Optimal. Leaf size=30 \[ \frac {2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d} \]
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Rubi [A] time = 0.06, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2673} \[ \frac {2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 2673
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac {2 a \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}\\ \end {align*}
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Mathematica [B] time = 5.16, size = 69, normalized size = 2.30 \[ \frac {2 (a (\sin (c+d x)+1))^{5/2}}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 43, normalized size = 1.43 \[ -\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{2}}{3 \, {\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\right )}} \]
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 = 0.16, size = 47, normalized size = 1.57 \[ -\frac {2 a^{3} \left (1+\sin \left (d x +c \right )\right )}{3 \left (\sin \left (d x +c \right )-1\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 184, normalized size = 6.13 \[ -\frac {2 \, {\left (a^{\frac {5}{2}} + \frac {4 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{\frac {5}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )}}{3 \, d {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.59, size = 225, normalized size = 7.50 \[ \frac {4\,a^2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left ({\sin \left (c+d\,x\right )}^2\,4{}\mathrm {i}+\sin \left (c+d\,x\right )\,1{}\mathrm {i}-2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2-2\,\sin \left (2\,c+2\,d\,x\right )+\sin \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}-4{}\mathrm {i}\right )}{3\,d\,\left (8\,{\sin \left (c+d\,x\right )}^2+4\,\sin \left (c+d\,x\right )-2\,{\sin \left (2\,c+2\,d\,x\right )}^2+4\,\sin \left (3\,c+3\,d\,x\right )-8\right )}+\frac {4\,a^2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (\sin \left (2\,c+2\,d\,x\right )+4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-{\sin \left (c+d\,x\right )}^2\,2{}\mathrm {i}-2+2{}\mathrm {i}\right )}{3\,d\,\left (4\,{\sin \left (c+d\,x\right )}^2+\sin \left (c+d\,x\right )+\sin \left (3\,c+3\,d\,x\right )-4\right )} \]
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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