Optimal. Leaf size=487 \[ \frac {3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left (\frac {b \sin (c+d x)}{a}+1\right )^{-p} F_1\left (n+1;-p,1;n+2;-\frac {b \sin (c+d x)}{a},-\sin (c+d x)\right )}{16 d (n+1)}+\frac {3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left (\frac {b \sin (c+d x)}{a}+1\right )^{-p} F_1\left (n+1;-p,1;n+2;-\frac {b \sin (c+d x)}{a},\sin (c+d x)\right )}{16 d (n+1)}+\frac {3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left (\frac {b \sin (c+d x)}{a}+1\right )^{-p} F_1\left (n+1;-p,2;n+2;-\frac {b \sin (c+d x)}{a},-\sin (c+d x)\right )}{16 d (n+1)}+\frac {3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left (\frac {b \sin (c+d x)}{a}+1\right )^{-p} F_1\left (n+1;-p,2;n+2;-\frac {b \sin (c+d x)}{a},\sin (c+d x)\right )}{16 d (n+1)}+\frac {\sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left (\frac {b \sin (c+d x)}{a}+1\right )^{-p} F_1\left (n+1;-p,3;n+2;-\frac {b \sin (c+d x)}{a},-\sin (c+d x)\right )}{8 d (n+1)}+\frac {\sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left (\frac {b \sin (c+d x)}{a}+1\right )^{-p} F_1\left (n+1;-p,3;n+2;-\frac {b \sin (c+d x)}{a},\sin (c+d x)\right )}{8 d (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.57, antiderivative size = 487, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2837, 961, 135, 133, 912} \[ \frac {3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left (\frac {b \sin (c+d x)}{a}+1\right )^{-p} F_1\left (n+1;-p,1;n+2;-\frac {b \sin (c+d x)}{a},-\sin (c+d x)\right )}{16 d (n+1)}+\frac {3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left (\frac {b \sin (c+d x)}{a}+1\right )^{-p} F_1\left (n+1;-p,1;n+2;-\frac {b \sin (c+d x)}{a},\sin (c+d x)\right )}{16 d (n+1)}+\frac {3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left (\frac {b \sin (c+d x)}{a}+1\right )^{-p} F_1\left (n+1;-p,2;n+2;-\frac {b \sin (c+d x)}{a},-\sin (c+d x)\right )}{16 d (n+1)}+\frac {3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left (\frac {b \sin (c+d x)}{a}+1\right )^{-p} F_1\left (n+1;-p,2;n+2;-\frac {b \sin (c+d x)}{a},\sin (c+d x)\right )}{16 d (n+1)}+\frac {\sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left (\frac {b \sin (c+d x)}{a}+1\right )^{-p} F_1\left (n+1;-p,3;n+2;-\frac {b \sin (c+d x)}{a},-\sin (c+d x)\right )}{8 d (n+1)}+\frac {\sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left (\frac {b \sin (c+d x)}{a}+1\right )^{-p} F_1\left (n+1;-p,3;n+2;-\frac {b \sin (c+d x)}{a},\sin (c+d x)\right )}{8 d (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 133
Rule 135
Rule 912
Rule 961
Rule 2837
Rubi steps
\begin {align*} \int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n (a+x)^p}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b^5 \operatorname {Subst}\left (\int \left (\frac {\left (\frac {x}{b}\right )^n (a+x)^p}{8 b^3 (b-x)^3}+\frac {3 \left (\frac {x}{b}\right )^n (a+x)^p}{16 b^4 (b-x)^2}+\frac {\left (\frac {x}{b}\right )^n (a+x)^p}{8 b^3 (b+x)^3}+\frac {3 \left (\frac {x}{b}\right )^n (a+x)^p}{16 b^4 (b+x)^2}+\frac {3 \left (\frac {x}{b}\right )^n (a+x)^p}{8 b^4 \left (b^2-x^2\right )}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {(3 b) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n (a+x)^p}{(b-x)^2} \, dx,x,b \sin (c+d x)\right )}{16 d}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n (a+x)^p}{(b+x)^2} \, dx,x,b \sin (c+d x)\right )}{16 d}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n (a+x)^p}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 d}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n (a+x)^p}{(b-x)^3} \, dx,x,b \sin (c+d x)\right )}{8 d}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n (a+x)^p}{(b+x)^3} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac {(3 b) \operatorname {Subst}\left (\int \left (\frac {\left (\frac {x}{b}\right )^n (a+x)^p}{2 b (b-x)}+\frac {\left (\frac {x}{b}\right )^n (a+x)^p}{2 