Optimal. Leaf size=61 \[ \frac {\, _2F_1\left (\frac {1}{2},-\frac {n}{2};\frac {2-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d n \sqrt {\sin ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {16, 3857, 2722}
\begin {gather*} \frac {\sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac {1}{2},-\frac {n}{2};\frac {2-n}{2};\cos ^2(c+d x)\right )}{d n \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 2722
Rule 3857
Rubi steps
\begin {align*} \int \sec (c+d x) (b \sec (c+d x))^n \, dx &=\frac {\int (b \sec (c+d x))^{1+n} \, dx}{b}\\ &=\frac {\left (\left (\frac {\cos (c+d x)}{b}\right )^n (b \sec (c+d x))^n\right ) \int \left (\frac {\cos (c+d x)}{b}\right )^{-1-n} \, dx}{b}\\ &=\frac {\, _2F_1\left (\frac {1}{2},-\frac {n}{2};\frac {2-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d n \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 65, normalized size = 1.07 \begin {gather*} \frac {\csc (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\sec ^2(c+d x)\right ) (b \sec (c+d x))^n \sqrt {-\tan ^2(c+d x)}}{d (1+n)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.31, size = 0, normalized size = 0.00 \[\int \sec \left (d x +c \right ) \left (b \sec \left (d x +c \right )\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \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 (b \sec {\left (c + d x \right )}\right )^{n} \sec {\left (c + d x \right )}\, 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 = -0.02 \begin {gather*} \int \frac {{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{\cos \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________