Optimal. Leaf size=650 \[ \frac {2 (a-b) \sqrt {a+b} \left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \cot (c+d x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{45045 b^5 d}+\frac {2 (a-b) \sqrt {a+b} \left (240 a^5 C+180 a^4 b C+1617 b^5 (13 A+11 C)+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)\right ) \cot (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{45045 b^4 d}+\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{45045 b^3 d}-\frac {2 \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{45045 b^2 d}+\frac {2 a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{9009 b d}+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d} \]
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Rubi [A]
time = 1.90, antiderivative size = 650, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4182, 4181,
4187, 4177, 4167, 4090, 3917, 4089} \begin {gather*} \frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{1287 d}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{9009 b d}-\frac {2 \left (90 a^4 C-15 a^2 b^2 (715 A+543 C)-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{45045 b^2 d}+\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{45045 b^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{45045 b^5 d}+\frac {2 (a-b) \sqrt {a+b} \left (240 a^5 C+180 a^4 b C+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)+1617 b^5 (13 A+11 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{45045 b^4 d}+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{143 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3917
Rule 4089
Rule 4090
Rule 4167
Rule 4177
Rule 4181
Rule 4182
Rule 4187
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac {2}{13} \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {1}{2} a (13 A+6 C)+\frac {1}{2} b (13 A+11 C) \sec (c+d x)+\frac {5}{2} a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac {4}{143} \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (\frac {1}{4} a^2 (143 A+96 C)+\frac {1}{2} a b (143 A+116 C) \sec (c+d x)+\frac {1}{4} \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac {8 \int \frac {\sec ^3(c+d x) \left (\frac {3}{8} a \left (22 b^2 (13 A+11 C)+a^2 (429 A+318 C)\right )+\frac {1}{8} b \left (77 b^2 (13 A+11 C)+a^2 (3861 A+3057 C)\right ) \sec (c+d x)+\frac {1}{8} a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{1287}\\ &=\frac {2 a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{9009 b d}+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac {16 \int \frac {\sec ^2(c+d x) \left (\frac {1}{4} a^2 \left (2717 A b^2+15 a^2 C+2209 b^2 C\right )+\frac {1}{16} a b \left (a^2 (9009 A+6753 C)+b^2 (19591 A+16127 C)\right ) \sec (c+d x)-\frac {1}{16} \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{9009 b}\\ &=-\frac {2 \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{45045 b^2 d}+\frac {2 a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{9009 b d}+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac {32 \int \frac {\sec (c+d x) \left (-\frac {1}{16} a \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right )+\frac {1}{32} b \left (30 a^4 C+1617 b^4 (13 A+11 C)+5 a^2 b^2 (17303 A+13723 C)\right ) \sec (c+d x)+\frac {3}{32} a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{45045 b^2}\\ &=\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{45045 b^3 d}-\frac {2 \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{45045 b^2 d}+\frac {2 a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{9009 b d}+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac {64 \int \frac {\sec (c+d x) \left (-\frac {3}{64} a b \left (60 a^4 C-5 a^2 b^2 (4433 A+3337 C)-3 b^4 (12441 A+10277 C)\right )-\frac {3}{64} \left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{135135 b^3}\\ &=\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{45045 b^3 d}-\frac {2 \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{45045 b^2 d}+\frac {2 a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{9009 b d}+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac {\left ((a-b) \left (240 a^5 C+180 a^4 b C+1617 b^5 (13 A+11 C)+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{45045 b^3}-\frac {\left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{45045 b^3}\\ &=\frac {2 (a-b) \sqrt {a+b} \left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{45045 b^5 d}+\frac {2 (a-b) \sqrt {a+b} \left (240 a^5 C+180 a^4 b C+1617 b^5 (13 A+11 C)+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{45045 b^4 d}+\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{45045 b^3 d}-\frac {2 \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{45045 b^2 d}+\frac {2 a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{9009 b d}+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(4418\) vs. \(2(650)=1300\).
time = 25.94, size = 4418, normalized size = 6.80 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(6076\) vs.
\(2(604)=1208\).
time = 1.93, size = 6077, normalized size = 9.35
method | result | size |
default | \(\text {Expression too large to display}\) | \(6077\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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