Optimal. Leaf size=110 \[ \frac {4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}+\frac {10 a^3 \sqrt {e \cos (c+d x)}}{3 d e^3}-\frac {10 a^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2670, 2680, 2682, 2642, 2641} \[ \frac {10 a^3 \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}-\frac {10 a^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2641
Rule 2642
Rule 2670
Rule 2680
Rule 2682
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{5/2}} \, dx &=\frac {a^6 \int \frac {(e \cos (c+d x))^{7/2}}{(a-a \sin (c+d x))^3} \, dx}{e^6}\\ &=\frac {4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}-\frac {\left (5 a^4\right ) \int \frac {(e \cos (c+d x))^{3/2}}{a-a \sin (c+d x)} \, dx}{3 e^4}\\ &=\frac {10 a^3 \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}-\frac {\left (5 a^3\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{3 e^2}\\ &=\frac {10 a^3 \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}-\frac {\left (5 a^3 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 e^2 \sqrt {e \cos (c+d x)}}\\ &=\frac {10 a^3 \sqrt {e \cos (c+d x)}}{3 d e^3}-\frac {10 a^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.05, size = 66, normalized size = 0.60 \[ \frac {8 \sqrt [4]{2} a^3 (\sin (c+d x)+1)^{3/4} \, _2F_1\left (-\frac {5}{4},-\frac {3}{4};\frac {1}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{3} \cos \left (d x + c\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.27, size = 219, normalized size = 1.99 \[ \frac {2 \left (10 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{3 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________