Optimal. Leaf size=159 \[ \frac {\cos (c+d x) \sqrt {1-\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}} \sqrt {1-\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}} (a+b \tan (c+d x))^{n+1} F_1\left (n+1;\frac {1}{2},\frac {1}{2};n+2;\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}},\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right )}{b d (n+1)} \]
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Rubi [A] time = 0.11, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3512, 760, 133} \[ \frac {\cos (c+d x) \sqrt {1-\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}} \sqrt {1-\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}} (a+b \tan (c+d x))^{n+1} F_1\left (n+1;\frac {1}{2},\frac {1}{2};n+2;\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}},\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 133
Rule 760
Rule 3512
Rubi steps
\begin {align*} \int \sec (c+d x) (a+b \tan (c+d x))^n \, dx &=\frac {\sec (c+d x) \operatorname {Subst}\left (\int \frac {(a+x)^n}{\sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{b d \sqrt {\sec ^2(c+d x)}}\\ &=\frac {\left (\cos (c+d x) \sqrt {1-\frac {a+b \tan (c+d x)}{a-\frac {b^2}{\sqrt {-b^2}}}} \sqrt {1-\frac {a+b \tan (c+d x)}{a+\frac {b^2}{\sqrt {-b^2}}}}\right ) \operatorname {Subst}\left (\int \frac {x^n}{\sqrt {1-\frac {x}{a-\sqrt {-b^2}}} \sqrt {1-\frac {x}{a+\sqrt {-b^2}}}} \, dx,x,a+b \tan (c+d x)\right )}{b d}\\ &=\frac {F_1\left (1+n;\frac {1}{2},\frac {1}{2};2+n;\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}},\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right ) \cos (c+d x) (a+b \tan (c+d x))^{1+n} \sqrt {1-\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}} \sqrt {1-\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}}}{b d (1+n)}\\ \end {align*}
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Mathematica [C] time = 3.87, size = 340, normalized size = 2.14 \[ \frac {2 (n+2) \left (a^2+b^2\right )^2 \cos ^3(c+d x) (\tan (c+d x)-i) (\tan (c+d x)+i) (a+b \tan (c+d x))^{n+1} F_1\left (n+1;\frac {1}{2},\frac {1}{2};n+2;\frac {a+b \tan (c+d x)}{a-i b},\frac {a+b \tan (c+d x)}{a+i b}\right )}{b d (n+1) (a-i b) (a+i b) \left (2 (n+2) \left (a^2+b^2\right ) F_1\left (n+1;\frac {1}{2},\frac {1}{2};n+2;\frac {a+b \tan (c+d x)}{a-i b},\frac {a+b \tan (c+d x)}{a+i b}\right )+(a+b \tan (c+d x)) \left ((a-i b) F_1\left (n+2;\frac {1}{2},\frac {3}{2};n+3;\frac {a+b \tan (c+d x)}{a-i b},\frac {a+b \tan (c+d x)}{a+i b}\right )+(a+i b) F_1\left (n+2;\frac {3}{2},\frac {1}{2};n+3;\frac {a+b \tan (c+d x)}{a-i b},\frac {a+b \tan (c+d x)}{a+i b}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right ), x\right ) \]
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 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.93, size = 0, normalized size = 0.00 \[ \int \sec \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n}{\cos \left (c+d\,x\right )} \,d x \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{n} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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