Optimal. Leaf size=114 \[ \frac {\left (a+b \sec ^2(e+f x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sec ^2(e+f x)}{a}+1\right )}{2 a f (p+1)}-\frac {\left (a+b \sec ^2(e+f x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sec ^2(e+f x)+a}{a+b}\right )}{2 f (p+1) (a+b)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4139, 446, 86, 68, 65} \[ \frac {\left (a+b \sec ^2(e+f x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sec ^2(e+f x)}{a}+1\right )}{2 a f (p+1)}-\frac {\left (a+b \sec ^2(e+f x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sec ^2(e+f x)+a}{a+b}\right )}{2 f (p+1) (a+b)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 68
Rule 86
Rule 446
Rule 4139
Rubi steps
\begin {align*} \int \cot (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{x \left (-1+x^2\right )} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^p}{(-1+x) x} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^p}{-1+x} \, dx,x,\sec ^2(e+f x)\right )}{2 f}-\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=-\frac {\, _2F_1\left (1,1+p;2+p;\frac {a+b \sec ^2(e+f x)}{a+b}\right ) \left (a+b \sec ^2(e+f x)\right )^{1+p}}{2 (a+b) f (1+p)}+\frac {\, _2F_1\left (1,1+p;2+p;1+\frac {b \sec ^2(e+f x)}{a}\right ) \left (a+b \sec ^2(e+f x)\right )^{1+p}}{2 a f (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.13, size = 115, normalized size = 1.01 \[ \frac {\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (a+b \sec ^2(e+f x)\right )^p \left ((a+b) \, _2F_1\left (1,p+1;p+2;\frac {b \tan ^2(e+f x)+a+b}{a}\right )-a \, _2F_1\left (1,p+1;p+2;\frac {b \tan ^2(e+f x)}{a+b}+1\right )\right )}{4 a f (p+1) (a+b)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \cot \left (f x + e\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \cot \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 2.03, size = 0, normalized size = 0.00 \[ \int \cot \left (f x +e \right ) \left (a +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \cot \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {cot}\left (e+f\,x\right )\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________