b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 d}+\frac {\left (3 b (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n \left (1+\frac {x}{a}\right )^p}{(b-x)^2} \, dx,x,b \sin (c+d x)\right )}{16 d}+\frac {\left (3 b (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n \left (1+\frac {x}{a}\right )^p}{(b+x)^2} \, dx,x,b \sin (c+d x)\right )}{16 d}+\frac {\left (b^2 (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n \left (1+\frac {x}{a}\right )^p}{(b-x)^3} \, dx,x,b \sin (c+d x)\right )}{8 d}+\frac {\left (b^2 (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n \left (1+\frac {x}{a}\right )^p}{(b+x)^3} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac {3 F_1\left (1+n;-p,2;2+n;-\frac {b \sin (c+d x)}{a},-\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{16 d (1+n)}+\frac {3 F_1\left (1+n;-p,2;2+n;-\frac {b \sin (c+d x)}{a},\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{16 d (1+n)}+\frac {F_1\left (1+n;-p,3;2+n;-\frac {b \sin (c+d x)}{a},-\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{8 d (1+n)}+\frac {F_1\left (1+n;-p,3;2+n;-\frac {b \sin (c+d x)}{a},\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{8 d (1+n)}+\frac {3 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n (a+x)^p}{b-x} \, dx,x,b \sin (c+d x)\right )}{16 d}+\frac {3 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n (a+x)^p}{b+x} \, dx,x,b \sin (c+d x)\right )}{16 d}\\ &=\frac {3 F_1\left (1+n;-p,2;2+n;-\frac {b \sin (c+d x)}{a},-\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{16 d (1+n)}+\frac {3 F_1\left (1+n;-p,2;2+n;-\frac {b \sin (c+d x)}{a},\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{16 d (1+n)}+\frac {F_1\left (1+n;-p,3;2+n;-\frac {b \sin (c+d x)}{a},-\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{8 d (1+n)}+\frac {F_1\left (1+n;-p,3;2+n;-\frac {b \sin (c+d x)}{a},\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{8 d (1+n)}+\frac {\left (3 (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n \left (1+\frac {x}{a}\right )^p}{b-x} \, dx,x,b \sin (c+d x)\right )}{16 d}+\frac {\left (3 (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n \left (1+\frac {x}{a}\right )^p}{b+x} \, dx,x,b \sin (c+d x)\right )}{16 d}\\ &=\frac {3 F_1\left (1+n;-p,1;2+n;-\frac {b \sin (c+d x)}{a},-\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{16 d (1+n)}+\frac {3 F_1\left (1+n;-p,1;2+n;-\frac {b \sin (c+d x)}{a},\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{16 d (1+n)}+\frac {3 F_1\left (1+n;-p,2;2+n;-\frac {b \sin (c+d x)}{a},-\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{16 d (1+n)}+\frac {3 F_1\left (1+n;-p,2;2+n;-\frac {b \sin (c+d x)}{a},\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{16 d (1+n)}+\frac {F_1\left (1+n;-p,3;2+n;-\frac {b \sin (c+d x)}{a},-\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{8 d (1+n)}+\frac {F_1\left (1+n;-p,3;2+n;-\frac {b \sin (c+d x)}{a},\sin (c+d x)\right ) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^p \left (1+\frac {b \sin (c+d x)}{a}\right )^{-p}}{8 d (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 15.01, size = 0, normalized size = 0.00 \[ \int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{p} \sin \left (d x + c\right )^{n} \sec \left (d x + c\right )^{5}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{p} \sin \left (d x + c\right )^{n} \sec \left (d x + c\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 2.26, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{5}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [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]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\sin \left (c+d\,x\right )}^n\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^p}{{\cos \left (c+d\,x\right )}^5} \,d x \]
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]
________________________________________________________________________________________