Optimal. Leaf size=237 \[ \frac {2 a^3 (803 A+710 B) \tan (c+d x)}{495 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (209 A+194 B) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a^2 (803 A+710 B) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {2 a^2 (11 A+14 B) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 a (803 A+710 B) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac {2 a B \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d} \]
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Rubi [A]
time = 0.46, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps
used = 6, 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^3 (209 A+194 B) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^3 (803 A+710 B) \tan (c+d x)}{495 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (11 A+14 B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{99 d}-\frac {4 a^2 (803 A+710 B) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3465 d}+\frac {2 a (803 A+710 B) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 d}+\frac {2 a B \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 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))^{5/2} (A+B \sec (c+d x)) \, dx &=\frac {2 a B \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {2}{11} \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{2} a (11 A+6 B)+\frac {1}{2} a (11 A+14 B) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a^2 (11 A+14 B) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 a B \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {4}{99} \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {3}{4} a^2 (55 A+46 B)+\frac {1}{4} a^2 (209 A+194 B) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a^3 (209 A+194 B) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (11 A+14 B) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 a B \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {1}{231} \left (a^2 (803 A+710 B)\right ) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a^3 (209 A+194 B) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (11 A+14 B) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 a (803 A+710 B) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac {2 a B \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {(2 a (803 A+710 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}{1155}\\ &=\frac {2 a^3 (209 A+194 B) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a^2 (803 A+710 B) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {2 a^2 (11 A+14 B) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 a (803 A+710 B) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac {2 a B \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {1}{495} \left (a^2 (803 A+710 B)\right ) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a^3 (803 A+710 B) \tan (c+d x)}{495 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (209 A+194 B) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a^2 (803 A+710 B) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {2 a^2 (11 A+14 B) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 a (803 A+710 B) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac {2 a B \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}\\ \end {align*}
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Mathematica [A]
time = 5.84, size = 117, normalized size = 0.49 \begin {gather*} \frac {2 a^3 \left (8 (803 A+710 B)+4 (803 A+710 B) \sec (c+d x)+3 (803 A+710 B) \sec ^2(c+d x)+5 (286 A+355 B) \sec ^3(c+d x)+35 (11 A+32 B) \sec ^4(c+d x)+315 B \sec ^5(c+d x)\right ) \tan (c+d x)}{3465 d \sqrt {a (1+\sec (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.96, size = 163, normalized size = 0.69
method | result | size |
default | \(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (6424 A \left (\cos ^{5}\left (d x +c \right )\right )+5680 B \left (\cos ^{5}\left (d x +c \right )\right )+3212 A \left (\cos ^{4}\left (d x +c \right )\right )+2840 B \left (\cos ^{4}\left (d x +c \right )\right )+2409 A \left (\cos ^{3}\left (d x +c \right )\right )+2130 B \left (\cos ^{3}\left (d x +c \right )\right )+1430 A \left (\cos ^{2}\left (d x +c \right )\right )+1775 B \left (\cos ^{2}\left (d x +c \right )\right )+385 A \cos \left (d x +c \right )+1120 B \cos \left (d x +c \right )+315 B \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{3465 d \cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}\) | \(163\) |
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 [A]
time = 4.24, size = 157, normalized size = 0.66 \begin {gather*} \frac {2 \, {\left (8 \, {\left (803 \, A + 710 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} + 4 \, {\left (803 \, A + 710 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 3 \, {\left (803 \, A + 710 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 5 \, {\left (286 \, A + 355 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 35 \, {\left (11 \, A + 32 \, B\right )} a^{2} \cos \left (d x + c\right ) + 315 \, B a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}} \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 [A]
time = 1.37, size = 306, normalized size = 1.29 \begin {gather*} \frac {8 \, {\left ({\left ({\left ({\left (4 \, {\left (2 \, \sqrt {2} {\left (143 \, A a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 125 \, B a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 11 \, \sqrt {2} {\left (143 \, A a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 125 \, B a^{8} \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} + 99 \, \sqrt {2} {\left (143 \, A a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 125 \, B a^{8} \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} - 231 \, \sqrt {2} {\left (69 \, A a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 65 \, B a^{8} \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} + 1155 \, \sqrt {2} {\left (9 \, A a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 7 \, B a^{8} \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} - 3465 \, \sqrt {2} {\left (A a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + B a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{3465 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{5} \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 = 13.51, size = 856, normalized size = 3.61 \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\,a^2\,8{}\mathrm {i}}{3\,d}-\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (803\,A+710\,B\right )\,8{}\mathrm {i}}{3465\,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^2\,\left (5\,A+2\,B\right )\,8{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (5\,A+16\,B\right )\,8{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (11\,A+50\,B\right )\,32{}\mathrm {i}}{693\,d}\right )+\frac {A\,a^2\,24{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (9\,A+10\,B\right )\,8{}\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\,a^2\,8{}\mathrm {i}}{11\,d}+\frac {a^2\,\left (3\,A+4\,B\right )\,40{}\mathrm {i}}{11\,d}-\frac {a^2\,\left (5\,A+2\,B\right )\,8{}\mathrm {i}}{11\,d}-\frac {a^2\,\left (11\,A+10\,B\right )\,8{}\mathrm {i}}{11\,d}\right )+\frac {A\,a^2\,8{}\mathrm {i}}{11\,d}+\frac {a^2\,\left (3\,A+4\,B\right )\,40{}\mathrm {i}}{11\,d}-\frac {a^2\,\left (5\,A+2\,B\right )\,8{}\mathrm {i}}{11\,d}-\frac {a^2\,\left (11\,A+10\,B\right )\,8{}\mathrm {i}}{11\,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 )}^5}-\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^2\,\left (A-8\,B\right )\,8{}\mathrm {i}}{9\,d}-\frac {B\,a^2\,64{}\mathrm {i}}{99\,d}+\frac {a^2\,\left (5\,A+2\,B\right )\,8{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (5\,A+9\,B\right )\,16{}\mathrm {i}}{9\,d}\right )+\frac {A\,a^2\,8{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (A+2\,B\right )\,40{}\mathrm {i}}{9\,d}+\frac {B\,a^2\,64{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (A+B\right )\,80{}\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 ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {a^2\,\left (5\,A+2\,B\right )\,8{}\mathrm {i}}{5\,d}+\frac {a^2\,\left (44\,A-31\,B\right )\,16{}\mathrm {i}}{1155\,d}\right )-\frac {A\,a^2\,8{}\mathrm {i}}{5\,d}+\frac {a^2\,\left (4\,A+5\,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^2\,{\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 (803\,A+710\,B\right )\,16{}\mathrm {i}}{3465\,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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