Optimal. Leaf size=159 \[ \frac {35 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} d}-\frac {35 a^3}{128 d \sqrt {a \sin (c+d x)+a}}+\frac {35 a^2 \sec ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{192 d}+\frac {\sec ^6(c+d x) (a \sin (c+d x)+a)^{5/2}}{6 d}+\frac {7 a \sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{48 d} \]
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Rubi [A] time = 0.24, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {35 a^3}{128 d \sqrt {a \sin (c+d x)+a}}+\frac {35 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} d}+\frac {35 a^2 \sec ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{192 d}+\frac {\sec ^6(c+d x) (a \sin (c+d x)+a)^{5/2}}{6 d}+\frac {7 a \sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{48 d} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 2667
Rule 2675
Rubi steps
\begin {align*} \int \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac {1}{12} (7 a) \int \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac {7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac {1}{96} \left (35 a^2\right ) \int \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {35 a^2 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{192 d}+\frac {7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac {1}{128} \left (35 a^3\right ) \int \frac {\sec (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=\frac {35 a^2 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{192 d}+\frac {7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac {\left (35 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{128 d}\\ &=-\frac {35 a^3}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {35 a^2 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{192 d}+\frac {7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac {\left (35 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{256 d}\\ &=-\frac {35 a^3}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {35 a^2 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{192 d}+\frac {7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}+\frac {\left (35 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{128 d}\\ &=\frac {35 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} d}-\frac {35 a^3}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {35 a^2 \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{192 d}+\frac {7 a \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{48 d}+\frac {\sec ^6(c+d x) (a+a \sin (c+d x))^{5/2}}{6 d}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 44, normalized size = 0.28 \[ -\frac {a^3 \, _2F_1\left (-\frac {1}{2},4;\frac {1}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{8 d \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 208, normalized size = 1.31 \[ \frac {105 \, {\left (\sqrt {2} a^{2} \cos \left (d x + c\right )^{4} + 2 \, \sqrt {2} a^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, \sqrt {2} a^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, {\left (245 \, a^{2} \cos \left (d x + c\right )^{2} - 160 \, a^{2} - 7 \, {\left (15 \, a^{2} \cos \left (d x + c\right )^{2} - 32 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{1536 \, {\left (d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )^{2}\right )}} \]
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 = 0.50, size = 113, normalized size = 0.71 \[ \frac {2 a^{7} \left (-\frac {1}{16 a^{4} \sqrt {a +a \sin \left (d x +c \right )}}-\frac {-\frac {a^{2} \sqrt {a +a \sin \left (d x +c \right )}\, \left (57 \left (\cos ^{2}\left (d x +c \right )\right )+158 \sin \left (d x +c \right )-190\right )}{48 \left (a \sin \left (d x +c \right )-a \right )^{3}}-\frac {35 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{32 \sqrt {a}}}{16 a^{4}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 185, normalized size = 1.16 \[ -\frac {105 \, \sqrt {2} a^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (105 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{3} a^{4} - 560 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a^{5} + 924 \, {\left (a \sin \left (d x + c\right ) + a\right )} a^{6} - 384 \, a^{7}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} - 6 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a + 12 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{2} - 8 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{3}}}{1536 \, a d} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^7} \,d x \]
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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