Optimal. Leaf size=185 \[ \frac {9 \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} a^{5/2} d}-\frac {9}{32 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {9 \sec ^2(c+d x)}{40 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {3}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac {9 \sec ^2(c+d x)}{70 a d (a \sin (c+d x)+a)^{3/2}}-\frac {\sec ^2(c+d x)}{7 d (a \sin (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.27, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2681, 2687, 2667, 51, 63, 206} \[ -\frac {9}{32 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {9 \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} a^{5/2} d}+\frac {9 \sec ^2(c+d x)}{40 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {3}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac {9 \sec ^2(c+d x)}{70 a d (a \sin (c+d x)+a)^{3/2}}-\frac {\sec ^2(c+d x)}{7 d (a \sin (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 2667
Rule 2681
Rule 2687
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=-\frac {\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}+\frac {9 \int \frac {\sec ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{14 a}\\ &=-\frac {\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac {9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}+\frac {9 \int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{20 a^2}\\ &=-\frac {\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac {9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}+\frac {9 \sec ^2(c+d x)}{40 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {9 \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{16 a}\\ &=-\frac {\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac {9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}+\frac {9 \sec ^2(c+d x)}{40 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,a \sin (c+d x)\right )}{16 d}\\ &=-\frac {\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac {3}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}+\frac {9 \sec ^2(c+d x)}{40 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{32 a d}\\ &=-\frac {\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac {3}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}-\frac {9}{32 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {9 \sec ^2(c+d x)}{40 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{64 a^2 d}\\ &=-\frac {\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac {3}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}-\frac {9}{32 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {9 \sec ^2(c+d x)}{40 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{32 a^2 d}\\ &=\frac {9 \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} a^{5/2} d}-\frac {\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac {3}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}-\frac {9}{32 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {9 \sec ^2(c+d x)}{40 a^2 d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 42, normalized size = 0.23 \[ -\frac {a \, _2F_1\left (-\frac {7}{2},2;-\frac {5}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{14 d (a \sin (c+d x)+a)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 225, normalized size = 1.22 \[ \frac {315 \, \sqrt {2} {\left (3 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, {\left (315 \, \cos \left (d x + c\right )^{4} - 1092 \, \cos \left (d x + c\right )^{2} - 120 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 3\right )} \sin \left (d x + c\right ) + 200\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{4480 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2} + {\left (a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 11.92, size = 916, normalized size = 4.95 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 141, normalized size = 0.76 \[ \frac {2 a^{3} \left (-\frac {1}{8 a^{5} \sqrt {a +a \sin \left (d x +c \right )}}-\frac {1}{16 a^{4} \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {1}{20 a^{3} \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {1}{28 a^{2} \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {\frac {\sqrt {a +a \sin \left (d x +c \right )}}{4 a \sin \left (d x +c \right )-4 a}-\frac {9 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 \sqrt {a}}}{16 a^{5}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 167, normalized size = 0.90 \[ -\frac {\frac {4 \, {\left (315 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} - 420 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{3} a - 168 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a^{2} - 144 \, {\left (a \sin \left (d x + c\right ) + a\right )} a^{3} - 160 \, a^{4}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a - 2 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2}} + \frac {315 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right )}{a^{\frac {3}{2}}}}{4480 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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