Optimal. Leaf size=89 \[ \frac {64 a^3 \sec (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {16 a^2 \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac {2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d} \]
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Rubi [A] time = 0.17, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2674, 2673} \[ \frac {64 a^3 \sec (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {16 a^2 \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac {2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 2673
Rule 2674
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=-\frac {2 a \sec (c+d x) (a+a \sin (c+d x))^{5/2}}{3 d}+\frac {1}{3} (8 a) \int \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx\\ &=-\frac {16 a^2 \sec (c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}-\frac {2 a \sec (c+d x) (a+a \sin (c+d x))^{5/2}}{3 d}+\frac {1}{3} \left (32 a^2\right ) \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac {64 a^3 \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {16 a^2 \sec (c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}-\frac {2 a \sec (c+d x) (a+a \sin (c+d x))^{5/2}}{3 d}\\ \end {align*}
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Mathematica [A] time = 5.48, size = 48, normalized size = 0.54 \[ \frac {a^3 \sec (c+d x) \sqrt {a (\sin (c+d x)+1)} (-20 \sin (c+d x)+\cos (2 (c+d x))+45)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 54, normalized size = 0.61 \[ \frac {2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - 10 \, a^{3} \sin \left (d x + c\right ) + 22 \, a^{3}\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3 \, d \cos \left (d x + c\right )} \]
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 = 0.18, size = 55, normalized size = 0.62 \[ -\frac {2 a^{4} \left (1+\sin \left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )+10 \sin \left (d x +c \right )-23\right )}{3 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 237, normalized size = 2.66 \[ -\frac {2 \, {\left (23 \, a^{\frac {7}{2}} - \frac {20 \, a^{\frac {7}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {88 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {60 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {130 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {60 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {88 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {20 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {23 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )}}{3 \, d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac {7}{2}}} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2}}{{\cos \left (c+d\,x\right )}^2} \,d x \]
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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