Integrand size = 27, antiderivative size = 239 \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{3/2}} \, dx=\frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a+a \sin (c+d x)}}-\frac {5 a^2 \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d e^{3/2} (1+\cos (c+d x)+\sin (c+d x))}-\frac {5 a^2 \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d e^{3/2} (1+\cos (c+d x)+\sin (c+d x))}+\frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}} \]
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Time = 0.27 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2755, 2757, 2763, 2854, 209, 2912, 65, 221} \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{3/2}} \, dx=\frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a \sin (c+d x)+a}}-\frac {5 a^2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{d e^{3/2} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {5 a^2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d e^{3/2} (\sin (c+d x)+\cos (c+d x)+1)}+\frac {4 a (a \sin (c+d x)+a)^{3/2}}{d e \sqrt {e \cos (c+d x)}} \]
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Rule 65
Rule 209
Rule 221
Rule 2755
Rule 2757
Rule 2763
Rule 2854
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (5 a^2\right ) \int \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)} \, dx}{e^2} \\ & = \frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a+a \sin (c+d x)}}+\frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (5 a^3\right ) \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx}{2 e^2} \\ & = \frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a+a \sin (c+d x)}}+\frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (5 a^3 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sqrt {1+\cos (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx}{2 e (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (5 a^3 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}} \, dx}{2 e (a+a \cos (c+d x)+a \sin (c+d x))} \\ & = \frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a+a \sin (c+d x)}}+\frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (5 a^3 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{2 d e (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (5 a^3 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+e x^2} \, dx,x,-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right )}{d e (a+a \cos (c+d x)+a \sin (c+d x))} \\ & = \frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a+a \sin (c+d x)}}+\frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}}-\frac {5 a^3 \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d e^{3/2} (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (5 a^3 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{e}}} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{d e^2 (a+a \cos (c+d x)+a \sin (c+d x))} \\ & = \frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a+a \sin (c+d x)}}+\frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}}-\frac {5 a^3 \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d e^{3/2} (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {5 a^3 \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d e^{3/2} (a+a \cos (c+d x)+a \sin (c+d x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.31 \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{3/2}} \, dx=\frac {8 \sqrt [4]{2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {1}{4},\frac {3}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (a (1+\sin (c+d x)))^{5/2}}{d e \sqrt {e \cos (c+d x)} (1+\sin (c+d x))^{9/4}} \]
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Time = 6.72 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.28
method | result | size |
default | \(\frac {\left (5 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )-5 \,\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+\cos ^{2}\left (d x +c \right )-\cos \left (d x +c \right ) \sin \left (d x +c \right )+5 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-5 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+9 \cos \left (d x +c \right )+8 \sin \left (d x +c \right )+8\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, a^{2}}{d \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) e \sqrt {e \cos \left (d x +c \right )}}\) | \(307\) |
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Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 1093, normalized size of antiderivative = 4.57 \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{3/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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