Optimal. Leaf size=156 \[ -\frac {(4 B-7 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac {2 (5 B-8 C) \tan (c+d x)}{3 a^2 d}-\frac {(4 B-7 C) \sec (c+d x) \tan (c+d x)}{2 a^2 d}+\frac {(5 B-8 C) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d (1+\sec (c+d x))}+\frac {(B-C) \sec ^3(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2} \]
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Rubi [A]
time = 0.27, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {4157, 4104,
3872, 3852, 8, 3853, 3855} \begin {gather*} \frac {2 (5 B-8 C) \tan (c+d x)}{3 a^2 d}-\frac {(4 B-7 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac {(5 B-8 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac {(4 B-7 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac {(B-C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rule 4104
Rule 4157
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\sec ^4(c+d x) (B+C \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx\\ &=\frac {(B-C) \sec ^3(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \frac {\sec ^3(c+d x) (3 a (B-C)-a (2 B-5 C) \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2}\\ &=\frac {(5 B-8 C) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d (1+\sec (c+d x))}+\frac {(B-C) \sec ^3(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \sec ^2(c+d x) \left (2 a^2 (5 B-8 C)-3 a^2 (4 B-7 C) \sec (c+d x)\right ) \, dx}{3 a^4}\\ &=\frac {(5 B-8 C) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d (1+\sec (c+d x))}+\frac {(B-C) \sec ^3(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {(2 (5 B-8 C)) \int \sec ^2(c+d x) \, dx}{3 a^2}-\frac {(4 B-7 C) \int \sec ^3(c+d x) \, dx}{a^2}\\ &=-\frac {(4 B-7 C) \sec (c+d x) \tan (c+d x)}{2 a^2 d}+\frac {(5 B-8 C) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d (1+\sec (c+d x))}+\frac {(B-C) \sec ^3(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {(4 B-7 C) \int \sec (c+d x) \, dx}{2 a^2}-\frac {(2 (5 B-8 C)) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 a^2 d}\\ &=-\frac {(4 B-7 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac {2 (5 B-8 C) \tan (c+d x)}{3 a^2 d}-\frac {(4 B-7 C) \sec (c+d x) \tan (c+d x)}{2 a^2 d}+\frac {(5 B-8 C) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d (1+\sec (c+d x))}+\frac {(B-C) \sec ^3(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(379\) vs. \(2(156)=312\).
time = 1.59, size = 379, normalized size = 2.43 \begin {gather*} \frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (3 (4 B-7 C) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+8 (B+5 C) \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-64 (B-C) \csc ^5(c+d x) \sin ^8\left (\frac {1}{2} (c+d x)\right )-128 C \csc ^7(c+d x) \sin ^{12}\left (\frac {1}{2} (c+d x)\right )+(26 B-44 C) \tan \left (\frac {1}{2} (c+d x)\right )-6 (4 B-7 C) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )-8 (5 B-8 C) \tan ^3\left (\frac {1}{2} (c+d x)\right )+3 (4 B-7 C) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \tan ^4\left (\frac {1}{2} (c+d x)\right )+\left (14 B-20 C+B \sec ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan ^5\left (\frac {1}{2} (c+d x)\right )\right )}{6 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 177, normalized size = 1.13
method | result | size |
derivativedivides | \(\frac {\frac {B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+5 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-7 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2 B -5 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (4 B -7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 B -5 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (7 C -4 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{2 d \,a^{2}}\) | \(177\) |
default | \(\frac {\frac {B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+5 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-7 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2 B -5 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (4 B -7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 B -5 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (7 C -4 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{2 d \,a^{2}}\) | \(177\) |
norman | \(\frac {\frac {\left (B -C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}+\frac {\left (9 B -13 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}-\frac {\left (11 B -18 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {\left (11 B -17 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}+\frac {\left (61 B -100 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {\left (95 B -149 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4} a}+\frac {\left (4 B -7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{2} d}-\frac {\left (4 B -7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{2} d}\) | \(228\) |
risch | \(\frac {i \left (12 B \,{\mathrm e}^{6 i \left (d x +c \right )}-21 C \,{\mathrm e}^{6 i \left (d x +c \right )}+36 B \,{\mathrm e}^{5 i \left (d x +c \right )}-63 C \,{\mathrm e}^{5 i \left (d x +c \right )}+56 B \,{\mathrm e}^{4 i \left (d x +c \right )}-98 C \,{\mathrm e}^{4 i \left (d x +c \right )}+84 B \,{\mathrm e}^{3 i \left (d x +c \right )}-126 C \,{\mathrm e}^{3 i \left (d x +c \right )}+64 B \,{\mathrm e}^{2 i \left (d x +c \right )}-97 C \,{\mathrm e}^{2 i \left (d x +c \right )}+48 B \,{\mathrm e}^{i \left (d x +c \right )}-75 C \,{\mathrm e}^{i \left (d x +c \right )}+20 B -32 C \right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{a^{2} d}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 a^{2} d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{a^{2} d}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 a^{2} d}\) | \(276\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 336 vs.
\(2 (146) = 292\).
time = 0.28, size = 336, normalized size = 2.15 \begin {gather*} -\frac {C {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {21 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {21 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}\right )} - B {\left (\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}} + \frac {12 \, \sin \left (d x + c\right )}{{\left (a^{2} - \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.88, size = 228, normalized size = 1.46 \begin {gather*} -\frac {3 \, {\left ({\left (4 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (4 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (4 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (4 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (4 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (4 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, {\left (5 \, B - 8 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (28 \, B - 43 \, C\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left (B - C\right )} \cos \left (d x + c\right ) + 3 \, C\right )} \sin \left (d x + c\right )}{12 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} + a^{2} d \cos \left (d x + c\right )^{2}\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*} \frac {\int \frac {B \sec ^{4}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{5}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 198, normalized size = 1.27 \begin {gather*} -\frac {\frac {3 \, {\left (4 \, B - 7 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} - \frac {3 \, {\left (4 \, B - 7 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} + \frac {6 \, {\left (2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{2}} - \frac {B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.90, size = 166, normalized size = 1.06 \begin {gather*} \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (B-C\right )}{2\,a^2}+\frac {2\,B-4\,C}{2\,a^2}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,B-5\,C\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,B-3\,C\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (B-C\right )}{6\,a^2\,d}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,B-7\,C\right )}{a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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