Optimal. Leaf size=233 \[ -\frac {1155 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{4096 \sqrt {2} a^{5/2} d}+\frac {11 \sec ^3(c+d x)}{64 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {385 \sec (c+d x)}{1024 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {1155 \cos (c+d x)}{4096 a d (a \sin (c+d x)+a)^{3/2}}-\frac {11 \sec ^3(c+d x)}{96 a d (a \sin (c+d x)+a)^{3/2}}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}-\frac {77 \sec (c+d x)}{512 a d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.36, 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 = {2681, 2687, 2650, 2649, 206} \[ \frac {11 \sec ^3(c+d x)}{64 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {385 \sec (c+d x)}{1024 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {1155 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{4096 \sqrt {2} a^{5/2} d}-\frac {1155 \cos (c+d x)}{4096 a d (a \sin (c+d x)+a)^{3/2}}-\frac {11 \sec ^3(c+d x)}{96 a d (a \sin (c+d x)+a)^{3/2}}-\frac {\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}-\frac {77 \sec (c+d x)}{512 a d (a \sin (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2650
Rule 2681
Rule 2687
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=-\frac {\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}+\frac {11 \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{16 a}\\ &=-\frac {\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac {11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac {33 \int \frac {\sec ^4(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{64 a^2}\\ &=-\frac {\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac {11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac {11 \sec ^3(c+d x)}{64 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {77 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{128 a}\\ &=-\frac {\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac {77 \sec (c+d x)}{512 a d (a+a \sin (c+d x))^{3/2}}-\frac {11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac {11 \sec ^3(c+d x)}{64 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {385 \int \frac {\sec ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{1024 a^2}\\ &=-\frac {\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac {77 \sec (c+d x)}{512 a d (a+a \sin (c+d x))^{3/2}}-\frac {11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac {385 \sec (c+d x)}{1024 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{64 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {1155 \int \frac {1}{(a+a \sin (c+d x))^{3/2}} \, dx}{2048 a}\\ &=-\frac {\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac {1155 \cos (c+d x)}{4096 a d (a+a \sin (c+d x))^{3/2}}-\frac {77 \sec (c+d x)}{512 a d (a+a \sin (c+d x))^{3/2}}-\frac {11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac {385 \sec (c+d x)}{1024 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{64 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {1155 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{8192 a^2}\\ &=-\frac {\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac {1155 \cos (c+d x)}{4096 a d (a+a \sin (c+d x))^{3/2}}-\frac {77 \sec (c+d x)}{512 a d (a+a \sin (c+d x))^{3/2}}-\frac {11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac {385 \sec (c+d x)}{1024 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{64 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {1155 \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 )}{4096 a^2 d}\\ &=-\frac {1155 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{4096 \sqrt {2} a^{5/2} d}-\frac {\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac {1155 \cos (c+d x)}{4096 a d (a+a \sin (c+d x))^{3/2}}-\frac {77 \sec (c+d x)}{512 a d (a+a \sin (c+d x))^{3/2}}-\frac {11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac {385 \sec (c+d x)}{1024 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{64 a^2 d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.55, size = 394, normalized size = 1.69 \[ \frac {\frac {1920 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {256 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-1545 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4+3090 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3-1036 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2+2072 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {1472 \sin \left (\frac {1}{2} (c+d x)\right )}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}-\frac {384}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {768 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}+(3465+3465 i) (-1)^{3/4} \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5 \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac {1}{4} (c+d x)\right )-1\right )\right )-736}{12288 d (a (\sin (c+d x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 308, normalized size = 1.32 \[ \frac {3465 \, \sqrt {2} {\left (3 \, \cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3} + {\left (\cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \, {\left (8085 \, \cos \left (d x + c\right )^{4} - 5280 \, \cos \left (d x + c\right )^{2} + 11 \, {\left (315 \, \cos \left (d x + c\right )^{4} - 672 \, \cos \left (d x + c\right )^{2} - 256\right )} \sin \left (d x + c\right ) - 1280\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{49152 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} + {\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 9.34, size = 1074, normalized size = 4.61 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 355, normalized size = 1.52 \[ \frac {6930 a^{\frac {11}{2}} \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )-924 \left (16 a^{\frac {11}{2}}+15 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{4}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (-5632 a^{\frac {11}{2}}+27720 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{4}\right ) \sin \left (d x +c \right )+\left (16170 a^{\frac {11}{2}}+3465 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{4}\right ) \left (\cos ^{4}\left (d x +c \right )\right )-1320 \left (8 a^{\frac {11}{2}}+21 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{4}\right ) \left (\cos ^{2}\left (d x +c \right )\right )-2560 a^{\frac {11}{2}}+27720 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{4}}{24576 a^{\frac {15}{2}} \left (\sin \left (d x +c \right )-1\right ) \left (1+\sin \left (d x +c \right )\right )^{3} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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