Optimal. Leaf size=108 \[ -\frac {(2 B-3 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac {2 (B-C) \tan (c+d x)}{a d}-\frac {(2 B-3 C) \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {(B-C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))} \]
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Rubi [A]
time = 0.17, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 (B-C) \tan (c+d x)}{a d}-\frac {(2 B-3 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac {(B-C) \tan (c+d x) \sec ^2(c+d x)}{d (a \sec (c+d x)+a)}-\frac {(2 B-3 C) \tan (c+d x) \sec (c+d x)}{2 a d} \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 ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx &=\int \frac {\sec ^3(c+d x) (B+C \sec (c+d x))}{a+a \sec (c+d x)} \, dx\\ &=\frac {(B-C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}+\frac {\int \sec ^2(c+d x) (2 a (B-C)-a (2 B-3 C) \sec (c+d x)) \, dx}{a^2}\\ &=\frac {(B-C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}-\frac {(2 B-3 C) \int \sec ^3(c+d x) \, dx}{a}+\frac {(2 (B-C)) \int \sec ^2(c+d x) \, dx}{a}\\ &=-\frac {(2 B-3 C) \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {(B-C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}-\frac {(2 B-3 C) \int \sec (c+d x) \, dx}{2 a}-\frac {(2 (B-C)) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{a d}\\ &=-\frac {(2 B-3 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac {2 (B-C) \tan (c+d x)}{a d}-\frac {(2 B-3 C) \sec (c+d x) \tan (c+d x)}{2 a d}+\frac {(B-C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(383\) vs. \(2(108)=216\).
time = 0.69, size = 383, normalized size = 3.55 \begin {gather*} \frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (2 B \cos \left (\frac {5}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-3 C \cos \left (\frac {5}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 (2 B-3 C) \cos \left (\frac {1}{2} (c+d x)\right ) \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 )+(2 B-3 C) \cos \left (\frac {3}{2} (c+d x)\right ) \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 )-2 B \cos \left (\frac {5}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 C \cos \left (\frac {5}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 B \sin \left (\frac {1}{2} (c+d x)\right )-2 C \sin \left (\frac {1}{2} (c+d x)\right )+2 C \sin \left (\frac {3}{2} (c+d x)\right )+4 B \sin \left (\frac {5}{2} (c+d x)\right )-4 C \sin \left (\frac {5}{2} (c+d x)\right )\right )}{4 a d (1+\cos (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 142, normalized size = 1.31
method | result | size |
derivativedivides | \(\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {C}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\left (\frac {3 C}{2}-B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {-\frac {3 C}{2}+B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {C}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-\frac {3 C}{2}+B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-\frac {3 C}{2}+B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d a}\) | \(142\) |
default | \(\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {C}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\left (\frac {3 C}{2}-B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {-\frac {3 C}{2}+B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {C}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-\frac {3 C}{2}+B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-\frac {3 C}{2}+B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d a}\) | \(142\) |
norman | \(\frac {\frac {\left (B -C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {7 \left (B -C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {\left (3 B -2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {\left (5 B -6 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {\left (2 B -3 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a d}-\frac {\left (2 B -3 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a d}\) | \(170\) |
risch | \(\frac {i \left (2 B \,{\mathrm e}^{4 i \left (d x +c \right )}-3 C \,{\mathrm e}^{4 i \left (d x +c \right )}+2 B \,{\mathrm e}^{3 i \left (d x +c \right )}-3 C \,{\mathrm e}^{3 i \left (d x +c \right )}+6 B \,{\mathrm e}^{2 i \left (d x +c \right )}-5 C \,{\mathrm e}^{2 i \left (d x +c \right )}+2 B \,{\mathrm e}^{i \left (d x +c \right )}-C \,{\mathrm e}^{i \left (d x +c \right )}+4 B -4 C \right )}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{a d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 a d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{a d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 a d}\) | \(227\) |
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. 282 vs.
\(2 (104) = 208\).
time = 0.28, size = 282, normalized size = 2.61 \begin {gather*} -\frac {C {\left (\frac {2 \, {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a - \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac {2 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + 2 \, B {\left (\frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{{\left (a - \frac {a \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 )}} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.48, size = 156, normalized size = 1.44 \begin {gather*} -\frac {{\left ({\left (2 \, B - 3 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, B - 3 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (2 \, B - 3 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, B - 3 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, {\left (B - C\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, B - C\right )} \cos \left (d x + c\right ) + C\right )} \sin \left (d x + c\right )}{4 \, {\left (a d \cos \left (d x + c\right )^{3} + a 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 ^{3}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{4}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 156, normalized size = 1.44 \begin {gather*} -\frac {\frac {{\left (2 \, B - 3 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac {{\left (2 \, B - 3 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac {2 \, {\left (B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} + \frac {2 \, {\left (2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, 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 ) + 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 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.04, size = 119, normalized size = 1.10 \begin {gather*} \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (B-C\right )}{a\,d}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (B-\frac {3\,C}{2}\right )}{a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,B-3\,C\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,B-C\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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