Optimal. Leaf size=71 \[ -\frac {\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {\sec (c+d x)}{5 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {2 \tan (c+d x)}{5 a^2 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2751, 3852, 8}
\begin {gather*} \frac {2 \tan (c+d x)}{5 a^2 d}-\frac {\sec (c+d x)}{5 d \left (a^2 \sin (c+d x)+a^2\right )}-\frac {\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2751
Rule 3852
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=-\frac {\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}+\frac {3 \int \frac {\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx}{5 a}\\ &=-\frac {\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {\sec (c+d x)}{5 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {2 \int \sec ^2(c+d x) \, dx}{5 a^2}\\ &=-\frac {\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {\sec (c+d x)}{5 d \left (a^2+a^2 \sin (c+d x)\right )}-\frac {2 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{5 a^2 d}\\ &=-\frac {\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac {\sec (c+d x)}{5 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {2 \tan (c+d x)}{5 a^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 53, normalized size = 0.75 \begin {gather*} -\frac {\sec (c+d x) (4 \cos (2 (c+d x))-5 \sin (c+d x)+\sin (3 (c+d x)))}{10 a^2 d (1+\sin (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 98, normalized size = 1.38
method | result | size |
risch | \(-\frac {4 i \left (4 i {\mathrm e}^{i \left (d x +c \right )}+5 \,{\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{5 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} d}\) | \(63\) |
derivativedivides | \(\frac {-\frac {4}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {3}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {5}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {7}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{a^{2} d}\) | \(98\) |
default | \(\frac {-\frac {4}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {3}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {5}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {7}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{a^{2} d}\) | \(98\) |
norman | \(\frac {-\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {4}{5 a d}-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5 d a}}{a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(114\) |
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. 204 vs.
\(2 (65) = 130\).
time = 0.30, size = 204, normalized size = 2.87 \begin {gather*} -\frac {2 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {10 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {5 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 2\right )}}{5 \, {\left (a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {5 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 79, normalized size = 1.11 \begin {gather*} \frac {4 \, \cos \left (d x + c\right )^{2} + {\left (2 \, \cos \left (d x + c\right )^{2} - 3\right )} \sin \left (d x + c\right ) - 2}{5 \, {\left (a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )\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 {\sec ^{2}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\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 = 5.94, size = 93, normalized size = 1.31 \begin {gather*} -\frac {\frac {5}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} + \frac {35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 90 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 70 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{20 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.77, size = 156, normalized size = 2.20 \begin {gather*} \frac {2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+10\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+10\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}{5\,a^2\,d\,\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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