Optimal. Leaf size=230 \[ \frac {32 a (11 B+10 C) \tan (c+d x)}{495 d \sqrt {a+a \sec (c+d x)}}+\frac {16 a (11 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+10 C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}-\frac {64 (11 B+10 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {32 (11 B+10 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d} \]
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Rubi [A]
time = 0.33, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4157, 4101,
3888, 3885, 4086, 3877} \begin {gather*} \frac {2 a (11 B+10 C) \tan (c+d x) \sec ^4(c+d x)}{99 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a (11 B+10 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt {a \sec (c+d x)+a}}+\frac {32 (11 B+10 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 a d}-\frac {64 (11 B+10 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3465 d}+\frac {32 a (11 B+10 C) \tan (c+d x)}{495 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a C \tan (c+d x) \sec ^5(c+d x)}{11 d \sqrt {a \sec (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3877
Rule 3885
Rule 3888
Rule 4086
Rule 4101
Rule 4157
Rubi steps
\begin {align*} \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sec ^5(c+d x) \sqrt {a+a \sec (c+d x)} (B+C \sec (c+d x)) \, dx\\ &=\frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}+\frac {1}{11} (11 B+10 C) \int \sec ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {2 a (11 B+10 C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}+\frac {1}{99} (8 (11 B+10 C)) \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {16 a (11 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+10 C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}+\frac {1}{231} (16 (11 B+10 C)) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {16 a (11 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+10 C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}+\frac {32 (11 B+10 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d}+\frac {(32 (11 B+10 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 a}\\ &=\frac {16 a (11 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+10 C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}-\frac {64 (11 B+10 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {32 (11 B+10 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d}+\frac {1}{495} (16 (11 B+10 C)) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {32 a (11 B+10 C) \tan (c+d x)}{495 d \sqrt {a+a \sec (c+d x)}}+\frac {16 a (11 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (11 B+10 C) \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a C \sec ^5(c+d x) \tan (c+d x)}{11 d \sqrt {a+a \sec (c+d x)}}-\frac {64 (11 B+10 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac {32 (11 B+10 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d}\\ \end {align*}
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Mathematica [A]
time = 6.08, size = 283, normalized size = 1.23 \begin {gather*} \frac {2 B \left (315 \sqrt {1-\sec (c+d x)}-420 (1-\sec (c+d x))^{3/2}+378 (1-\sec (c+d x))^{5/2}-180 (1-\sec (c+d x))^{7/2}+35 (1-\sec (c+d x))^{9/2}\right ) \sqrt {a (1+\sec (c+d x))} \tan (c+d x)}{315 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))}+\frac {2 C \left (693 \sqrt {1-\sec (c+d x)}-1155 (1-\sec (c+d x))^{3/2}+1386 (1-\sec (c+d x))^{5/2}-990 (1-\sec (c+d x))^{7/2}+385 (1-\sec (c+d x))^{9/2}-63 (1-\sec (c+d x))^{11/2}\right ) \sqrt {a (1+\sec (c+d x))} \tan (c+d x)}{693 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 13.40, size = 160, normalized size = 0.70
method | result | size |
default | \(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (1408 B \left (\cos ^{5}\left (d x +c \right )\right )+1280 C \left (\cos ^{5}\left (d x +c \right )\right )+704 B \left (\cos ^{4}\left (d x +c \right )\right )+640 C \left (\cos ^{4}\left (d x +c \right )\right )+528 B \left (\cos ^{3}\left (d x +c \right )\right )+480 C \left (\cos ^{3}\left (d x +c \right )\right )+440 B \left (\cos ^{2}\left (d x +c \right )\right )+400 C \left (\cos ^{2}\left (d x +c \right )\right )+385 B \cos \left (d x +c \right )+350 C \cos \left (d x +c \right )+315 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{3465 d \cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}\) | \(160\) |
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 = 2.52, size = 139, normalized size = 0.60 \begin {gather*} \frac {2 \, {\left (128 \, {\left (11 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{5} + 64 \, {\left (11 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{4} + 48 \, {\left (11 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{3} + 40 \, {\left (11 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{2} + 35 \, {\left (11 \, B + 10 \, C\right )} \cos \left (d x + c\right ) + 315 \, C\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 \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \left (B + C \sec {\left (c + d x \right )}\right ) \sec ^{5}{\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 = 1.16, size = 314, normalized size = 1.37 \begin {gather*} -\frac {2 \, {\left (3465 \, \sqrt {2} B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 3465 \, \sqrt {2} C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (8085 \, \sqrt {2} B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 5775 \, \sqrt {2} C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (14322 \, \sqrt {2} B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 16170 \, \sqrt {2} C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (13266 \, \sqrt {2} B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 8910 \, \sqrt {2} C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (4741 \, \sqrt {2} B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 5885 \, \sqrt {2} C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - {\left (1177 \, \sqrt {2} B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 755 \, \sqrt {2} C 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}\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 = 12.33, size = 626, normalized size = 2.72 \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 {B\,32{}\mathrm {i}}{9\,d}+\frac {C\,128{}\mathrm {i}}{9\,d}+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {C\,256{}\mathrm {i}}{33\,d}+\frac {\left (352\,B+704\,C\right )\,1{}\mathrm {i}}{99\,d}\right )\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 {B\,32{}\mathrm {i}}{11\,d}+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {B\,32{}\mathrm {i}}{11\,d}-\frac {\left (32\,B+64\,C\right )\,1{}\mathrm {i}}{11\,d}\right )-\frac {\left (32\,B+64\,C\right )\,1{}\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 {\left (\frac {B\,32{}\mathrm {i}}{5\,d}-\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (352\,B+320\,C\right )\,1{}\mathrm {i}}{1155\,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 )}^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 {\left (352\,B+896\,C\right )\,1{}\mathrm {i}}{693\,d}+\frac {\left (3168\,B+6336\,C\right )\,1{}\mathrm {i}}{693\,d}\right )+\frac {B\,32{}\mathrm {i}}{7\,d}+\frac {\left (3168\,B-6336\,C\right )\,1{}\mathrm {i}}{693\,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 {{\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 (2816\,B+2560\,C\right )\,1{}\mathrm {i}}{3465\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )}-\frac {{\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 (1408\,B+1280\,C\right )\,1{}\mathrm {i}}{3465\,d\,\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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