Optimal. Leaf size=115 \[ \frac {\, _2F_1\left (1,1+n;2+n;\frac {a+b \sec (c+d x)}{a-b}\right ) (a+b \sec (c+d x))^{1+n}}{2 (a-b) d (1+n)}-\frac {\, _2F_1\left (1,1+n;2+n;\frac {a+b \sec (c+d x)}{a+b}\right ) (a+b \sec (c+d x))^{1+n}}{2 (a+b) d (1+n)} \]
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Rubi [A]
time = 0.08, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3959, 88, 70}
\begin {gather*} \frac {(a+b \sec (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \sec (c+d x)}{a-b}\right )}{2 d (n+1) (a-b)}-\frac {(a+b \sec (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \sec (c+d x)}{a+b}\right )}{2 d (n+1) (a+b)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 88
Rule 3959
Rubi steps
\begin {align*} \int \csc (c+d x) (a+b \sec (c+d x))^n \, dx &=-\frac {\text {Subst}\left (\int \frac {(a-b x)^n}{(-1+x) (1+x)} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {(a-b x)^n}{-1+x^2} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {(a-b x)^n}{2 (1-x)}-\frac {(a-b x)^n}{2 (1+x)}\right ) \, dx,x,-\sec (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {(a-b x)^n}{1-x} \, dx,x,-\sec (c+d x)\right )}{2 d}+\frac {\text {Subst}\left (\int \frac {(a-b x)^n}{1+x} \, dx,x,-\sec (c+d x)\right )}{2 d}\\ &=\frac {\, _2F_1\left (1,1+n;2+n;\frac {a+b \sec (c+d x)}{a-b}\right ) (a+b \sec (c+d x))^{1+n}}{2 (a-b) d (1+n)}-\frac {\, _2F_1\left (1,1+n;2+n;\frac {a+b \sec (c+d x)}{a+b}\right ) (a+b \sec (c+d x))^{1+n}}{2 (a+b) d (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.66, size = 132, normalized size = 1.15 \begin {gather*} \frac {\left (\, _2F_1\left (1,-n;1-n;\frac {(a+b) \cos (c+d x)}{b+a \cos (c+d x)}\right )-2^n \, _2F_1\left (-n,-n;1-n;\frac {(-a+b) \cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{2 b}\right ) \left (\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{b}\right )^{-n}\right ) (a+b \sec (c+d x))^n}{2 d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \csc \left (d x +c \right ) \left (a +b \sec \left (d x +c \right )\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sec {\left (c + d x \right )}\right )^{n} \csc {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{\sin \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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