Optimal. Leaf size=259 \[ \frac {2^{n+\frac {1}{2}} \tan (c+d x) (A (m+n+1)-B (m+n+1)+C (m-n)) (\sec (c+d x)+1)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n F_1\left (\frac {1}{2};1-m,\frac {1}{2}-n;\frac {3}{2};1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right )}{d (m+n+1)}+\frac {2^{n+\frac {3}{2}} (B (m+n+1)+C n) \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n F_1\left (\frac {1}{2};1-m,-n-\frac {1}{2};\frac {3}{2};1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right )}{d (m+n+1)}+\frac {C \sin (c+d x) \sec ^{m+1}(c+d x) (a \sec (c+d x)+a)^n}{d (m+n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.61, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {4088, 4023, 3828, 3825, 133} \[ \frac {2^{n+\frac {1}{2}} \tan (c+d x) (A (m+n+1)-B (m+n+1)+C (m-n)) (\sec (c+d x)+1)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n F_1\left (\frac {1}{2};1-m,\frac {1}{2}-n;\frac {3}{2};1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right )}{d (m+n+1)}+\frac {2^{n+\frac {3}{2}} (B (m+n+1)+C n) \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n F_1\left (\frac {1}{2};1-m,-n-\frac {1}{2};\frac {3}{2};1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right )}{d (m+n+1)}+\frac {C \sin (c+d x) \sec ^{m+1}(c+d x) (a \sec (c+d x)+a)^n}{d (m+n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 133
Rule 3825
Rule 3828
Rule 4023
Rule 4088
Rubi steps
\begin {align*} \int \sec ^m(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {C \sec ^{1+m}(c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1+m+n)}+\frac {\int \sec ^m(c+d x) (a+a \sec (c+d x))^n (a (C m+A (1+m+n))+a (C n+B (1+m+n)) \sec (c+d x)) \, dx}{a (1+m+n)}\\ &=\frac {C \sec ^{1+m}(c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1+m+n)}+\left (A-B+\frac {C (m-n)}{1+m+n}\right ) \int \sec ^m(c+d x) (a+a \sec (c+d x))^n \, dx+\frac {(C n+B (1+m+n)) \int \sec ^m(c+d x) (a+a \sec (c+d x))^{1+n} \, dx}{a (1+m+n)}\\ &=\frac {C \sec ^{1+m}(c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1+m+n)}+\left (\left (A-B+\frac {C (m-n)}{1+m+n}\right ) (1+\sec (c+d x))^{-n} (a+a \sec (c+d x))^n\right ) \int \sec ^m(c+d x) (1+\sec (c+d x))^n \, dx+\frac {\left ((C n+B (1+m+n)) (1+\sec (c+d x))^{-n} (a+a \sec (c+d x))^n\right ) \int \sec ^m(c+d x) (1+\sec (c+d x))^{1+n} \, dx}{1+m+n}\\ &=\frac {C \sec ^{1+m}(c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1+m+n)}+\frac {\left (\left (A-B+\frac {C (m-n)}{1+m+n}\right ) (1+\sec (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {(1-x)^{-1+m} (2-x)^{-\frac {1}{2}+n}}{\sqrt {x}} \, dx,x,1-\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)}}+\frac {\left ((C n+B (1+m+n)) (1+\sec (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {(1-x)^{-1+m} (2-x)^{\frac {1}{2}+n}}{\sqrt {x}} \, dx,x,1-\sec (c+d x)\right )}{d (1+m+n) \sqrt {1-\sec (c+d x)}}\\ &=\frac {C \sec ^{1+m}(c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1+m+n)}+\frac {2^{\frac {3}{2}+n} (C n+B (1+m+n)) F_1\left (\frac {1}{2};1-m,-\frac {1}{2}-n;\frac {3}{2};1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right ) (1+\sec (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \tan (c+d x)}{d (1+m+n)}+\frac {2^{\frac {1}{2}+n} \left (A-B+\frac {C (m-n)}{1+m+n}\right ) F_1\left (\frac {1}{2};1-m,\frac {1}{2}-n;\frac {3}{2};1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right ) (1+\sec (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \tan (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 4.32, size = 0, normalized size = 0.00 \[ \int \sec ^m(c+d x) (a+a \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 3.59, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{m}\left (d x +c \right )\right ) \left (a +a \sec \left (d x +c \right )\right )^{n} \left (A +B \sec \left (d x +c \right )+C \left (\sec ^{2}\left (d x +c \right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
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]
________________________________________________________________________________________