Optimal. Leaf size=171 \[ \frac {3}{64 d \left (a^3-a^3 \sin (c+d x)\right )}-\frac {15}{128 d \left (a^3 \sin (c+d x)+a^3\right )}+\frac {21 \tanh ^{-1}(\sin (c+d x))}{128 a^3 d}-\frac {a^2}{40 d (a \sin (c+d x)+a)^5}-\frac {3 a}{64 d (a \sin (c+d x)+a)^4}-\frac {1}{16 d (a \sin (c+d x)+a)^3}+\frac {1}{128 a d (a-a \sin (c+d x))^2}-\frac {5}{64 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.13, antiderivative size = 171, 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 = {2667, 44, 206} \[ -\frac {a^2}{40 d (a \sin (c+d x)+a)^5}+\frac {3}{64 d \left (a^3-a^3 \sin (c+d x)\right )}-\frac {15}{128 d \left (a^3 \sin (c+d x)+a^3\right )}+\frac {21 \tanh ^{-1}(\sin (c+d x))}{128 a^3 d}-\frac {3 a}{64 d (a \sin (c+d x)+a)^4}-\frac {1}{16 d (a \sin (c+d x)+a)^3}+\frac {1}{128 a d (a-a \sin (c+d x))^2}-\frac {5}{64 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {1}{(a-x)^3 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^5 \operatorname {Subst}\left (\int \left (\frac {1}{64 a^6 (a-x)^3}+\frac {3}{64 a^7 (a-x)^2}+\frac {1}{8 a^3 (a+x)^6}+\frac {3}{16 a^4 (a+x)^5}+\frac {3}{16 a^5 (a+x)^4}+\frac {5}{32 a^6 (a+x)^3}+\frac {15}{128 a^7 (a+x)^2}+\frac {21}{128 a^7 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {1}{128 a d (a-a \sin (c+d x))^2}-\frac {a^2}{40 d (a+a \sin (c+d x))^5}-\frac {3 a}{64 d (a+a \sin (c+d x))^4}-\frac {1}{16 d (a+a \sin (c+d x))^3}-\frac {5}{64 a d (a+a \sin (c+d x))^2}+\frac {3}{64 d \left (a^3-a^3 \sin (c+d x)\right )}-\frac {15}{128 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {21 \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{128 a^2 d}\\ &=\frac {21 \tanh ^{-1}(\sin (c+d x))}{128 a^3 d}+\frac {1}{128 a d (a-a \sin (c+d x))^2}-\frac {a^2}{40 d (a+a \sin (c+d x))^5}-\frac {3 a}{64 d (a+a \sin (c+d x))^4}-\frac {1}{16 d (a+a \sin (c+d x))^3}-\frac {5}{64 a d (a+a \sin (c+d x))^2}+\frac {3}{64 d \left (a^3-a^3 \sin (c+d x)\right )}-\frac {15}{128 d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 145, normalized size = 0.85 \[ \frac {\sec ^4(c+d x) \left (-105 \sin ^6(c+d x)-315 \sin ^5(c+d x)-140 \sin ^4(c+d x)+420 \sin ^3(c+d x)+469 \sin ^2(c+d x)+7 \sin (c+d x)+105 \tanh ^{-1}(\sin (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^{10}-176\right )}{640 a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 248, normalized size = 1.45 \[ -\frac {210 \, \cos \left (d x + c\right )^{6} - 910 \, \cos \left (d x + c\right )^{4} + 252 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (3 \, \cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4} + {\left (\cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, {\left (3 \, \cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4} + {\left (\cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 14 \, {\left (45 \, \cos \left (d x + c\right )^{4} - 30 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 96}{1280 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{6} - 4 \, a^{3} d \cos \left (d x + c\right )^{4} + {\left (a^{3} d \cos \left (d x + c\right )^{6} - 4 \, a^{3} d \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.71, size = 136, normalized size = 0.80 \[ \frac {\frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} - \frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3}} + \frac {10 \, {\left (63 \, \sin \left (d x + c\right )^{2} - 150 \, \sin \left (d x + c\right ) + 91\right )}}{a^{3} {\left (\sin \left (d x + c\right ) - 1\right )}^{2}} - \frac {959 \, \sin \left (d x + c\right )^{5} + 5395 \, \sin \left (d x + c\right )^{4} + 12390 \, \sin \left (d x + c\right )^{3} + 14710 \, \sin \left (d x + c\right )^{2} + 9275 \, \sin \left (d x + c\right ) + 2647}{a^{3} {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{5120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 162, normalized size = 0.95 \[ \frac {1}{128 a^{3} d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3}{64 a^{3} d \left (\sin \left (d x +c \right )-1\right )}-\frac {21 \ln \left (\sin \left (d x +c \right )-1\right )}{256 a^{3} d}-\frac {1}{40 a^{3} d \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {3}{64 a^{3} d \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{16 a^{3} d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {5}{64 a^{3} d \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {15}{128 a^{3} d \left (1+\sin \left (d x +c \right )\right )}+\frac {21 \ln \left (1+\sin \left (d x +c \right )\right )}{256 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 188, normalized size = 1.10 \[ -\frac {\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} + 315 \, \sin \left (d x + c\right )^{5} + 140 \, \sin \left (d x + c\right )^{4} - 420 \, \sin \left (d x + c\right )^{3} - 469 \, \sin \left (d x + c\right )^{2} - 7 \, \sin \left (d x + c\right ) + 176\right )}}{a^{3} \sin \left (d x + c\right )^{7} + 3 \, a^{3} \sin \left (d x + c\right )^{6} + a^{3} \sin \left (d x + c\right )^{5} - 5 \, a^{3} \sin \left (d x + c\right )^{4} - 5 \, a^{3} \sin \left (d x + c\right )^{3} + a^{3} \sin \left (d x + c\right )^{2} + 3 \, a^{3} \sin \left (d x + c\right ) + a^{3}} - \frac {105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {105 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3}}}{1280 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.77, size = 173, normalized size = 1.01 \[ \frac {21\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{128\,a^3\,d}-\frac {\frac {21\,{\sin \left (c+d\,x\right )}^6}{128}+\frac {63\,{\sin \left (c+d\,x\right )}^5}{128}+\frac {7\,{\sin \left (c+d\,x\right )}^4}{32}-\frac {21\,{\sin \left (c+d\,x\right )}^3}{32}-\frac {469\,{\sin \left (c+d\,x\right )}^2}{640}-\frac {7\,\sin \left (c+d\,x\right )}{640}+\frac {11}{40}}{d\,\left (a^3\,{\sin \left (c+d\,x\right )}^7+3\,a^3\,{\sin \left (c+d\,x\right )}^6+a^3\,{\sin \left (c+d\,x\right )}^5-5\,a^3\,{\sin \left (c+d\,x\right )}^4-5\,a^3\,{\sin \left (c+d\,x\right )}^3+a^3\,{\sin \left (c+d\,x\right )}^2+3\,a^3\,\sin \left (c+d\,x\right )+a^3\right )} \]
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 \[ \frac {\int \frac {\sec ^{5}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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