Optimal. Leaf size=69 \[ \frac {\sqrt {2} 7^n F_1\left (\frac {1}{2};-n,\frac {1}{2};\frac {3}{2};\frac {3}{7} (1+\sin (c+d x)),\frac {1}{2} (1+\sin (c+d x))\right ) \cos (c+d x)}{d \sqrt {1-\sin (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2744, 143}
\begin {gather*} \frac {\sqrt {2} 7^n \cos (c+d x) F_1\left (\frac {1}{2};-n,\frac {1}{2};\frac {3}{2};\frac {3}{7} (\sin (c+d x)+1),\frac {1}{2} (\sin (c+d x)+1)\right )}{d \sqrt {1-\sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 143
Rule 2744
Rubi steps
\begin {align*} \int (4-3 \sin (c+d x))^n \, dx &=\frac {\cos (c+d x) \text {Subst}\left (\int \frac {(4-3 x)^n}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sin (c+d x)\right )}{d \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}}\\ &=\frac {\sqrt {2} 7^n F_1\left (\frac {1}{2};-n,\frac {1}{2};\frac {3}{2};\frac {3}{7} (1+\sin (c+d x)),\frac {1}{2} (1+\sin (c+d x))\right ) \cos (c+d x)}{d \sqrt {1-\sin (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 96, normalized size = 1.39 \begin {gather*} -\frac {F_1\left (1+n;\frac {1}{2},\frac {1}{2};2+n;\frac {1}{7} (4-3 \sin (c+d x)),4-3 \sin (c+d x)\right ) \sec (c+d x) (4-3 \sin (c+d x))^{1+n} \sqrt {-1+\sin (c+d x)} \sqrt {1+\sin (c+d x)}}{\sqrt {7} d (1+n)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (4-3 \sin \left (d x +c \right )\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 (4 - 3 \sin {\left (c + d x \right )}\right )^{n}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (4-3\,\sin \left (c+d\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________