Optimal. Leaf size=132 \[ \frac {a^2 (12 A+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a^2 (12 A+7 C) \tan (c+d x)}{6 d}+\frac {a^2 (12 A+7 C) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {C (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d} \]
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Rubi [A]
time = 0.15, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4168, 4086,
3873, 3852, 8, 4131, 3855} \begin {gather*} \frac {a^2 (12 A+7 C) \tan (c+d x)}{6 d}+\frac {a^2 (12 A+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a^2 (12 A+7 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 a d}-\frac {C \tan (c+d x) (a \sec (c+d x)+a)^2}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3855
Rule 3873
Rule 4086
Rule 4131
Rule 4168
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^2 (a (4 A+3 C)-a C \sec (c+d x)) \, dx}{4 a}\\ &=-\frac {C (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}+\frac {1}{12} (12 A+7 C) \int \sec (c+d x) (a+a \sec (c+d x))^2 \, dx\\ &=-\frac {C (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}+\frac {1}{12} (12 A+7 C) \int \sec (c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx+\frac {1}{6} \left (a^2 (12 A+7 C)\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac {a^2 (12 A+7 C) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {C (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}+\frac {1}{8} \left (a^2 (12 A+7 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (a^2 (12 A+7 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d}\\ &=\frac {a^2 (12 A+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a^2 (12 A+7 C) \tan (c+d x)}{6 d}+\frac {a^2 (12 A+7 C) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {C (a+a \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 a d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(291\) vs. \(2(132)=264\).
time = 1.51, size = 291, normalized size = 2.20 \begin {gather*} -\frac {a^2 (1+\cos (c+d x))^2 \left (C+A \cos ^2(c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^4(c+d x) \left (24 (12 A+7 C) \cos ^4(c+d x) \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 )-\sec (c) (-48 (3 A+2 C) \sin (c)+3 (4 A+15 C) \sin (d x)+12 A \sin (2 c+d x)+45 C \sin (2 c+d x)+144 A \sin (c+2 d x)+128 C \sin (c+2 d x)-48 A \sin (3 c+2 d x)+12 A \sin (2 c+3 d x)+21 C \sin (2 c+3 d x)+12 A \sin (4 c+3 d x)+21 C \sin (4 c+3 d x)+48 A \sin (3 c+4 d x)+32 C \sin (3 c+4 d x))\right )}{384 d (A+2 C+A \cos (2 (c+d x)))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.63, size = 182, normalized size = 1.38
method | result | size |
norman | \(\frac {\frac {5 a^{2} \left (4 A +5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {11 a^{2} \left (12 A +7 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {a^{2} \left (12 A +7 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {a^{2} \left (156 A +83 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {a^{2} \left (12 A +7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {a^{2} \left (12 A +7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(175\) |
derivativedivides | \(\frac {a^{2} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{2} C \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+2 a^{2} A \tan \left (d x +c \right )-2 a^{2} C \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+a^{2} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{2} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(182\) |
default | \(\frac {a^{2} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{2} C \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+2 a^{2} A \tan \left (d x +c \right )-2 a^{2} C \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+a^{2} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{2} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(182\) |
risch | \(-\frac {i a^{2} \left (12 A \,{\mathrm e}^{7 i \left (d x +c \right )}+21 C \,{\mathrm e}^{7 i \left (d x +c \right )}-48 A \,{\mathrm e}^{6 i \left (d x +c \right )}+12 A \,{\mathrm e}^{5 i \left (d x +c \right )}+45 C \,{\mathrm e}^{5 i \left (d x +c \right )}-144 A \,{\mathrm e}^{4 i \left (d x +c \right )}-96 C \,{\mathrm e}^{4 i \left (d x +c \right )}-12 A \,{\mathrm e}^{3 i \left (d x +c \right )}-45 C \,{\mathrm e}^{3 i \left (d x +c \right )}-144 A \,{\mathrm e}^{2 i \left (d x +c \right )}-128 C \,{\mathrm e}^{2 i \left (d x +c \right )}-12 \,{\mathrm e}^{i \left (d x +c \right )} A -21 C \,{\mathrm e}^{i \left (d x +c \right )}-48 A -32 C \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 d}-\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 d}+\frac {7 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}\) | \(275\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 227, normalized size = 1.72 \begin {gather*} \frac {32 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} - 3 \, C a^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, A a^{2} {\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 \, C a^{2} {\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 )} + 48 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 96 \, A a^{2} \tan \left (d x + c\right )}{48 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.94, size = 141, normalized size = 1.07 \begin {gather*} \frac {3 \, {\left (12 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (12 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (3 \, A + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 16 \, C a^{2} \cos \left (d x + c\right ) + 6 \, C a^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \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*} a^{2} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 2 A \sec ^{2}{\left (c + d x \right )}\, dx + \int A \sec ^{3}{\left (c + d x \right )}\, dx + \int C \sec ^{3}{\left (c + d x \right )}\, dx + \int 2 C \sec ^{4}{\left (c + d x \right )}\, dx + \int C \sec ^{5}{\left (c + d x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 212, normalized size = 1.61 \begin {gather*} \frac {3 \, {\left (12 \, A a^{2} + 7 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (12 \, A a^{2} + 7 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (36 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 21 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 132 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 77 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 156 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 83 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 60 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 75 \, C a^{2} \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 )}^{4}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.00, size = 185, normalized size = 1.40 \begin {gather*} \frac {\left (-3\,A\,a^2-\frac {7\,C\,a^2}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (11\,A\,a^2+\frac {77\,C\,a^2}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-13\,A\,a^2-\frac {83\,C\,a^2}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (5\,A\,a^2+\frac {25\,C\,a^2}{4}\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 )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (12\,A+7\,C\right )}{4\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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