Optimal. Leaf size=62 \[ \frac {2 \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {2 a \sqrt {a+a \sin (c+d x)}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2746, 52, 65,
212} \begin {gather*} \frac {2 \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {2 a \sqrt {a \sin (c+d x)+a}}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 212
Rule 2746
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac {a \text {Subst}\left (\int \frac {\sqrt {a+x}}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {2 a \sqrt {a+a \sin (c+d x)}}{d}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {2 a \sqrt {a+a \sin (c+d x)}}{d}+\frac {\left (4 a^2\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{d}\\ &=\frac {2 \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {2 a \sqrt {a+a \sin (c+d x)}}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 60, normalized size = 0.97 \begin {gather*} \frac {2 a \left (\sqrt {2} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )-\sqrt {a+a \sin (c+d x)}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.20, size = 49, normalized size = 0.79
method | result | size |
default | \(-\frac {2 a \left (\sqrt {a +a \sin \left (d x +c \right )}-\sqrt {a}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )\right )}{d}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.56, size = 80, normalized size = 1.29 \begin {gather*} -\frac {\sqrt {2} a^{\frac {5}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + 2 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{2}}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 72, normalized size = 1.16 \begin {gather*} \frac {\sqrt {2} a^{\frac {3}{2}} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 2 \, \sqrt {a \sin \left (d x + c\right ) + a} a}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \sec {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 6.77, size = 73, normalized size = 1.18 \begin {gather*} -\frac {\sqrt {2} a^{\frac {3}{2}} {\left (2 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \log \left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{\cos \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________