Integrand size = 21, antiderivative size = 186 \[ \int (a \cos (c+d x)+b \sin (c+d x))^{7/2} \, dx=-\frac {10 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) \sqrt {a \cos (c+d x)+b \sin (c+d x)}}{21 d}-\frac {2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^{5/2}}{7 d}+\frac {10 \left (a^2+b^2\right )^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right ),2\right ) \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}}{21 d \sqrt {a \cos (c+d x)+b \sin (c+d x)}} \]
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Time = 0.10 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3152, 3157, 2720} \[ \int (a \cos (c+d x)+b \sin (c+d x))^{7/2} \, dx=\frac {10 \left (a^2+b^2\right )^2 \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right ),2\right )}{21 d \sqrt {a \cos (c+d x)+b \sin (c+d x)}}-\frac {10 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) \sqrt {a \cos (c+d x)+b \sin (c+d x)}}{21 d}-\frac {2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^{5/2}}{7 d} \]
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Rule 2720
Rule 3152
Rule 3157
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^{5/2}}{7 d}+\frac {1}{7} \left (5 \left (a^2+b^2\right )\right ) \int (a \cos (c+d x)+b \sin (c+d x))^{3/2} \, dx \\ & = -\frac {10 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) \sqrt {a \cos (c+d x)+b \sin (c+d x)}}{21 d}-\frac {2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^{5/2}}{7 d}+\frac {1}{21} \left (5 \left (a^2+b^2\right )^2\right ) \int \frac {1}{\sqrt {a \cos (c+d x)+b \sin (c+d x)}} \, dx \\ & = -\frac {10 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) \sqrt {a \cos (c+d x)+b \sin (c+d x)}}{21 d}-\frac {2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^{5/2}}{7 d}+\frac {\left (5 \left (a^2+b^2\right )^2 \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}\right ) \int \frac {1}{\sqrt {\cos \left (c+d x-\tan ^{-1}(a,b)\right )}} \, dx}{21 \sqrt {a \cos (c+d x)+b \sin (c+d x)}} \\ & = -\frac {10 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) \sqrt {a \cos (c+d x)+b \sin (c+d x)}}{21 d}-\frac {2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^{5/2}}{7 d}+\frac {10 \left (a^2+b^2\right )^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right ),2\right ) \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}}{21 d \sqrt {a \cos (c+d x)+b \sin (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.43 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.10 \[ \int (a \cos (c+d x)+b \sin (c+d x))^{7/2} \, dx=\frac {\sqrt {a \cos (c+d x)+b \sin (c+d x)} \left (-23 b \left (a^2+b^2\right ) \cos (c+d x)+\left (-9 a^2 b+3 b^3\right ) \cos (3 (c+d x))+2 a \left (13 a^2+7 b^2+3 \left (a^2-3 b^2\right ) \cos (2 (c+d x))\right ) \sin (c+d x)\right )+\frac {20 \left (a^2+b^2\right )^2 \sqrt {\cos ^2\left (c+d x+\arctan \left (\frac {a}{b}\right )\right )} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2\left (c+d x+\arctan \left (\frac {a}{b}\right )\right )\right ) \tan \left (c+d x+\arctan \left (\frac {a}{b}\right )\right )}{\sqrt {\sqrt {1+\frac {a^2}{b^2}} b \sin \left (c+d x+\arctan \left (\frac {a}{b}\right )\right )}}}{42 d} \]
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Time = 1.36 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {\left (a^{2}+b^{2}\right )^{2} \left (-6 \sin \left (d x +c -\arctan \left (-a , b\right )\right )^{5}+5 \sqrt {-\sin \left (d x +c -\arctan \left (-a , b\right )\right )+1}\, \sqrt {2 \sin \left (d x +c -\arctan \left (-a , b\right )\right )+2}\, \sqrt {\sin \left (d x +c -\arctan \left (-a , b\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c -\arctan \left (-a , b\right )\right )+1}, \frac {\sqrt {2}}{2}\right )-4 \sin \left (d x +c -\arctan \left (-a , b\right )\right )^{3}+10 \sin \left (d x +c -\arctan \left (-a , b\right )\right )\right )}{21 \cos \left (d x +c -\arctan \left (-a , b\right )\right ) \sqrt {\sin \left (d x +c -\arctan \left (-a , b\right )\right ) \sqrt {a^{2}+b^{2}}}\, d}\) | \(185\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.40 \[ \int (a \cos (c+d x)+b \sin (c+d x))^{7/2} \, dx=-\frac {5 \, \sqrt {2} {\left (i \, a^{3} - a^{2} b + i \, a b^{2} - b^{3}\right )} \sqrt {a - i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a^{2} + 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-i \, a^{3} - a^{2} b - i \, a b^{2} - b^{3}\right )} \sqrt {a + i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a^{2} - 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (3 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} b - 8 \, b^{3}\right )} \cos \left (d x + c\right ) - {\left (5 \, a^{3} + 8 \, a b^{2} + 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}}{21 \, d} \]
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Timed out. \[ \int (a \cos (c+d x)+b \sin (c+d x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int (a \cos (c+d x)+b \sin (c+d x))^{7/2} \, dx=\int { {\left (a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )\right )}^{\frac {7}{2}} \,d x } \]
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\[ \int (a \cos (c+d x)+b \sin (c+d x))^{7/2} \, dx=\int { {\left (a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )\right )}^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int (a \cos (c+d x)+b \sin (c+d x))^{7/2} \, dx=\int {\left (a\,\cos \left (c+d\,x\right )+b\,\sin \left (c+d\,x\right )\right )}^{7/2} \,d x \]
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