Optimal. Leaf size=192 \[ -\frac {a (a+b \tan (c+d x))^{n+1}}{b^2 d (n+1) (n+2)}+\frac {(a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (a-i b)}+\frac {(a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (a+i b)}+\frac {\tan (c+d x) (a+b \tan (c+d x))^{n+1}}{b d (n+2)} \]
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Rubi [A] time = 0.28, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3566, 3630, 12, 3539, 3537, 68} \[ -\frac {a (a+b \tan (c+d x))^{n+1}}{b^2 d (n+1) (n+2)}+\frac {(a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (a-i b)}+\frac {(a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (a+i b)}+\frac {\tan (c+d x) (a+b \tan (c+d x))^{n+1}}{b d (n+2)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 68
Rule 3537
Rule 3539
Rule 3566
Rule 3630
Rubi steps
\begin {align*} \int \tan ^3(c+d x) (a+b \tan (c+d x))^n \, dx &=\frac {\tan (c+d x) (a+b \tan (c+d x))^{1+n}}{b d (2+n)}+\frac {\int (a+b \tan (c+d x))^n \left (-a-b (2+n) \tan (c+d x)-a \tan ^2(c+d x)\right ) \, dx}{b (2+n)}\\ &=-\frac {a (a+b \tan (c+d x))^{1+n}}{b^2 d (1+n) (2+n)}+\frac {\tan (c+d x) (a+b \tan (c+d x))^{1+n}}{b d (2+n)}-\frac {\int b (2+n) \tan (c+d x) (a+b \tan (c+d x))^n \, dx}{b (2+n)}\\ &=-\frac {a (a+b \tan (c+d x))^{1+n}}{b^2 d (1+n) (2+n)}+\frac {\tan (c+d x) (a+b \tan (c+d x))^{1+n}}{b d (2+n)}-\int \tan (c+d x) (a+b \tan (c+d x))^n \, dx\\ &=-\frac {a (a+b \tan (c+d x))^{1+n}}{b^2 d (1+n) (2+n)}+\frac {\tan (c+d x) (a+b \tan (c+d x))^{1+n}}{b d (2+n)}-\frac {1}{2} i \int (1-i \tan (c+d x)) (a+b \tan (c+d x))^n \, dx+\frac {1}{2} i \int (1+i \tan (c+d x)) (a+b \tan (c+d x))^n \, dx\\ &=-\frac {a (a+b \tan (c+d x))^{1+n}}{b^2 d (1+n) (2+n)}+\frac {\tan (c+d x) (a+b \tan (c+d x))^{1+n}}{b d (2+n)}-\frac {\operatorname {Subst}\left (\int \frac {(a-i b x)^n}{-1+x} \, dx,x,i \tan (c+d x)\right )}{2 d}-\frac {\operatorname {Subst}\left (\int \frac {(a+i b x)^n}{-1+x} \, dx,x,-i \tan (c+d x)\right )}{2 d}\\ &=-\frac {a (a+b \tan (c+d x))^{1+n}}{b^2 d (1+n) (2+n)}+\frac {\, _2F_1\left (1,1+n;2+n;\frac {a+b \tan (c+d x)}{a-i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (a-i b) d (1+n)}+\frac {\, _2F_1\left (1,1+n;2+n;\frac {a+b \tan (c+d x)}{a+i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (a+i b) d (1+n)}+\frac {\tan (c+d x) (a+b \tan (c+d x))^{1+n}}{b d (2+n)}\\ \end {align*}
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Mathematica [A] time = 1.05, size = 135, normalized size = 0.70 \[ \frac {(a+b \tan (c+d x))^{n+1} \left (\frac {2 \left (b \tan (c+d x)-\frac {a}{n+1}\right )}{b^2 (n+2)}+\frac {\, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a-i b}\right )}{(n+1) (a-i b)}+\frac {\, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a+i b}\right )}{(n+1) (a+i b)}\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \tan \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{3}, x\right ) \]
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 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.83, size = 0, normalized size = 0.00 \[ \int \left (\tan ^{3}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^3\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{n} \tan ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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