Optimal. Leaf size=150 \[ \frac {7 \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} a^{3/2} d}-\frac {7}{16 a d \sqrt {a \sin (c+d x)+a}}-\frac {7}{24 d (a \sin (c+d x)+a)^{3/2}}+\frac {7 \sec ^2(c+d x)}{20 a d \sqrt {a \sin (c+d x)+a}}-\frac {\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2681, 2687, 2667, 51, 63, 206} \[ \frac {7 \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} a^{3/2} d}-\frac {7}{16 a d \sqrt {a \sin (c+d x)+a}}-\frac {7}{24 d (a \sin (c+d x)+a)^{3/2}}+\frac {7 \sec ^2(c+d x)}{20 a d \sqrt {a \sin (c+d x)+a}}-\frac {\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}} \]
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 ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=-\frac {\sec ^2(c+d x)}{5 d (a+a \sin (c+d x))^{3/2}}+\frac {7 \int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{10 a}\\ &=-\frac {\sec ^2(c+d x)}{5 d (a+a \sin (c+d x))^{3/2}}+\frac {7 \sec ^2(c+d x)}{20 a d \sqrt {a+a \sin (c+d x)}}+\frac {7}{8} \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {\sec ^2(c+d x)}{5 d (a+a \sin (c+d x))^{3/2}}+\frac {7 \sec ^2(c+d x)}{20 a d \sqrt {a+a \sin (c+d x)}}+\frac {(7 a) \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,a \sin (c+d x)\right )}{8 d}\\ &=-\frac {7}{24 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^2(c+d x)}{5 d (a+a \sin (c+d x))^{3/2}}+\frac {7 \sec ^2(c+d x)}{20 a d \sqrt {a+a \sin (c+d x)}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{16 d}\\ &=-\frac {7}{24 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^2(c+d x)}{5 d (a+a \sin (c+d x))^{3/2}}-\frac {7}{16 a d \sqrt {a+a \sin (c+d x)}}+\frac {7 \sec ^2(c+d x)}{20 a d \sqrt {a+a \sin (c+d x)}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{32 a d}\\ &=-\frac {7}{24 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^2(c+d x)}{5 d (a+a \sin (c+d x))^{3/2}}-\frac {7}{16 a d \sqrt {a+a \sin (c+d x)}}+\frac {7 \sec ^2(c+d x)}{20 a d \sqrt {a+a \sin (c+d x)}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{16 a d}\\ &=\frac {7 \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} a^{3/2} d}-\frac {7}{24 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^2(c+d x)}{5 d (a+a \sin (c+d x))^{3/2}}-\frac {7}{16 a d \sqrt {a+a \sin (c+d x)}}+\frac {7 \sec ^2(c+d x)}{20 a d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 42, normalized size = 0.28 \[ -\frac {a \, _2F_1\left (-\frac {5}{2},2;-\frac {3}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{10 d (a \sin (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 187, normalized size = 1.25 \[ \frac {105 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )^{2}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) + 4 \, {\left (175 \, \cos \left (d x + c\right )^{2} + 21 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 36\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{960 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-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 = 0.27, size = 124, normalized size = 0.83 \[ \frac {2 a^{3} \left (-\frac {3}{16 a^{4} \sqrt {a +a \sin \left (d x +c \right )}}-\frac {1}{12 a^{3} \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {1}{20 a^{2} \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {\frac {\sqrt {a +a \sin \left (d x +c \right )}}{2 a \sin \left (d x +c \right )-2 a}-\frac {7 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{4 \sqrt {a}}}{16 a^{4}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 146, normalized size = 0.97 \[ -\frac {\frac {105 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right )}{\sqrt {a}} + \frac {4 \, {\left (105 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{3} - 140 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a - 56 \, {\left (a \sin \left (d x + c\right ) + a\right )} a^{2} - 48 \, a^{3}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} - 2 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a}}{960 \, a d} \]
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 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sec ^{3}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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