Optimal. Leaf size=99 \[ -\frac {\sec (c+d x)}{7 d (a+a \sin (c+d x))^3}-\frac {4 \sec (c+d x)}{35 a d (a+a \sin (c+d x))^2}-\frac {4 \sec (c+d x)}{35 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {8 \tan (c+d x)}{35 a^3 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {8 \tan (c+d x)}{35 a^3 d}-\frac {4 \sec (c+d x)}{35 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {4 \sec (c+d x)}{35 a d (a \sin (c+d x)+a)^2}-\frac {\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3} \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))^3} \, dx &=-\frac {\sec (c+d x)}{7 d (a+a \sin (c+d x))^3}+\frac {4 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{7 a}\\ &=-\frac {\sec (c+d x)}{7 d (a+a \sin (c+d x))^3}-\frac {4 \sec (c+d x)}{35 a d (a+a \sin (c+d x))^2}+\frac {12 \int \frac {\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx}{35 a^2}\\ &=-\frac {\sec (c+d x)}{7 d (a+a \sin (c+d x))^3}-\frac {4 \sec (c+d x)}{35 a d (a+a \sin (c+d x))^2}-\frac {4 \sec (c+d x)}{35 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {8 \int \sec ^2(c+d x) \, dx}{35 a^3}\\ &=-\frac {\sec (c+d x)}{7 d (a+a \sin (c+d x))^3}-\frac {4 \sec (c+d x)}{35 a d (a+a \sin (c+d x))^2}-\frac {4 \sec (c+d x)}{35 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {8 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{35 a^3 d}\\ &=-\frac {\sec (c+d x)}{7 d (a+a \sin (c+d x))^3}-\frac {4 \sec (c+d x)}{35 a d (a+a \sin (c+d x))^2}-\frac {4 \sec (c+d x)}{35 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {8 \tan (c+d x)}{35 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 63, normalized size = 0.64 \begin {gather*} \frac {\sec (c+d x) (-14 \cos (2 (c+d x))+\cos (4 (c+d x))+14 \sin (c+d x)-6 \sin (3 (c+d x)))}{35 a^3 d (1+\sin (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 130, normalized size = 1.31
method | result | size |
risch | \(\frac {-\frac {96 \,{\mathrm e}^{i \left (d x +c \right )}}{35}-\frac {16 i}{35}+\frac {32 \,{\mathrm e}^{3 i \left (d x +c \right )}}{5}+\frac {32 i {\mathrm e}^{2 i \left (d x +c \right )}}{5}}{\left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) d \,a^{3}}\) | \(74\) |
derivativedivides | \(\frac {-\frac {8}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {38}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {9}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {15}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {17}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {15}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d \,a^{3}}\) | \(130\) |
default | \(\frac {-\frac {8}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {38}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {9}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {15}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {17}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {15}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d \,a^{3}}\) | \(130\) |
norman | \(\frac {-\frac {6 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {26}{35 a d}-\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {10 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {6 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {22 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {86 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{35 d a}}{a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}\) | \(171\) |
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. 310 vs.
\(2 (91) = 182\).
time = 0.30, size = 310, normalized size = 3.13 \begin {gather*} -\frac {2 \, {\left (\frac {43 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {105 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {175 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {105 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {35 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 13\right )}}{35 \, {\left (a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 106, normalized size = 1.07 \begin {gather*} -\frac {8 \, \cos \left (d x + c\right )^{4} - 36 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (6 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) + 15}{35 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right ) + {\left (a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right )\right )} \sin \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 ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.53, size = 119, normalized size = 1.20 \begin {gather*} -\frac {\frac {35}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} + \frac {525 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4025 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3143 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1176 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 243}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{280 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.08, size = 228, normalized size = 2.30 \begin {gather*} -\frac {2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (13\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+43\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+77\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+7\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-175\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-105\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}{35\,a^3\,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 )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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