Optimal. Leaf size=191 \[ \frac {315 a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{2048 \sqrt {2} d}-\frac {315 a^4}{2048 d \sqrt {a \sin (c+d x)+a}}+\frac {105 a^3 \sec ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{1024 d}+\frac {21 a^2 \sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{256 d}+\frac {\sec ^8(c+d x) (a \sin (c+d x)+a)^{7/2}}{8 d}+\frac {3 a \sec ^6(c+d x) (a \sin (c+d x)+a)^{5/2}}{32 d} \]
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Rubi [A] time = 0.31, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2675, 2667, 51, 63, 206} \[ -\frac {315 a^4}{2048 d \sqrt {a \sin (c+d x)+a}}+\frac {315 a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{2048 \sqrt {2} d}+\frac {21 a^2 \sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{256 d}+\frac {105 a^3 \sec ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{1024 d}+\frac {\sec ^8(c+d x) (a \sin (c+d x)+a)^{7/2}}{8 d}+\frac {3 a \sec ^6(c+d x) (a \sin (c+d x)+a)^{5/2}}{32 d} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 2667
Rule 2675
Rubi steps
\begin {align*} \int \sec ^9(c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=\frac {\sec ^8(c+d x) (a+a \sin (c+d x))^{7/2}}{8 d}+\frac {1}{16} (9 a) \int \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2} \, dx\\ &=\frac {3 a \sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{32 d}+\frac {\sec ^8(c+d x) (a+a \sin (c+d x))^{7/2}}{8 d}+\frac {1}{64} \left (21 a^2\right ) \int \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac {21 a^2 \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{256 d}+\frac {3 a \sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{32 d}+\frac {\sec ^8(c+d x) (a+a \sin (c+d x))^{7/2}}{8 d}+\frac {1}{512} \left (105 a^3\right ) \int \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {105 a^3 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{1024 d}+\frac {21 a^2 \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{256 d}+\frac {3 a \sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{32 d}+\frac {\sec ^8(c+d x) (a+a \sin (c+d x))^{7/2}}{8 d}+\frac {\left (315 a^4\right ) \int \frac {\sec (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{2048}\\ &=\frac {105 a^3 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{1024 d}+\frac {21 a^2 \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{256 d}+\frac {3 a \sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{32 d}+\frac {\sec ^8(c+d x) (a+a \sin (c+d x))^{7/2}}{8 d}+\frac {\left (315 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{2048 d}\\ &=-\frac {315 a^4}{2048 d \sqrt {a+a \sin (c+d x)}}+\frac {105 a^3 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{1024 d}+\frac {21 a^2 \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{256 d}+\frac {3 a \sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{32 d}+\frac {\sec ^8(c+d x) (a+a \sin (c+d x))^{7/2}}{8 d}+\frac {\left (315 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{4096 d}\\ &=-\frac {315 a^4}{2048 d \sqrt {a+a \sin (c+d x)}}+\frac {105 a^3 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{1024 d}+\frac {21 a^2 \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{256 d}+\frac {3 a \sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{32 d}+\frac {\sec ^8(c+d x) (a+a \sin (c+d x))^{7/2}}{8 d}+\frac {\left (315 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{2048 d}\\ &=\frac {315 a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2048 \sqrt {2} d}-\frac {315 a^4}{2048 d \sqrt {a+a \sin (c+d x)}}+\frac {105 a^3 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{1024 d}+\frac {21 a^2 \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{256 d}+\frac {3 a \sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{32 d}+\frac {\sec ^8(c+d x) (a+a \sin (c+d x))^{7/2}}{8 d}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 44, normalized size = 0.23 \[ -\frac {a^4 \, _2F_1\left (-\frac {1}{2},5;\frac {1}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{16 d \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 254, normalized size = 1.33 \[ \frac {315 \, {\left (3 \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{4} - 4 \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{2} - {\left (\sqrt {2} a^{3} \cos \left (d x + c\right )^{4} - 4 \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) + 4 \, {\left (315 \, a^{3} \cos \left (d x + c\right )^{4} - 1722 \, a^{3} \cos \left (d x + c\right )^{2} + 896 \, a^{3} + 6 \, {\left (175 \, a^{3} \cos \left (d x + c\right )^{2} - 192 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{8192 \, {\left (3 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.42, size = 129, normalized size = 0.68 \[ -\frac {2 a^{9} \left (\frac {1}{32 a^{5} \sqrt {a +a \sin \left (d x +c \right )}}+\frac {-\frac {\sqrt {a +a \sin \left (d x +c \right )}\, a^{3} \left (187 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-725 \left (\cos ^{2}\left (d x +c \right )\right )-1236 \sin \left (d x +c \right )+1364\right )}{128 \left (a \sin \left (d x +c \right )-a \right )^{4}}-\frac {315 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{256 \sqrt {a}}}{32 a^{5}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 219, normalized size = 1.15 \[ -\frac {315 \, \sqrt {2} a^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (315 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} a^{5} - 2310 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{3} a^{6} + 6132 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a^{7} - 6696 \, {\left (a \sin \left (d x + c\right ) + a\right )} a^{8} + 2048 \, a^{9}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 8 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 24 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 32 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + 16 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{4}}}{8192 \, a d} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2}}{{\cos \left (c+d\,x\right )}^9} \,d x \]
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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