Optimal. Leaf size=206 \[ -\frac {b d^2 \left (3 a^2-b^2\right ) (d \cot (e+f x))^{n-2} \, _2F_1\left (1,\frac {n-2}{2};\frac {n}{2};-\cot ^2(e+f x)\right )}{f (2-n)}-\frac {a d \left (a^2-3 b^2\right ) (d \cot (e+f x))^{n-1} \, _2F_1\left (1,\frac {n-1}{2};\frac {n+1}{2};-\cot ^2(e+f x)\right )}{f (1-n)}+\frac {a^2 d^2 (a \cot (e+f x)+b) (d \cot (e+f x))^{n-2}}{f (1-n)}+\frac {a^2 b d^2 (1-2 n) (d \cot (e+f x))^{n-2}}{f (1-n) (2-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.43, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3673, 3566, 3630, 3538, 3476, 364} \[ -\frac {b d^2 \left (3 a^2-b^2\right ) (d \cot (e+f x))^{n-2} \, _2F_1\left (1,\frac {n-2}{2};\frac {n}{2};-\cot ^2(e+f x)\right )}{f (2-n)}-\frac {a d \left (a^2-3 b^2\right ) (d \cot (e+f x))^{n-1} \, _2F_1\left (1,\frac {n-1}{2};\frac {n+1}{2};-\cot ^2(e+f x)\right )}{f (1-n)}+\frac {a^2 d^2 (a \cot (e+f x)+b) (d \cot (e+f x))^{n-2}}{f (1-n)}+\frac {a^2 b d^2 (1-2 n) (d \cot (e+f x))^{n-2}}{f (1-n) (2-n)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 3476
Rule 3538
Rule 3566
Rule 3630
Rule 3673
Rubi steps
\begin {align*} \int (d \cot (e+f x))^n (a+b \tan (e+f x))^3 \, dx &=d^3 \int (d \cot (e+f x))^{-3+n} (b+a \cot (e+f x))^3 \, dx\\ &=\frac {a^2 d^2 (d \cot (e+f x))^{-2+n} (b+a \cot (e+f x))}{f (1-n)}+\frac {d^2 \int (d \cot (e+f x))^{-3+n} \left (b d \left (b^2 (1-n)-a^2 (2-n)\right )-a \left (a^2-3 b^2\right ) d (1-n) \cot (e+f x)+a^2 b d (1-2 n) \cot ^2(e+f x)\right ) \, dx}{1-n}\\ &=\frac {a^2 b d^2 (1-2 n) (d \cot (e+f x))^{-2+n}}{f (1-n) (2-n)}+\frac {a^2 d^2 (d \cot (e+f x))^{-2+n} (b+a \cot (e+f x))}{f (1-n)}+\frac {d^2 \int (d \cot (e+f x))^{-3+n} \left (-b \left (3 a^2-b^2\right ) d (1-n)-a \left (a^2-3 b^2\right ) d (1-n) \cot (e+f x)\right ) \, dx}{1-n}\\ &=\frac {a^2 b d^2 (1-2 n) (d \cot (e+f x))^{-2+n}}{f (1-n) (2-n)}+\frac {a^2 d^2 (d \cot (e+f x))^{-2+n} (b+a \cot (e+f x))}{f (1-n)}-\left (a \left (a^2-3 b^2\right ) d^2\right ) \int (d \cot (e+f x))^{-2+n} \, dx-\left (b \left (3 a^2-b^2\right ) d^3\right ) \int (d \cot (e+f x))^{-3+n} \, dx\\ &=\frac {a^2 b d^2 (1-2 n) (d \cot (e+f x))^{-2+n}}{f (1-n) (2-n)}+\frac {a^2 d^2 (d \cot (e+f x))^{-2+n} (b+a \cot (e+f x))}{f (1-n)}+\frac {\left (a \left (a^2-3 b^2\right ) d^3\right ) \operatorname {Subst}\left (\int \frac {x^{-2+n}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{f}+\frac {\left (b \left (3 a^2-b^2\right ) d^4\right ) \operatorname {Subst}\left (\int \frac {x^{-3+n}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{f}\\ &=\frac {a^2 b d^2 (1-2 n) (d \cot (e+f x))^{-2+n}}{f (1-n) (2-n)}+\frac {a^2 d^2 (d \cot (e+f x))^{-2+n} (b+a \cot (e+f x))}{f (1-n)}-\frac {b \left (3 a^2-b^2\right ) d^2 (d \cot (e+f x))^{-2+n} \, _2F_1\left (1,\frac {1}{2} (-2+n);\frac {n}{2};-\cot ^2(e+f x)\right )}{f (2-n)}-\frac {a \left (a^2-3 b^2\right ) d (d \cot (e+f x))^{-1+n} \, _2F_1\left (1,\frac {1}{2} (-1+n);\frac {1+n}{2};-\cot ^2(e+f x)\right )}{f (1-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.92, size = 141, normalized size = 0.68 \[ \frac {\tan ^2(e+f x) (d \cot (e+f x))^n \left (a \left ((n-2) \left (a^2-3 b^2\right ) \cot (e+f x) \, _2F_1\left (1,\frac {n-1}{2};\frac {n+1}{2};-\cot ^2(e+f x)\right )+a (-a (n-2) \cot (e+f x)-3 b (n-1))\right )-b (n-1) \left (b^2-3 a^2\right ) \, _2F_1\left (1,\frac {n-2}{2};\frac {n}{2};-\cot ^2(e+f x)\right )\right )}{f (n-2) (n-1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.37, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \tan \left (f x + e\right )^{3} + 3 \, a b^{2} \tan \left (f x + e\right )^{2} + 3 \, a^{2} b \tan \left (f x + e\right ) + a^{3}\right )} \left (d \cot \left (f x + e\right )\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (f x + e\right ) + a\right )}^{3} \left (d \cot \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.49, size = 0, normalized size = 0.00 \[ \int \left (d \cot \left (f x +e \right )\right )^{n} \left (a +b \tan \left (f x +e \right )\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (f x + e\right ) + a\right )}^{3} \left (d \cot \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cot {\left (e + f x \right )}\right )^{n} \left (a + b \tan {\left (e + f x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________