Optimal. Leaf size=91 \[ -\frac {3 \sqrt {2} a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {3 a^3 \sqrt {a \sin (c+d x)+a}}{d}+\frac {a \sec ^2(c+d x) (a \sin (c+d x)+a)^{5/2}}{d} \]
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Rubi [A] time = 0.13, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2676, 2667, 50, 63, 206} \[ \frac {3 a^3 \sqrt {a \sin (c+d x)+a}}{d}-\frac {3 \sqrt {2} a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a \sec ^2(c+d x) (a \sin (c+d x)+a)^{5/2}}{d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 2667
Rule 2676
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=\frac {a \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2}}{d}-\frac {1}{2} \left (3 a^2\right ) \int \sec (c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac {a \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2}}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+x}}{a-x} \, dx,x,a \sin (c+d x)\right )}{2 d}\\ &=\frac {3 a^3 \sqrt {a+a \sin (c+d x)}}{d}+\frac {a \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2}}{d}-\frac {\left (3 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {3 a^3 \sqrt {a+a \sin (c+d x)}}{d}+\frac {a \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2}}{d}-\frac {\left (6 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{d}\\ &=-\frac {3 \sqrt {2} a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {3 a^3 \sqrt {a+a \sin (c+d x)}}{d}+\frac {a \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2}}{d}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 42, normalized size = 0.46 \[ \frac {a (a \sin (c+d x)+a)^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{10 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 116, normalized size = 1.27 \[ \frac {3 \, \sqrt {2} {\left (a^{3} \sin \left (d x + c\right ) - a^{3}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) + 4 \, {\left (a^{3} \sin \left (d x + c\right ) - 2 \, a^{3}\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{2 \, {\left (d \sin \left (d x + c\right ) - d\right )}} \]
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 = 0.27, size = 83, normalized size = 0.91 \[ \frac {2 a^{3} \left (\sqrt {a +a \sin \left (d x +c \right )}+4 a \left (-\frac {\sqrt {a +a \sin \left (d x +c \right )}}{4 \left (a \sin \left (d x +c \right )-a \right )}-\frac {3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 \sqrt {a}}\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 112, normalized size = 1.23 \[ \frac {3 \, \sqrt {2} a^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + 4 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{4} - \frac {4 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{5}}{a \sin \left (d x + c\right ) - a}}{2 \, a d} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2}}{{\cos \left (c+d\,x\right )}^3} \,d x \]
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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