Optimal. Leaf size=211 \[ \frac {99 \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {99}{256 a d \sqrt {a \sin (c+d x)+a}}-\frac {33}{128 d (a \sin (c+d x)+a)^{3/2}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a \sin (c+d x)+a}}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a \sin (c+d x)+a}}-\frac {99 \sec ^2(c+d x)}{560 d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.34, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {99 \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {99}{256 a d \sqrt {a \sin (c+d x)+a}}-\frac {33}{128 d (a \sin (c+d x)+a)^{3/2}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a \sin (c+d x)+a}}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a \sin (c+d x)+a}}-\frac {99 \sec ^2(c+d x)}{560 d (a \sin (c+d x)+a)^{3/2}} \]
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 ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac {11 \int \frac {\sec ^5(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{14 a}\\ &=-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {99}{112} \int \frac {\sec ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{160 a}\\ &=-\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {99}{128} \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {(99 a) \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,a \sin (c+d x)\right )}{128 d}\\ &=-\frac {33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{256 d}\\ &=-\frac {33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}-\frac {99}{256 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{512 a d}\\ &=-\frac {33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}-\frac {99}{256 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{256 a d}\\ &=\frac {99 \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}-\frac {99}{256 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 44, normalized size = 0.21 \[ -\frac {a^2 \, _2F_1\left (-\frac {7}{2},3;-\frac {5}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{28 d (a \sin (c+d x)+a)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 207, normalized size = 0.98 \[ \frac {3465 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{6} - 2 \, \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )^{4}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) + 4 \, {\left (5775 \, \cos \left (d x + c\right )^{4} - 1188 \, \cos \left (d x + c\right )^{2} + 11 \, {\left (315 \, \cos \left (d x + c\right )^{4} - 252 \, \cos \left (d x + c\right )^{2} - 160\right )} \sin \left (d x + c\right ) - 480\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{35840 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 2 \, a^{2} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 7.41, size = 1076, normalized size = 5.10 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 152, normalized size = 0.72 \[ -\frac {2 a^{5} \left (\frac {5}{32 a^{6} \sqrt {a +a \sin \left (d x +c \right )}}+\frac {1}{16 a^{5} \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {3}{80 a^{4} \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {1}{56 a^{3} \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}}}+\frac {\frac {\sqrt {a +a \sin \left (d x +c \right )}\, a \left (19 \sin \left (d x +c \right )-23\right )}{16 \left (a \sin \left (d x +c \right )-a \right )^{2}}-\frac {99 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{32 \sqrt {a}}}{32 a^{6}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 197, normalized size = 0.93 \[ -\frac {\frac {3465 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right )}{\sqrt {a}} + \frac {4 \, {\left (3465 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} - 11550 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} a + 7392 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{3} a^{2} + 2112 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a^{3} + 1408 \, {\left (a \sin \left (d x + c\right ) + a\right )} a^{4} + 1280 \, a^{5}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 4 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 4 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2}}}{35840 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^5\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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