Optimal. Leaf size=189 \[ \frac {2 a^2 (39 A+34 B) \tan (c+d x)}{45 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (9 A+10 B) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a (39 A+34 B) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a B \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (39 A+34 B) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d} \]
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Rubi [A]
time = 0.31, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {4103, 4101,
3885, 4086, 3877} \begin {gather*} \frac {2 a^2 (9 A+10 B) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (39 A+34 B) \tan (c+d x)}{45 d \sqrt {a \sec (c+d x)+a}}+\frac {2 (39 A+34 B) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}-\frac {4 a (39 A+34 B) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {2 a B \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 4103
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx &=\frac {2 a B \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2}{9} \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {3}{2} a (3 A+2 B)+\frac {1}{2} a (9 A+10 B) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a^2 (9 A+10 B) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a B \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {1}{21} (a (39 A+34 B)) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a^2 (9 A+10 B) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a B \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (39 A+34 B) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac {1}{105} (2 (39 A+34 B)) \int \sec (c+d x) \left (\frac {3 a}{2}-a \sec (c+d x)\right ) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a^2 (9 A+10 B) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a (39 A+34 B) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a B \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (39 A+34 B) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac {1}{45} (a (39 A+34 B)) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a^2 (39 A+34 B) \tan (c+d x)}{45 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (9 A+10 B) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a (39 A+34 B) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a B \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{9 d}+\frac {2 (39 A+34 B) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}\\ \end {align*}
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Mathematica [A]
time = 0.74, size = 100, normalized size = 0.53 \begin {gather*} \frac {2 a^2 \left (8 (39 A+34 B)+4 (39 A+34 B) \sec (c+d x)+3 (39 A+34 B) \sec ^2(c+d x)+5 (9 A+17 B) \sec ^3(c+d x)+35 B \sec ^4(c+d x)\right ) \tan (c+d x)}{315 d \sqrt {a (1+\sec (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.04, size = 139, normalized size = 0.74
method | result | size |
default | \(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (312 A \left (\cos ^{4}\left (d x +c \right )\right )+272 B \left (\cos ^{4}\left (d x +c \right )\right )+156 A \left (\cos ^{3}\left (d x +c \right )\right )+136 B \left (\cos ^{3}\left (d x +c \right )\right )+117 A \left (\cos ^{2}\left (d x +c \right )\right )+102 B \left (\cos ^{2}\left (d x +c \right )\right )+45 A \cos \left (d x +c \right )+85 B \cos \left (d x +c \right )+35 B \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a}{315 d \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )}\) | \(139\) |
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 = 3.24, size = 127, normalized size = 0.67 \begin {gather*} \frac {2 \, {\left (8 \, {\left (39 \, A + 34 \, B\right )} a \cos \left (d x + c\right )^{4} + 4 \, {\left (39 \, A + 34 \, B\right )} a \cos \left (d x + c\right )^{3} + 3 \, {\left (39 \, A + 34 \, B\right )} a \cos \left (d x + c\right )^{2} + 5 \, {\left (9 \, A + 17 \, B\right )} a \cos \left (d x + c\right ) + 35 \, B a\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 \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \left (A + B \sec {\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 = 1.28, size = 258, normalized size = 1.37 \begin {gather*} \frac {4 \, {\left ({\left ({\left ({\left (2 \, \sqrt {2} {\left (57 \, A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 47 \, B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \sqrt {2} {\left (57 \, A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 47 \, B a^{6} \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} + 819 \, \sqrt {2} {\left (A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + B a^{6} \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} - 105 \, \sqrt {2} {\left (7 \, A a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 5 \, B a^{6} \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^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + B a^{6} \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 = 9.60, size = 596, normalized size = 3.15 \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 ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (-\frac {a\,\left (3\,A+2\,B\right )\,8{}\mathrm {i}}{7\,d}+\frac {a\,\left (A+4\,B\right )\,8{}\mathrm {i}}{7\,d}+\frac {B\,a\,32{}\mathrm {i}}{63\,d}\right )+\frac {A\,a\,8{}\mathrm {i}}{7\,d}-\frac {a\,\left (A+2\,B\right )\,24{}\mathrm {i}}{7\,d}-\frac {B\,a\,32{}\mathrm {i}}{7\,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 )}^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 {a\,\left (2\,A+3\,B\right )\,16{}\mathrm {i}}{9\,d}+\frac {a\,\left (3\,A+2\,B\right )\,8{}\mathrm {i}}{9\,d}+\frac {A\,a\,8{}\mathrm {i}}{9\,d}\right )+\frac {a\,\left (2\,A+3\,B\right )\,16{}\mathrm {i}}{9\,d}-\frac {a\,\left (3\,A+2\,B\right )\,8{}\mathrm {i}}{9\,d}-\frac {A\,a\,8{}\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 {\left (\frac {A\,a\,8{}\mathrm {i}}{3\,d}-\frac {a\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (39\,A+34\,B\right )\,8{}\mathrm {i}}{315\,d}\right )\,\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}}+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\,\left (3\,A+2\,B\right )\,8{}\mathrm {i}}{5\,d}+\frac {a\,\left (3\,A+B\right )\,16{}\mathrm {i}}{105\,d}\right )-\frac {A\,a\,8{}\mathrm {i}}{5\,d}+\frac {a\,\left (A+3\,B\right )\,16{}\mathrm {i}}{5\,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 {a\,{\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 (39\,A+34\,B\right )\,16{}\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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