Optimal. Leaf size=78 \[ \frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a d \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2683, 2642, 2641} \[ \frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a d \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2641
Rule 2642
Rule 2683
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))} \, dx &=-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a+a \sin (c+d x))}+\frac {\int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{3 a}\\ &=-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a+a \sin (c+d x))}+\frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 a \sqrt {e \cos (c+d x)}}\\ &=\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a d \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a+a \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 64, normalized size = 0.82 \[ -\frac {\sqrt [4]{2} \sqrt {e \cos (c+d x)} \, _2F_1\left (\frac {1}{4},\frac {7}{4};\frac {5}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{a d e \sqrt [4]{\sin (c+d x)+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )}}{a e \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a e \cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.48, size = 190, normalized size = 2.44 \[ -\frac {2 \left (2 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}+2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{\sqrt {e \cos {\left (c + d x \right )}} \sin {\left (c + d x \right )} + \sqrt {e \cos {\left (c + d x \right )}}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________