Optimal. Leaf size=40 \[ -\frac {(a \sec (c+d x)+a)^n \, _2F_1\left (1,n;n+1;\frac {1}{2} (\sec (c+d x)+1)\right )}{2 d n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3873, 68} \[ -\frac {(a \sec (c+d x)+a)^n \, _2F_1\left (1,n;n+1;\frac {1}{2} (\sec (c+d x)+1)\right )}{2 d n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 68
Rule 3873
Rubi steps
\begin {align*} \int \csc (c+d x) (a+a \sec (c+d x))^n \, dx &=-\frac {a^2 \operatorname {Subst}\left (\int \frac {(a-a x)^{-1+n}}{-a-a x} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=-\frac {\, _2F_1\left (1,n;1+n;\frac {1}{2} (1+\sec (c+d x))\right ) (a+a \sec (c+d x))^n}{2 d n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.68, size = 92, normalized size = 2.30 \[ \frac {2^{n-1} (\sec (c+d x)+1)^{-n} (a (\sec (c+d x)+1))^n \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^{n-1} \, _2F_1\left (1,1-n;2-n;\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )}{d (n-1)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.31, size = 0, normalized size = 0.00 \[ \int \csc \left (d x +c \right ) \left (a +a \sec \left (d x +c \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n}{\sin \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \csc {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________