Optimal. Leaf size=349 \[ -\frac {a^4 \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d (3-2 n) (a-a \cos (c+d x))^2 (a \cos (c+d x)+a)^2}-\frac {a^3 (4-n) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d \left (4 n^2-8 n+3\right ) (a-a \cos (c+d x))^2 (a \cos (c+d x)+a)}+\frac {n \left (-n^2-3 n+7\right ) \sin (c+d x) \cos (c+d x) \left (\frac {\cos (c+d x)+1}{1-\cos (c+d x)}\right )^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \, _2F_1\left (-n-\frac {1}{2},1-n;2-n;-\frac {2 \cos (c+d x)}{1-\cos (c+d x)}\right )}{d (1-2 n) (3-2 n) (1-n) (2 n+1) (1-\cos (c+d x))^2}+\frac {\left (n^2-n+2\right ) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d (3-2 n) \left (1-4 n^2\right ) (1-\cos (c+d x))^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.54, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3876, 2883, 129, 155, 12, 132} \[ -\frac {a^3 (4-n) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d \left (4 n^2-8 n+3\right ) (a-a \cos (c+d x))^2 (a \cos (c+d x)+a)}-\frac {a^4 \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d (3-2 n) (a-a \cos (c+d x))^2 (a \cos (c+d x)+a)^2}+\frac {n \left (-n^2-3 n+7\right ) \sin (c+d x) \cos (c+d x) \left (\frac {\cos (c+d x)+1}{1-\cos (c+d x)}\right )^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \, _2F_1\left (-n-\frac {1}{2},1-n;2-n;-\frac {2 \cos (c+d x)}{1-\cos (c+d x)}\right )}{d (1-2 n) (3-2 n) (1-n) (2 n+1) (1-\cos (c+d x))^2}+\frac {\left (n^2-n+2\right ) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d (3-2 n) \left (1-4 n^2\right ) (1-\cos (c+d x))^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 129
Rule 132
Rule 155
Rule 2883
Rule 3876
Rubi steps
\begin {align*} \int \csc ^4(c+d x) (a+a \sec (c+d x))^n \, dx &=\left ((-\cos (c+d x))^n (-a-a \cos (c+d x))^{-n} (a+a \sec (c+d x))^n\right ) \int (-\cos (c+d x))^{-n} (-a-a \cos (c+d x))^n \csc ^4(c+d x) \, dx\\ &=-\frac {\left (a^6 (-\cos (c+d x))^n (-a-a \cos (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \sin (c+d x)\right ) \operatorname {Subst}\left (\int \frac {(-x)^{-n} (-a-a x)^{-\frac {5}{2}+n}}{(-a+a x)^{5/2}} \, dx,x,\cos (c+d x)\right )}{d \sqrt {-a+a \cos (c+d x)}}\\ &=-\frac {a^4 \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (3-2 n) (a-a \cos (c+d x))^2 (a+a \cos (c+d x))^2}-\frac {\left (a^3 (-\cos (c+d x))^n (-a-a \cos (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \sin (c+d x)\right ) \operatorname {Subst}\left (\int \frac {(-x)^{-n} (-a-a x)^{-\frac {3}{2}+n} \left (-a^2 (2-n)+2 a^2 x\right )}{(-a+a x)^{5/2}} \, dx,x,\cos (c+d x)\right )}{d (3-2 n) \sqrt {-a+a \cos (c+d x)}}\\ &=-\frac {a^4 \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (3-2 n) (a-a \cos (c+d x))^2 (a+a \cos (c+d x))^2}-\frac {a^3 (4-n) \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) (3-2 n) (a-a \cos (c+d x))^2 (a+a \cos (c+d x))}-\frac {\left ((-\cos (c+d x))^n (-a-a \cos (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \sin (c+d x)\right ) \operatorname {Subst}\left (\int \frac {(-x)^{-n} (-a-a x)^{-\frac {1}{2}+n} \left (-a^4 \left (2-n^2\right )-a^4 (4-n) x\right )}{(-a+a x)^{5/2}} \, dx,x,\cos (c+d x)\right )}{d (1-2 n) (3-2 n) \sqrt {-a+a \cos (c+d x)}}\\ &=\frac {\left (2-n+n^2\right ) \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) (3-2 