Optimal. Leaf size=233 \[ -\frac {11 a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{64 \sqrt {2} d}-\frac {11 a^5 \cos (c+d x)}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac {11 a^4 \sec (c+d x)}{48 d \sqrt {a \sin (c+d x)+a}}+\frac {11 a^3 \sec ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{140 d}+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}+\frac {11 a \sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{126 d} \]
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Rubi [A] time = 0.35, antiderivative size = 233, 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, 2687, 2650, 2649, 206} \[ -\frac {11 a^5 \cos (c+d x)}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac {11 a^2 \sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{140 d}+\frac {11 a^3 \sec ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{120 d}+\frac {11 a^4 \sec (c+d x)}{48 d \sqrt {a \sin (c+d x)+a}}-\frac {11 a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{64 \sqrt {2} d}+\frac {\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}+\frac {11 a \sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{126 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2650
Rule 2675
Rule 2687
Rubi steps
\begin {align*} \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}+\frac {1}{18} (11 a) \int \sec ^8(c+d x) (a+a \sin (c+d x))^{5/2} \, dx\\ &=\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}+\frac {1}{28} \left (11 a^2\right ) \int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}+\frac {1}{40} \left (11 a^3\right ) \int \sec ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {11 a^3 \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}+\frac {1}{48} \left (11 a^4\right ) \int \frac {\sec ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=\frac {11 a^4 \sec (c+d x)}{48 d \sqrt {a+a \sin (c+d x)}}+\frac {11 a^3 \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}+\frac {1}{32} \left (11 a^5\right ) \int \frac {1}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {11 a^5 \cos (c+d x)}{64 d (a+a \sin (c+d x))^{3/2}}+\frac {11 a^4 \sec (c+d x)}{48 d \sqrt {a+a \sin (c+d x)}}+\frac {11 a^3 \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}+\frac {1}{128} \left (11 a^4\right ) \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {11 a^5 \cos (c+d x)}{64 d (a+a \sin (c+d x))^{3/2}}+\frac {11 a^4 \sec (c+d x)}{48 d \sqrt {a+a \sin (c+d x)}}+\frac {11 a^3 \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}-\frac {\left (11 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}\\ &=-\frac {11 a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{64 \sqrt {2} d}-\frac {11 a^5 \cos (c+d x)}{64 d (a+a \sin (c+d x))^{3/2}}+\frac {11 a^4 \sec (c+d x)}{48 d \sqrt {a+a \sin (c+d x)}}+\frac {11 a^3 \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{120 d}+\frac {11 a^2 \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{140 d}+\frac {11 a \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2}}{126 d}+\frac {\sec ^9(c+d x) (a+a \sin (c+d x))^{7/2}}{9 d}\\ \end {align*}
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Mathematica [C] time = 5.68, size = 388, normalized size = 1.67 \[ \frac {(a (\sin (c+d x)+1))^{7/2} \left (630 \sin \left (\frac {1}{2} (c+d x)\right )+\frac {3150 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {1680 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {1512 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5}+\frac {1440 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^7}+\frac {1120 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^9}-315 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+(3465+3465 i) (-1)^{3/4} \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2 \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac {1}{4} (c+d x)\right )-1\right )\right )\right )}{20160 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 346, normalized size = 1.48 \[ \frac {3465 \, {\left (3 \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{5} - 4 \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{3} - {\left (\sqrt {2} a^{3} \cos \left (d x + c\right )^{5} - 4 \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\sqrt {2} \cos \left (d x + c\right ) - \sqrt {2} \sin \left (d x + c\right ) + \sqrt {2}\right )} \sqrt {a} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \, {\left (12705 \, a^{3} \cos \left (d x + c\right )^{4} - 25212 \, a^{3} \cos \left (d x + c\right )^{2} + 3920 \, a^{3} - 77 \, {\left (45 \, a^{3} \cos \left (d x + c\right )^{4} - 276 \, a^{3} \cos \left (d x + c\right )^{2} + 80 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{80640 \, {\left (3 \, d \cos \left (d x + c\right )^{5} - 4 \, d \cos \left (d x + c\right )^{3} - {\left (d \cos \left (d x + c\right )^{5} - 4 \, d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
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 = 0.28, size = 205, normalized size = 0.88 \[ -\frac {-6930 a^{\frac {11}{2}} \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+42504 a^{\frac {11}{2}} \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+385 \left (9 \left (a -a \sin \left (d x +c \right )\right )^{\frac {9}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a -32 a^{\frac {11}{2}}\right ) \sin \left (d x +c \right )+25410 a^{\frac {11}{2}} \left (\cos ^{4}\left (d x +c \right )\right )-50424 a^{\frac {11}{2}} \left (\cos ^{2}\left (d x +c \right )\right )+3465 \left (a -a \sin \left (d x +c \right )\right )^{\frac {9}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a +7840 a^{\frac {11}{2}}}{40320 a^{\frac {3}{2}} \left (\sin \left (d x +c \right )-1\right )^{4} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2}}{{\cos \left (c+d\,x\right )}^{10}} \,d x \]
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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