Integrand size = 30, antiderivative size = 39 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\cos (e+f x)}{f \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.10, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2817} \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {a \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}} \]
[In]
[Out]
Rule 2817
Rubi steps \begin{align*} \text {integral}& = \frac {a \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(39)=78\).
Time = 1.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\sqrt {1+\sin (e+f x)} \sqrt {c-c \sin (e+f x)}}{\sqrt {3} c^4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs. \(2(37)=74\).
Time = 3.64 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.36
method | result | size |
default | \(\frac {\sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \left (\sin \left (f x +e \right ) \cos \left (f x +e \right )-3 \cos \left (f x +e \right )-4 \tan \left (f x +e \right )+3 \sec \left (f x +e \right )\right )}{3 f \left (2 \sin \left (f x +e \right )+\cos ^{2}\left (f x +e \right )-2\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{3}}\) | \(92\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (37) = 74\).
Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.31 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, {\left (3 \, c^{4} f \cos \left (f x + e\right )^{3} - 4 \, c^{4} f \cos \left (f x + e\right ) - {\left (c^{4} f \cos \left (f x + e\right )^{3} - 4 \, c^{4} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {\sqrt {a \sin \left (f x + e\right ) + a}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{24 \, c^{\frac {7}{2}} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}} \]
[In]
[Out]
Time = 11.07 (sec) , antiderivative size = 190, normalized size of antiderivative = 4.87 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\,\sqrt {c-c\,\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\,16{}\mathrm {i}}{3\,c^4\,f\,\left (1+14\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}-{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}-14\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,6{}\mathrm {i}-{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,14{}\mathrm {i}-{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,14{}\mathrm {i}+{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,6{}\mathrm {i}\right )} \]
[In]
[Out]