Optimal. Leaf size=193 \[ \frac {2 a (21 A+18 B+16 C) \tan (c+d x)}{45 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (9 B+C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 (21 A+18 B+16 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (21 A+18 B+16 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d} \]
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Rubi [A]
time = 0.31, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {4173, 4101,
3885, 4086, 3877} \begin {gather*} \frac {2 (21 A+18 B+16 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 a d}-\frac {4 (21 A+18 B+16 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {2 a (21 A+18 B+16 C) \tan (c+d x)}{45 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a (9 B+C) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt {a \sec (c+d x)+a}}+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{9 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3877
Rule 3885
Rule 4086
Rule 4101
Rule 4173
Rubi steps
\begin {align*} \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {3}{2} a (3 A+2 C)+\frac {1}{2} a (9 B+C) \sec (c+d x)\right ) \, dx}{9 a}\\ &=\frac {2 a (9 B+C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {1}{21} (21 A+18 B+16 C) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a (9 B+C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (21 A+18 B+16 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d}+\frac {(2 (21 A+18 B+16 C)) \int \sec (c+d x) \left (\frac {3 a}{2}-a \sec (c+d x)\right ) \sqrt {a+a \sec (c+d x)} \, dx}{105 a}\\ &=\frac {2 a (9 B+C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 (21 A+18 B+16 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (21 A+18 B+16 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d}+\frac {1}{45} (21 A+18 B+16 C) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a (21 A+18 B+16 C) \tan (c+d x)}{45 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (9 B+C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 (21 A+18 B+16 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 C \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (21 A+18 B+16 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 a d}\\ \end {align*}
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Mathematica [A]
time = 1.95, size = 153, normalized size = 0.79 \begin {gather*} \frac {(189 A+162 B+214 C+2 (63 A+99 B+88 C) \cos (c+d x)+11 (21 A+18 B+16 C) \cos (2 (c+d x))+42 A \cos (3 (c+d x))+36 B \cos (3 (c+d x))+32 C \cos (3 (c+d x))+42 A \cos (4 (c+d x))+36 B \cos (4 (c+d x))+32 C \cos (4 (c+d x))) \sec ^4(c+d x) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{315 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 13.99, size = 171, normalized size = 0.89
method | result | size |
default | \(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (168 A \left (\cos ^{4}\left (d x +c \right )\right )+144 B \left (\cos ^{4}\left (d x +c \right )\right )+128 C \left (\cos ^{4}\left (d x +c \right )\right )+84 A \left (\cos ^{3}\left (d x +c \right )\right )+72 B \left (\cos ^{3}\left (d x +c \right )\right )+64 C \left (\cos ^{3}\left (d x +c \right )\right )+63 A \left (\cos ^{2}\left (d x +c \right )\right )+54 B \left (\cos ^{2}\left (d x +c \right )\right )+48 C \left (\cos ^{2}\left (d x +c \right )\right )+45 B \cos \left (d x +c \right )+40 C \cos \left (d x +c \right )+35 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{315 d \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )}\) | \(171\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.11, size = 131, normalized size = 0.68 \begin {gather*} \frac {2 \, {\left (8 \, {\left (21 \, A + 18 \, B + 16 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (21 \, A + 18 \, B + 16 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 18 \, B + 16 \, C\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (9 \, B + 8 \, C\right )} \cos \left (d x + c\right ) + 35 \, C\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 323, normalized size = 1.67 \begin {gather*} \frac {2 \, {\left ({\left ({\left ({\left (\sqrt {2} {\left (147 \, A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 81 \, B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 107 \, C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 18 \, \sqrt {2} {\left (28 \, A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 29 \, B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 18 \, C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 126 \, \sqrt {2} {\left (7 \, A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 6 \, B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 7 \, C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 210 \, \sqrt {2} {\left (4 \, A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 3 \, B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 2 \, C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 315 \, \sqrt {2} {\left (A a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + B a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + C a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{315 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 11.43, size = 600, normalized size = 3.11 \begin {gather*} \frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {A\,8{}\mathrm {i}}{3\,d}-\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (168\,A+144\,B+128\,C\right )\,1{}\mathrm {i}}{315\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}+\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {A\,8{}\mathrm {i}}{9\,d}-\frac {\left (16\,A+16\,B+32\,C\right )\,1{}\mathrm {i}}{9\,d}+\frac {\left (8\,A+16\,B\right )\,1{}\mathrm {i}}{9\,d}\right )-\frac {A\,8{}\mathrm {i}}{9\,d}+\frac {\left (16\,A+16\,B+32\,C\right )\,1{}\mathrm {i}}{9\,d}-\frac {\left (8\,A+16\,B\right )\,1{}\mathrm {i}}{9\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {A\,8{}\mathrm {i}}{7\,d}-\frac {C\,32{}\mathrm {i}}{7\,d}-\frac {\left (72\,A+144\,B+288\,C\right )\,1{}\mathrm {i}}{63\,d}+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {C\,32{}\mathrm {i}}{63\,d}-\frac {\left (72\,A+144\,B\right )\,1{}\mathrm {i}}{63\,d}+\frac {\left (72\,A+288\,C\right )\,1{}\mathrm {i}}{63\,d}\right )\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}+\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {\left (168\,A+336\,B\right )\,1{}\mathrm {i}}{105\,d}+\frac {\left (48\,B-32\,C\right )\,1{}\mathrm {i}}{105\,d}\right )-\frac {A\,8{}\mathrm {i}}{5\,d}+\frac {\left (336\,B+672\,C\right )\,1{}\mathrm {i}}{105\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (336\,A+288\,B+256\,C\right )\,1{}\mathrm {i}}{315\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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