Integrand size = 14, antiderivative size = 256 \[ \int (a+b \sin (c+d x))^{7/2} \, dx=-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{105 d}-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d}-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac {32 a \left (11 a^2+13 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{105 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (71 a^4-46 a^2 b^2-25 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{105 d \sqrt {a+b \sin (c+d x)}} \]
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Time = 0.26 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2735, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int (a+b \sin (c+d x))^{7/2} \, dx=-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{105 d}+\frac {32 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{105 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (71 a^4-46 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{105 d \sqrt {a+b \sin (c+d x)}}-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d} \]
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Rule 2732
Rule 2734
Rule 2735
Rule 2740
Rule 2742
Rule 2831
Rule 2832
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac {2}{7} \int (a+b \sin (c+d x))^{3/2} \left (\frac {1}{2} \left (7 a^2+5 b^2\right )+6 a b \sin (c+d x)\right ) \, dx \\ & = -\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d}-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac {4}{35} \int \sqrt {a+b \sin (c+d x)} \left (\frac {1}{4} a \left (35 a^2+61 b^2\right )+\frac {1}{4} b \left (71 a^2+25 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{105 d}-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d}-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac {8}{105} \int \frac {\frac {1}{8} \left (105 a^4+254 a^2 b^2+25 b^4\right )+2 a b \left (11 a^2+13 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx \\ & = -\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{105 d}-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d}-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac {1}{105} \left (16 a \left (11 a^2+13 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx+\frac {1}{105} \left (-71 a^4+46 a^2 b^2+25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx \\ & = -\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{105 d}-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d}-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac {\left (16 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{105 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (\left (-71 a^4+46 a^2 b^2+25 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{105 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{105 d}-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d}-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac {32 a \left (11 a^2+13 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{105 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (71 a^4-46 a^2 b^2-25 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{105 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}
Time = 0.76 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.86 \[ \int (a+b \sin (c+d x))^{7/2} \, dx=\frac {-64 a \left (11 a^3+11 a^2 b+13 a b^2+13 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+4 \left (71 a^4-46 a^2 b^2-25 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-b \cos (c+d x) \left (488 a^3+262 a b^2-162 a b^2 \cos (2 (c+d x))+b \left (752 a^2+145 b^2\right ) \sin (c+d x)-15 b^3 \sin (3 (c+d x))\right )}{210 d \sqrt {a+b \sin (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1039\) vs. \(2(298)=596\).
Time = 1.19 (sec) , antiderivative size = 1040, normalized size of antiderivative = 4.06
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.92 \[ \int (a+b \sin (c+d x))^{7/2} \, dx=-\frac {\sqrt {2} {\left (37 \, a^{4} - 346 \, a^{2} b^{2} - 75 \, b^{4}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + \sqrt {2} {\left (37 \, a^{4} - 346 \, a^{2} b^{2} - 75 \, b^{4}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) + 48 \, \sqrt {2} {\left (11 i \, a^{3} b + 13 i \, a b^{3}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) + 48 \, \sqrt {2} {\left (-11 i \, a^{3} b - 13 i \, a b^{3}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) - 6 \, {\left (15 \, b^{4} \cos \left (d x + c\right )^{3} - 66 \, a b^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, {\left (61 \, a^{2} b^{2} + 20 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{315 \, b d} \]
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Timed out. \[ \int (a+b \sin (c+d x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int (a+b \sin (c+d x))^{7/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]
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\[ \int (a+b \sin (c+d x))^{7/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int (a+b \sin (c+d x))^{7/2} \, dx=\int {\left (a+b\,\sin \left (c+d\,x\right )\right )}^{7/2} \,d x \]
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