Optimal. Leaf size=86 \[ \frac {a (3 A+2 C) \tan (c+d x)}{3 d}+\frac {a (2 A+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a C \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac {a C \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4077, 4047, 3767, 8, 4046, 3770} \[ \frac {a (3 A+2 C) \tan (c+d x)}{3 d}+\frac {a (2 A+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a C \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac {a C \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 4046
Rule 4047
Rule 4077
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {a C \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{3} \int \sec (c+d x) \left (3 a A+a (3 A+2 C) \sec (c+d x)+3 a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a C \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{3} \int \sec (c+d x) \left (3 a A+3 a C \sec ^2(c+d x)\right ) \, dx+\frac {1}{3} (a (3 A+2 C)) \int \sec ^2(c+d x) \, dx\\ &=\frac {a C \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a C \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{2} (a (2 A+C)) \int \sec (c+d x) \, dx-\frac {(a (3 A+2 C)) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac {a (2 A+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a (3 A+2 C) \tan (c+d x)}{3 d}+\frac {a C \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a C \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 56, normalized size = 0.65 \[ \frac {a \left (3 (2 A+C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (6 (A+C)+2 C \tan ^2(c+d x)+3 C \sec (c+d x)\right )\right )}{6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 107, normalized size = 1.24 \[ \frac {3 \, {\left (2 \, A + C\right )} a \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (2 \, A + C\right )} a \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (3 \, A + 2 \, C\right )} a \cos \left (d x + c\right )^{2} + 3 \, C a \cos \left (d x + c\right ) + 2 \, C a\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 156, normalized size = 1.81 \[ \frac {3 \, {\left (2 \, A a + C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (2 \, A a + C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.35, size = 108, normalized size = 1.26 \[ \frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a C \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {a A \tan \left (d x +c \right )}{d}+\frac {2 a C \tan \left (d x +c \right )}{3 d}+\frac {a C \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 100, normalized size = 1.16 \[ \frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a - 3 \, C a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 12 \, A a \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.42, size = 129, normalized size = 1.50 \[ \frac {a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,A+C\right )}{d}-\frac {\left (2\,A\,a+C\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-4\,A\,a-\frac {4\,C\,a}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a+3\,C\,a\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ a \left (\int A \sec {\left (c + d x \right )}\, dx + \int A \sec ^{2}{\left (c + d x \right )}\, dx + \int C \sec ^{3}{\left (c + d x \right )}\, dx + \int C \sec ^{4}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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