Optimal. Leaf size=115 \[ \frac {(a+b \sec (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \sec (c+d x)}{a-b}\right )}{2 d (n+1) (a-b)}-\frac {(a+b \sec (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \sec (c+d x)}{a+b}\right )}{2 d (n+1) (a+b)} \]
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Rubi [A] time = 0.12, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3874, 73, 712, 68} \[ \frac {(a+b \sec (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \sec (c+d x)}{a-b}\right )}{2 d (n+1) (a-b)}-\frac {(a+b \sec (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \sec (c+d x)}{a+b}\right )}{2 d (n+1) (a+b)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 73
Rule 712
Rule 3874
Rubi steps
\begin {align*} \int \csc (c+d x) (a+b \sec (c+d x))^n \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-b x)^n}{(-1+x) (1+x)} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(a-b x)^n}{-1+x^2} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=-\frac {\operatorname {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 {\operatorname {Subst}\left (\int \frac {(a-b x)^n}{1-x} \, dx,x,-\sec (c+d x)\right )}{2 d}+\frac {\operatorname {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 = 0.98, size = 132, normalized size = 1.15 \[ \frac {(a+b \sec (c+d x))^n \left (\, _2F_1\left (1,-n;1-n;\frac {(a+b) \cos (c+d x)}{b+a \cos (c+d x)}\right )-2^n \left (\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{b}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac {(b-a) \cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{2 b}\right )\right )}{2 d n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.07, 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 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{\sin \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec {\left (c + d x \right )}\right )^{n} \csc {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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