Optimal. Leaf size=127 \[ \frac {15 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} d}-\frac {15 a^2}{32 d \sqrt {a \sin (c+d x)+a}}+\frac {\sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}+\frac {5 a \sec ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{16 d} \]
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Rubi [A] time = 0.18, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {15 a^2}{32 d \sqrt {a \sin (c+d x)+a}}+\frac {15 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} d}+\frac {\sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}+\frac {5 a \sec ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{16 d} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 2667
Rule 2675
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac {\sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d}+\frac {1}{8} (5 a) \int \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {5 a \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{16 d}+\frac {\sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d}+\frac {1}{32} \left (15 a^2\right ) \int \frac {\sec (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=\frac {5 a \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{16 d}+\frac {\sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d}+\frac {\left (15 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{32 d}\\ &=-\frac {15 a^2}{32 d \sqrt {a+a \sin (c+d x)}}+\frac {5 a \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{16 d}+\frac {\sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d}+\frac {\left (15 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{64 d}\\ &=-\frac {15 a^2}{32 d \sqrt {a+a \sin (c+d x)}}+\frac {5 a \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{16 d}+\frac {\sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d}+\frac {\left (15 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{32 d}\\ &=\frac {15 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} d}-\frac {15 a^2}{32 d \sqrt {a+a \sin (c+d x)}}+\frac {5 a \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{16 d}+\frac {\sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{4 d}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 44, normalized size = 0.35 \[ -\frac {a^2 \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{4 d \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 155, normalized size = 1.22 \[ \frac {15 \, {\left (\sqrt {2} a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - \sqrt {2} a \cos \left (d x + c\right )^{2}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, {\left (15 \, a \cos \left (d x + c\right )^{2} + 20 \, a \sin \left (d x + c\right ) - 12 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{128 \, {\left (d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{2}\right )}} \]
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 = 0.33, size = 101, normalized size = 0.80 \[ -\frac {2 a^{5} \left (\frac {1}{8 a^{3} \sqrt {a +a \sin \left (d x +c \right )}}+\frac {\frac {\sqrt {a +a \sin \left (d x +c \right )}\, a \left (7 \sin \left (d x +c \right )-11\right )}{8 \left (a \sin \left (d x +c \right )-a \right )^{2}}-\frac {15 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{16 \sqrt {a}}}{8 a^{3}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 151, normalized size = 1.19 \[ -\frac {15 \, \sqrt {2} a^{\frac {5}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (15 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a^{3} - 50 \, {\left (a \sin \left (d x + c\right ) + a\right )} a^{4} + 32 \, a^{5}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} - 4 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a + 4 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{2}}}{128 \, a d} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^5} \,d x \]
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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