Optimal. Leaf size=175 \[ \frac {64 a^3 (15 A+13 B) \tan (c+d x)}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 (15 A+13 B) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {2 (9 A-2 B) \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac {2 a (15 A+13 B) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac {2 B \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d} \]
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Rubi [A] time = 0.35, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4010, 4001, 3793, 3792} \[ \frac {16 a^2 (15 A+13 B) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {64 a^3 (15 A+13 B) \tan (c+d x)}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {2 (9 A-2 B) \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac {2 a (15 A+13 B) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac {2 B \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d} \]
Antiderivative was successfully verified.
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Rule 3792
Rule 3793
Rule 4001
Rule 4010
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx &=\frac {2 B (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {2 \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac {7 a B}{2}+\frac {1}{2} a (9 A-2 B) \sec (c+d x)\right ) \, dx}{9 a}\\ &=\frac {2 (9 A-2 B) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 B (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{21} (15 A+13 B) \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \, dx\\ &=\frac {2 a (15 A+13 B) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac {2 (9 A-2 B) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 B (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{105} (8 a (15 A+13 B)) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac {16 a^2 (15 A+13 B) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a (15 A+13 B) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac {2 (9 A-2 B) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 B (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{315} \left (32 a^2 (15 A+13 B)\right ) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {64 a^3 (15 A+13 B) \tan (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 (15 A+13 B) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a (15 A+13 B) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}+\frac {2 (9 A-2 B) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 B (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}\\ \end {align*}
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Mathematica [A] time = 0.64, size = 96, normalized size = 0.55 \[ \frac {2 a^3 \tan (c+d x) \left (5 (9 A+26 B) \sec ^3(c+d x)+3 (60 A+73 B) \sec ^2(c+d x)+(345 A+292 B) \sec (c+d x)+690 A+35 B \sec ^4(c+d x)+584 B\right )}{315 d \sqrt {a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 136, normalized size = 0.78 \[ \frac {2 \, {\left (2 \, {\left (345 \, A + 292 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + {\left (345 \, A + 292 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (60 \, A + 73 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 5 \, {\left (9 \, A + 26 \, B\right )} a^{2} \cos \left (d x + c\right ) + 35 \, B a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.06, size = 261, normalized size = 1.49 \[ \frac {8 \, {\left ({\left ({\left (4 \, {\left (2 \, \sqrt {2} {\left (15 \, A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13 \, B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \sqrt {2} {\left (15 \, A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13 \, B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 63 \, \sqrt {2} {\left (15 \, A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13 \, B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 210 \, \sqrt {2} {\left (4 \, A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 3 \, B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 315 \, \sqrt {2} {\left (A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{315 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.52, size = 141, normalized size = 0.81 \[ -\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (690 A \left (\cos ^{4}\left (d x +c \right )\right )+584 B \left (\cos ^{4}\left (d x +c \right )\right )+345 A \left (\cos ^{3}\left (d x +c \right )\right )+292 B \left (\cos ^{3}\left (d x +c \right )\right )+180 A \left (\cos ^{2}\left (d x +c \right )\right )+219 B \left (\cos ^{2}\left (d x +c \right )\right )+45 A \cos \left (d x +c \right )+130 B \cos \left (d x +c \right )+35 B \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{315 d \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [B] time = 10.81, size = 723, normalized size = 4.13 \[ \frac {\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {A\,a^2\,4{}\mathrm {i}}{3\,d}-\frac {a^2\,\left (60\,A+73\,B\right )\,8{}\mathrm {i}}{315\,d}\right )+\frac {a^2\,\left (5\,A+2\,B\right )\,4{}\mathrm {i}}{3\,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\,a^2\,4{}\mathrm {i}}{5\,d}+\frac {a^2\,\left (3\,A+4\,B\right )\,16{}\mathrm {i}}{105\,d}+\frac {a^2\,\left (9\,A+10\,B\right )\,4{}\mathrm {i}}{5\,d}\right )-\frac {a^2\,\left (5\,A+2\,B\right )\,4{}\mathrm {i}}{5\,d}+\frac {a^2\,\left (5\,A+16\,B\right )\,4{}\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 {\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\,4{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (A+2\,B\right )\,20{}\mathrm {i}}{7\,d}+\frac {B\,a^2\,32{}\mathrm {i}}{63\,d}-\frac {a^2\,\left (A+B\right )\,40{}\mathrm {i}}{7\,d}\right )+\frac {a^2\,\left (A-8\,B\right )\,4{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (5\,A+2\,B\right )\,4{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (5\,A+9\,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\,4{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (3\,A+4\,B\right )\,20{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (5\,A+2\,B\right )\,4{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (11\,A+10\,B\right )\,4{}\mathrm {i}}{9\,d}\right )-\frac {A\,a^2\,4{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (3\,A+4\,B\right )\,20{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (5\,A+2\,B\right )\,4{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (11\,A+10\,B\right )\,4{}\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 {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 (345\,A+292\,B\right )\,4{}\mathrm {i}}{315\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )} \]
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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