Optimal. Leaf size=193 \[ -\frac {b (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}\right )}{2 \sqrt {-b^2} d (n+1) \left (a-\sqrt {-b^2}\right )}+\frac {b (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right )}{2 \sqrt {-b^2} d (n+1) \left (a+\sqrt {-b^2}\right )}+\frac {(a+b \tan (c+d x))^{n+1}}{b d (n+1)} \]
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Rubi [A] time = 0.15, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3543, 3485, 712, 68} \[ -\frac {b (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}\right )}{2 \sqrt {-b^2} d (n+1) \left (a-\sqrt {-b^2}\right )}+\frac {b (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right )}{2 \sqrt {-b^2} d (n+1) \left (a+\sqrt {-b^2}\right )}+\frac {(a+b \tan (c+d x))^{n+1}}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 712
Rule 3485
Rule 3543
Rubi steps
\begin {align*} \int \tan ^2(c+d x) (a+b \tan (c+d x))^n \, dx &=\frac {(a+b \tan (c+d x))^{1+n}}{b d (1+n)}-\int (a+b \tan (c+d x))^n \, dx\\ &=\frac {(a+b \tan (c+d x))^{1+n}}{b d (1+n)}-\frac {b \operatorname {Subst}\left (\int \frac {(a+x)^n}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {(a+b \tan (c+d x))^{1+n}}{b d (1+n)}-\frac {b \operatorname {Subst}\left (\int \left (\frac {\sqrt {-b^2} (a+x)^n}{2 b^2 \left (\sqrt {-b^2}-x\right )}+\frac {\sqrt {-b^2} (a+x)^n}{2 b^2 \left (\sqrt {-b^2}+x\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {(a+b \tan (c+d x))^{1+n}}{b d (1+n)}+\frac {b \operatorname {Subst}\left (\int \frac {(a+x)^n}{\sqrt {-b^2}-x} \, dx,x,b \tan (c+d x)\right )}{2 \sqrt {-b^2} d}+\frac {b \operatorname {Subst}\left (\int \frac {(a+x)^n}{\sqrt {-b^2}+x} \, dx,x,b \tan (c+d x)\right )}{2 \sqrt {-b^2} d}\\ &=\frac {(a+b \tan (c+d x))^{1+n}}{b d (1+n)}-\frac {b \, _2F_1\left (1,1+n;2+n;\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}\right ) (a+b \tan (c+d x))^{1+n}}{2 \sqrt {-b^2} \left (a-\sqrt {-b^2}\right ) d (1+n)}+\frac {b \, _2F_1\left (1,1+n;2+n;\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right ) (a+b \tan (c+d x))^{1+n}}{2 \sqrt {-b^2} \left (a+\sqrt {-b^2}\right ) d (1+n)}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 138, normalized size = 0.72 \[ \frac {(a+b \tan (c+d x))^{n+1} \left (i b (a+i b) \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a-i b}\right )+(a-i b) \left (-i b \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a+i b}\right )+2 a+2 i b\right )\right )}{2 b d (n+1) (b-i a) (b+i a)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \tan \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{2}, x\right ) \]
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 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.71, size = 0, normalized size = 0.00 \[ \int \left (\tan ^{2}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^2\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{n} \tan ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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