Optimal. Leaf size=99 \[ \frac {8 \tan (c+d x)}{35 a^3 d}-\frac {4 \sec (c+d x)}{35 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {4 \sec (c+d x)}{35 a d (a \sin (c+d x)+a)^2}-\frac {\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.13, 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 = {2672, 3767, 8} \[ \frac {8 \tan (c+d x)}{35 a^3 d}-\frac {4 \sec (c+d x)}{35 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {4 \sec (c+d x)}{35 a d (a \sin (c+d x)+a)^2}-\frac {\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2672
Rule 3767
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 \operatorname {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 = 0.11, size = 63, normalized size = 0.64 \[ \frac {\sec (c+d x) (14 \sin (c+d x)-6 \sin (3 (c+d x))-14 \cos (2 (c+d x))+\cos (4 (c+d x)))}{35 a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 106, normalized size = 1.07 \[ -\frac {8 \, \cos \left (d x + c\right )^{4} - 36 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (6 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) + 15}{35 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right ) + {\left (a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 119, normalized size = 1.20 \[ -\frac {\frac {35}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} + \frac {525 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4025 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3143 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1176 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 243}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{280 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 130, normalized size = 1.31 \[ \frac {-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {8}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {38}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {9}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {15}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {17}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {15}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 310, normalized size = 3.13 \[ -\frac {2 \, {\left (\frac {43 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {105 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {175 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {105 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {35 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 13\right )}}{35 \, {\left (a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d} \]
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 \[ -\frac {2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (13\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+43\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+77\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+7\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-175\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-105\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}{35\,a^3\,d\,\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^7} \]
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 \[ \frac {\int \frac {\sec ^{2}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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