Optimal. Leaf size=243 \[ \frac {2 a^2 (429 A+374 B+336 C) \tan (c+d x)}{495 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a (429 A+374 B+336 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {2 a (11 B+3 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d} \]
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Rubi [A]
time = 0.48, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {4173, 4103,
4101, 3885, 4086, 3877} \begin {gather*} \frac {2 a^2 (99 A+110 B+84 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (429 A+374 B+336 C) \tan (c+d x)}{495 d \sqrt {a \sec (c+d x)+a}}+\frac {2 (429 A+374 B+336 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 d}-\frac {4 a (429 A+374 B+336 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3465 d}+\frac {2 a (11 B+3 C) \tan (c+d x) \sec ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{99 d}+\frac {2 C \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
Rule 4173
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {2 \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{2} a (11 A+6 C)+\frac {1}{2} a (11 B+3 C) \sec (c+d x)\right ) \, dx}{11 a}\\ &=\frac {2 a (11 B+3 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {4 \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {3}{4} a^2 (33 A+22 B+24 C)+\frac {1}{4} a^2 (99 A+110 B+84 C) \sec (c+d x)\right ) \, dx}{99 a}\\ &=\frac {2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+3 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {1}{231} (a (429 A+374 B+336 C)) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+3 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {(2 (429 A+374 B+336 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}{1155}\\ &=\frac {2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a (429 A+374 B+336 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {2 a (11 B+3 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac {2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac {1}{495} (a (429 A+374 B+336 C)) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a^2 (429 A+374 B+336 C) \tan (c+d x)}{495 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}-\frac {4 a (429 A+374 B+336 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {2 a (11 B+3 C) \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac {2 C \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 = 2.18, size = 185, normalized size = 0.76 \begin {gather*} \frac {a (3564 A+4114 B+4956 C+(12441 A+12386 B+12684 C) \cos (c+d x)+(4422 A+4862 B+4368 C) \cos (2 (c+d x))+5577 A \cos (3 (c+d x))+4862 B \cos (3 (c+d x))+4368 C \cos (3 (c+d x))+858 A \cos (4 (c+d x))+748 B \cos (4 (c+d x))+672 C \cos (4 (c+d x))+858 A \cos (5 (c+d x))+748 B \cos (5 (c+d x))+672 C \cos (5 (c+d x))) \sec ^5(c+d x) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{6930 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 15.49, size = 205, normalized size = 0.84
method | result | size |
default | \(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (3432 A \left (\cos ^{5}\left (d x +c \right )\right )+2992 B \left (\cos ^{5}\left (d x +c \right )\right )+2688 C \left (\cos ^{5}\left (d x +c \right )\right )+1716 A \left (\cos ^{4}\left (d x +c \right )\right )+1496 B \left (\cos ^{4}\left (d x +c \right )\right )+1344 C \left (\cos ^{4}\left (d x +c \right )\right )+1287 A \left (\cos ^{3}\left (d x +c \right )\right )+1122 B \left (\cos ^{3}\left (d x +c \right )\right )+1008 C \left (\cos ^{3}\left (d x +c \right )\right )+495 A \left (\cos ^{2}\left (d x +c \right )\right )+935 B \left (\cos ^{2}\left (d x +c \right )\right )+840 C \left (\cos ^{2}\left (d x +c \right )\right )+385 B \cos \left (d x +c \right )+735 C \cos \left (d x +c \right )+315 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a}{3465 d \cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}\) | \(205\) |
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 = 2.32, size = 157, normalized size = 0.65 \begin {gather*} \frac {2 \, {\left (8 \, {\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right )^{5} + 4 \, {\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right )^{4} + 3 \, {\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right )^{3} + 5 \, {\left (99 \, A + 187 \, B + 168 \, C\right )} a \cos \left (d x + c\right )^{2} + 35 \, {\left (11 \, B + 21 \, C\right )} a \cos \left (d x + c\right ) + 315 \, C a\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]
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 )} + 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 = 1.40, size = 410, normalized size = 1.69 \begin {gather*} -\frac {4 \, {\left (3465 \, \sqrt {2} A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 3465 \, \sqrt {2} B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 3465 \, \sqrt {2} C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (11550 \, \sqrt {2} A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 9240 \, \sqrt {2} B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 6930 \, \sqrt {2} C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (17094 \, \sqrt {2} A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 14784 \, \sqrt {2} B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 15246 \, \sqrt {2} C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (14652 \, \sqrt {2} A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13662 \, \sqrt {2} B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 11088 \, \sqrt {2} C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (6897 \, \sqrt {2} A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 5687 \, \sqrt {2} B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 5313 \, \sqrt {2} C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 2 \, {\left (627 \, \sqrt {2} A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 517 \, \sqrt {2} B a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 483 \, \sqrt {2} C 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}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\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 = 13.53, size = 852, normalized size = 3.51 \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}}{11\,d}-\frac {A\,a\,8{}\mathrm {i}}{11\,d}+\frac {a\,\left (5\,A+6\,B+4\,C\right )\,8{}\mathrm {i}}{11\,d}-\frac {a\,\left (7\,A+8\,B+12\,C\right )\,8{}\mathrm {i}}{11\,d}\right )+\frac {a\,\left (3\,A+2\,B\right )\,8{}\mathrm {i}}{11\,d}-\frac {A\,a\,8{}\mathrm {i}}{11\,d}+\frac {a\,\left (5\,A+6\,B+4\,C\right )\,8{}\mathrm {i}}{11\,d}-\frac {a\,\left (7\,A+8\,B+12\,C\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\,\left (3\,A+2\,B\right )\,8{}\mathrm {i}}{7\,d}+\frac {a\,\left (11\,B+39\,C\right )\,32{}\mathrm {i}}{693\,d}+\frac {a\,\left (A+4\,B+12\,C\right )\,8{}\mathrm {i}}{7\,d}\right )+\frac {A\,a\,8{}\mathrm {i}}{7\,d}-\frac {a\,\left (3\,A+6\,B+4\,C\right )\,8{}\mathrm {i}}{7\,d}-\frac {a\,\left (B-C\right )\,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 (3\,A+2\,B\right )\,8{}\mathrm {i}}{5\,d}+\frac {a\,\left (33\,A+11\,B-42\,C\right )\,16{}\mathrm {i}}{1155\,d}\right )+\frac {a\,\left (A+3\,B+2\,C\right )\,16{}\mathrm {i}}{5\,d}-\frac {A\,a\,8{}\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\,\left (3\,A+2\,B\right )\,8{}\mathrm {i}}{9\,d}+\frac {a\,\left (A-8\,C\right )\,8{}\mathrm {i}}{9\,d}-\frac {C\,a\,64{}\mathrm {i}}{99\,d}-\frac {a\,\left (2\,A+3\,B+6\,C\right )\,16{}\mathrm {i}}{9\,d}\right )+\frac {A\,a\,8{}\mathrm {i}}{9\,d}+\frac {C\,a\,64{}\mathrm {i}}{9\,d}-\frac {a\,\left (2\,A+3\,B+2\,C\right )\,16{}\mathrm {i}}{9\,d}+\frac {a\,\left (3\,A+2\,B+8\,C\right )\,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 {\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\,8{}\mathrm {i}}{3\,d}-\frac {a\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (429\,A+374\,B+336\,C\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 {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 (429\,A+374\,B+336\,C\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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