b \left (5 b^2 (2 A+3 C)+6 a^2 (4 A+5 C)\right )\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right )\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac {a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {b \left (A b^2+3 a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{15 d}+\frac {a \left (6 A b^2+5 a^2 (5 A+6 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {A b (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{16} \left (a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right )\right ) \int \sec (c+d x) \, dx-\frac {\left (b \left (5 b^2 (2 A+3 C)+6 a^2 (4 A+5 C)\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d}\\ &=\frac {a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {b \left (5 b^2 (2 A+3 C)+6 a^2 (4 A+5 C)\right ) \tan (c+d x)}{15 d}+\frac {a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {b \left (A b^2+3 a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{15 d}+\frac {a \left (6 A b^2+5 a^2 (5 A+6 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {A b (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 1.71, size = 184, normalized size = 0.67 \[ \frac {15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (40 a^3 A \sec ^5(c+d x)+16 b \left (5 \left (3 a^2 (2 A+C)+A b^2\right ) \tan ^2(c+d x)+15 \left (3 a^2+b^2\right ) (A+C)+9 a^2 A \tan ^4(c+d x)\right )+10 a \left (a^2 (5 A+6 C)+18 A b^2\right ) \sec ^3(c+d x)+15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \sec (c+d x)\right )}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 262, normalized size = 0.96 \[ \frac {15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (6 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 5 \, {\left (2 \, A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{5} + 144 \, A a^{2} b \cos \left (d x + c\right ) + 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{4} + 40 \, A a^{3} + 16 \, {\left (3 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 5 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.48, size = 932, normalized size = 3.41 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 430, normalized size = 1.58 \[ \frac {A \,a^{3} \tan \left (d x +c \right ) \left (\sec ^{5}\left (d x +c \right )\right )}{6 d}+\frac {5 A \,a^{3} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{24 d}+\frac {5 A \,a^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{16 d}+\frac {5 A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16 d}+\frac {C \,a^{3} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 C \,a^{3} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{8 d}+\frac {3 C \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {8 A \,a^{2} b \tan \left (d x +c \right )}{5 d}+\frac {3 A \,a^{2} b \tan \left (d x +c \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{5 d}+\frac {4 A \,a^{2} b \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{5 d}+\frac {2 C \,a^{2} b \tan \left (d x +c \right )}{d}+\frac {C \,a^{2} b \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{d}+\frac {3 A a \,b^{2} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {9 A a \,b^{2} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{8 d}+\frac {9 A a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {3 C a \,b^{2} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{2 d}+\frac {3 C a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {2 A \,b^{3} \tan \left (d x +c \right )}{3 d}+\frac {A \,b^{3} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {b^{3} C \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 386, normalized size = 1.41 \[ \frac {96 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{2} b + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} b + 160 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b^{3} - 5 \, A a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, C a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, A a b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 360 \, C a b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, C b^{3} \tan \left (d x + c\right )}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.67, size = 572, normalized size = 2.10 \[ \frac {\left (\frac {11\,A\,a^3}{8}-2\,A\,b^3+\frac {5\,C\,a^3}{4}-2\,C\,b^3+\frac {15\,A\,a\,b^2}{4}-6\,A\,a^2\,b+3\,C\,a\,b^2-6\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {5\,A\,a^3}{24}+\frac {22\,A\,b^3}{3}-\frac {7\,C\,a^3}{4}+10\,C\,b^3-\frac {21\,A\,a\,b^2}{4}+14\,A\,a^2\,b-9\,C\,a\,b^2+22\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {15\,A\,a^3}{4}-12\,A\,b^3+\frac {C\,a^3}{2}-20\,C\,b^3+\frac {3\,A\,a\,b^2}{2}-\frac {156\,A\,a^2\,b}{5}+6\,C\,a\,b^2-36\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {15\,A\,a^3}{4}+12\,A\,b^3+\frac {C\,a^3}{2}+20\,C\,b^3+\frac {3\,A\,a\,b^2}{2}+\frac {156\,A\,a^2\,b}{5}+6\,C\,a\,b^2+36\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {5\,A\,a^3}{24}-\frac {22\,A\,b^3}{3}-\frac {7\,C\,a^3}{4}-10\,C\,b^3-\frac {21\,A\,a\,b^2}{4}-14\,A\,a^2\,b-9\,C\,a\,b^2-22\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {11\,A\,a^3}{8}+2\,A\,b^3+\frac {5\,C\,a^3}{4}+2\,C\,b^3+\frac {15\,A\,a\,b^2}{4}+6\,A\,a^2\,b+3\,C\,a\,b^2+6\,C\,a^2\,b\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 {a\,\mathrm {atanh}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,A\,a^2+18\,A\,b^2+6\,C\,a^2+24\,C\,b^2\right )}{4\,\left (\frac {5\,A\,a^3}{4}+\frac {3\,C\,a^3}{2}+\frac {9\,A\,a\,b^2}{2}+6\,C\,a\,b^2\right )}\right )\,\left (5\,A\,a^2+18\,A\,b^2+6\,C\,a^2+24\,C\,b^2\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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