Integrand size = 10, antiderivative size = 70 \[ \int \sin ^{\frac {7}{2}}(a+b x) \, dx=\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right )}{21 b}-\frac {10 \cos (a+b x) \sqrt {\sin (a+b x)}}{21 b}-\frac {2 \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x)}{7 b} \]
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Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2715, 2720} \[ \int \sin ^{\frac {7}{2}}(a+b x) \, dx=\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{21 b}-\frac {2 \sin ^{\frac {5}{2}}(a+b x) \cos (a+b x)}{7 b}-\frac {10 \sqrt {\sin (a+b x)} \cos (a+b x)}{21 b} \]
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Rule 2715
Rule 2720
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x)}{7 b}+\frac {5}{7} \int \sin ^{\frac {3}{2}}(a+b x) \, dx \\ & = -\frac {10 \cos (a+b x) \sqrt {\sin (a+b x)}}{21 b}-\frac {2 \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x)}{7 b}+\frac {5}{21} \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx \\ & = \frac {10 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right )}{21 b}-\frac {10 \cos (a+b x) \sqrt {\sin (a+b x)}}{21 b}-\frac {2 \cos (a+b x) \sin ^{\frac {5}{2}}(a+b x)}{7 b} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.79 \[ \int \sin ^{\frac {7}{2}}(a+b x) \, dx=\frac {-20 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right )+(-23 \cos (a+b x)+3 \cos (3 (a+b x))) \sqrt {\sin (a+b x)}}{42 b} \]
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Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.49
method | result | size |
default | \(\frac {\frac {2 \left (\cos ^{4}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{7}+\frac {5 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, F\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{21}-\frac {16 \left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{21}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b}\) | \(104\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.24 \[ \int \sin ^{\frac {7}{2}}(a+b x) \, dx=\frac {5 \, \sqrt {2} \sqrt {-i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 \, \sqrt {2} \sqrt {i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + 2 \, {\left (3 \, \cos \left (b x + a\right )^{3} - 8 \, \cos \left (b x + a\right )\right )} \sqrt {\sin \left (b x + a\right )}}{21 \, b} \]
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Timed out. \[ \int \sin ^{\frac {7}{2}}(a+b x) \, dx=\text {Timed out} \]
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\[ \int \sin ^{\frac {7}{2}}(a+b x) \, dx=\int { \sin \left (b x + a\right )^{\frac {7}{2}} \,d x } \]
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\[ \int \sin ^{\frac {7}{2}}(a+b x) \, dx=\int { \sin \left (b x + a\right )^{\frac {7}{2}} \,d x } \]
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Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.60 \[ \int \sin ^{\frac {7}{2}}(a+b x) \, dx=-\frac {\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^{9/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ \frac {3}{2};\ {\cos \left (a+b\,x\right )}^2\right )}{b\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{9/4}} \]
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