Optimal. Leaf size=175 \[ -\frac {63}{128 d \sqrt {a \sin (c+d x)+a}}-\frac {21 a}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac {63 \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} \sqrt {a} d}+\frac {\sec ^4(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {63 \sec ^2(c+d x)}{160 d \sqrt {a \sin (c+d x)+a}}-\frac {9 a \sec ^2(c+d x)}{40 d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.26, antiderivative size = 175, 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 = {2687, 2681, 2667, 51, 63, 206} \[ -\frac {63}{128 d \sqrt {a \sin (c+d x)+a}}-\frac {21 a}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac {63 \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} \sqrt {a} d}+\frac {\sec ^4(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {63 \sec ^2(c+d x)}{160 d \sqrt {a \sin (c+d x)+a}}-\frac {9 a \sec ^2(c+d x)}{40 d (a \sin (c+d x)+a)^{3/2}} \]
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)}{\sqrt {a+a \sin (c+d x)}} \, dx &=\frac {\sec ^4(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{8} (9 a) \int \frac {\sec ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {9 a \sec ^2(c+d x)}{40 d (a+a \sin (c+d x))^{3/2}}+\frac {\sec ^4(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {63}{80} \int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {9 a \sec ^2(c+d x)}{40 d (a+a \sin (c+d x))^{3/2}}+\frac {63 \sec ^2(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^4(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{64} (63 a) \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {9 a \sec ^2(c+d x)}{40 d (a+a \sin (c+d x))^{3/2}}+\frac {63 \sec ^2(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^4(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {\left (63 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,a \sin (c+d x)\right )}{64 d}\\ &=-\frac {21 a}{64 d (a+a \sin (c+d x))^{3/2}}-\frac {9 a \sec ^2(c+d x)}{40 d (a+a \sin (c+d x))^{3/2}}+\frac {63 \sec ^2(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^4(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {(63 a) \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{128 d}\\ &=-\frac {21 a}{64 d (a+a \sin (c+d x))^{3/2}}-\frac {9 a \sec ^2(c+d x)}{40 d (a+a \sin (c+d x))^{3/2}}-\frac {63}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {63 \sec ^2(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^4(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {63 \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{256 d}\\ &=-\frac {21 a}{64 d (a+a \sin (c+d x))^{3/2}}-\frac {9 a \sec ^2(c+d x)}{40 d (a+a \sin (c+d x))^{3/2}}-\frac {63}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {63 \sec ^2(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^4(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {63 \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{128 d}\\ &=\frac {63 \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} \sqrt {a} d}-\frac {21 a}{64 d (a+a \sin (c+d x))^{3/2}}-\frac {9 a \sec ^2(c+d x)}{40 d (a+a \sin (c+d x))^{3/2}}-\frac {63}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {63 \sec ^2(c+d x)}{160 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^4(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 44, normalized size = 0.25 \[ -\frac {a^2 \, _2F_1\left (-\frac {5}{2},3;-\frac {3}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{20 d (a \sin (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 167, normalized size = 0.95 \[ \frac {315 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{4}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, {\left (315 \, \cos \left (d x + c\right )^{4} - 42 \, \cos \left (d x + c\right )^{2} - 6 \, {\left (35 \, \cos \left (d x + c\right )^{2} + 24\right )} \sin \left (d x + c\right ) - 16\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{2560 \, {\left (a d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 135, normalized size = 0.77 \[ -\frac {2 a^{5} \left (\frac {3}{16 a^{5} \sqrt {a +a \sin \left (d x +c \right )}}+\frac {1}{16 a^{4} \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {1}{40 a^{3} \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {\frac {\sqrt {a +a \sin \left (d x +c \right )}\, a \left (15 \sin \left (d x +c \right )-19\right )}{16 \left (a \sin \left (d x +c \right )-a \right )^{2}}-\frac {63 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{32 \sqrt {a}}}{16 a^{5}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 183, normalized size = 1.05 \[ -\frac {315 \, \sqrt {2} \sqrt {a} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (315 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} a - 1050 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{3} a^{2} + 672 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a^{3} + 192 \, {\left (a \sin \left (d x + c\right ) + a\right )} a^{4} + 128 \, a^{5}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 4 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 4 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2}}}{2560 \, a d} \]
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 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^5\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {\sec ^{5}{\left (c + d x \right )}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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