n) (1+2 n) (1-\cos (c+d x))^2}-\frac {a^4 \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (3-2 n) (a-a \cos (c+d x))^2 (a+a \cos (c+d x))^2}-\frac {a^3 (4-n) \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) (3-2 n) (a-a \cos (c+d x))^2 (a+a \cos (c+d x))}-\frac {\left ((-\cos (c+d x))^n (-a-a \cos (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \sin (c+d x)\right ) \operatorname {Subst}\left (\int \frac {a^6 n \left (7-3 n-n^2\right ) (-x)^{-n} (-a-a x)^{\frac {1}{2}+n}}{(-a+a x)^{5/2}} \, dx,x,\cos (c+d x)\right )}{a^3 d (1-2 n) (3-2 n) (1+2 n) \sqrt {-a+a \cos (c+d x)}}\\ &=\frac {\left (2-n+n^2\right ) \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) (3-2 n) (1+2 n) (1-\cos (c+d x))^2}-\frac {a^4 \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (3-2 n) (a-a \cos (c+d x))^2 (a+a \cos (c+d x))^2}-\frac {a^3 (4-n) \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) (3-2 n) (a-a \cos (c+d x))^2 (a+a \cos (c+d x))}-\frac {\left (a^3 n \left (7-3 n-n^2\right ) (-\cos (c+d x))^n (-a-a \cos (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \sin (c+d x)\right ) \operatorname {Subst}\left (\int \frac {(-x)^{-n} (-a-a x)^{\frac {1}{2}+n}}{(-a+a x)^{5/2}} \, dx,x,\cos (c+d x)\right )}{d (1-2 n) (3-2 n) (1+2 n) \sqrt {-a+a \cos (c+d x)}}\\ &=\frac {\left (2-n+n^2\right ) \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) (3-2 n) (1+2 n) (1-\cos (c+d x))^2}-\frac {a^4 \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (3-2 n) (a-a \cos (c+d x))^2 (a+a \cos (c+d x))^2}-\frac {a^3 (4-n) \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) (3-2 n) (a-a \cos (c+d x))^2 (a+a \cos (c+d x))}+\frac {n \left (7-3 n-n^2\right ) \cos (c+d x) \left (\frac {1+\cos (c+d x)}{1-\cos (c+d x)}\right )^{-\frac {1}{2}-n} \, _2F_1\left (-\frac {1}{2}-n,1-n;2-n;-\frac {2 \cos (c+d x)}{1-\cos (c+d x)}\right ) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) (3-2 n) (1-n) (1+2 n) (1-\cos (c+d x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 7.04, size = 350, normalized size = 1.00 \[ \frac {\tan \left (\frac {1}{2} (c+d x)\right ) (a (\sec (c+d x)+1))^n \left (\frac {24 (\sec (c+d x)+1)^{-n} \, _2F_1\left (\frac {1}{2},n;\frac {3}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^n \left (-2 (\sec (c+d x)+1)^n-3\ 2^n \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^n+n \left ((\sec (c+d x)+1)^n+2^{n+1} \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^n\right )\right )-\cos (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (4 n \cos (c+d x)+(n-3) (\cos (2 (c+d x))+3))}{4 (2 n-3)}-2 \cot ^2\left (\frac {1}{2} (c+d x)\right ) (\sec (c+d x)+1)^{-n} \, _2F_1\left (-\frac {1}{2},n;\frac {1}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^n \left (n (\sec (c+d x)+1)^n+2 (\sec (c+d x)+1)^n+3\ 2^n \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^n\right )\right )}{24 d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.39, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{4}\left (d x +c \right )\right ) \left (a +a \sec \left (d x +c \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n}{{\sin \left (c+d\,x\right )}^4} \,d x \]
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]
________________________________________________________________________________________