Optimal. Leaf size=360 \[ \frac {\left (3 a^2-9 a b+8 b^2\right ) \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{16 d (n+1) (a-b)^3}+\frac {\left (3 a^2+9 a b+8 b^2\right ) \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;\sin (c+d x))}{16 d (n+1) (a+b)^3}-\frac {b^6 \sin ^{n+1}(c+d x) \, _2F_1\left (1,n+1;n+2;-\frac {b \sin (c+d x)}{a}\right )}{a d (n+1) \left (a^2-b^2\right )^3}+\frac {(3 a-5 b) \sin ^{n+1}(c+d x) \, _2F_1(2,n+1;n+2;-\sin (c+d x))}{16 d (n+1) (a-b)^2}+\frac {(3 a+5 b) \sin ^{n+1}(c+d x) \, _2F_1(2,n+1;n+2;\sin (c+d x))}{16 d (n+1) (a+b)^2}+\frac {\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;-\sin (c+d x))}{8 d (n+1) (a-b)}+\frac {\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;\sin (c+d x))}{8 d (n+1) (a+b)} \]
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Rubi [A] time = 0.54, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2837, 961, 64} \[ \frac {\left (3 a^2-9 a b+8 b^2\right ) \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{16 d (n+1) (a-b)^3}+\frac {\left (3 a^2+9 a b+8 b^2\right ) \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;\sin (c+d x))}{16 d (n+1) (a+b)^3}-\frac {b^6 \sin ^{n+1}(c+d x) \, _2F_1\left (1,n+1;n+2;-\frac {b \sin (c+d x)}{a}\right )}{a d (n+1) \left (a^2-b^2\right )^3}+\frac {(3 a-5 b) \sin ^{n+1}(c+d x) \, _2F_1(2,n+1;n+2;-\sin (c+d x))}{16 d (n+1) (a-b)^2}+\frac {(3 a+5 b) \sin ^{n+1}(c+d x) \, _2F_1(2,n+1;n+2;\sin (c+d x))}{16 d (n+1) (a+b)^2}+\frac {\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;-\sin (c+d x))}{8 d (n+1) (a-b)}+\frac {\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;\sin (c+d x))}{8 d (n+1) (a+b)} \]
Antiderivative was successfully verified.
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Rule 64
Rule 961
Rule 2837
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x) \sin ^n(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n}{(a+x) \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}{8 b^3 (a+b) (b-x)^3}+\frac {(3 a+5 b) \left (\frac {x}{b}\right )^n}{16 b^4 (a+b)^2 (b-x)^2}+\frac {\left (3 a^2+9 a b+8 b^2\right ) \left (\frac {x}{b}\right )^n}{16 b^5 (a+b)^3 (b-x)}-\frac {\left (\frac {x}{b}\right )^n}{(a-b)^3 (a+b)^3 (a+x)}-\frac {\left (\frac {x}{b}\right )^n}{8 b^3 (-a+b) (b+x)^3}+\frac {(3 a-5 b) \left (\frac {x}{b}\right )^n}{16 (a-b)^2 b^4 (b+x)^2}+\frac {\left (3 a^2-9 a b+8 b^2\right ) \left (\frac {x}{b}\right )^n}{16 (a-b)^3 b^5 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {((3 a-5 b) b) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n}{(b+x)^2} \, dx,x,b \sin (c+d x)\right )}{16 (a-b)^2 d}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n}{(b+x)^3} \, dx,x,b \sin (c+d x)\right )}{8 (a-b) d}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n}{(b-x)^3} \, dx,x,b \sin (c+d x)\right )}{8 (a+b) d}+\frac {(b (3 a+5 b)) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n}{(b-x)^2} \, dx,x,b \sin (c+d x)\right )}{16 (a+b)^2 d}-\frac {b^5 \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n}{a+x} \, dx,x,b \sin (c+d x)\right )}{\left (a^2-b^2\right )^3 d}+\frac {\left (3 a^2-9 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n}{b+x} \, dx,x,b \sin (c+d x)\right )}{16 (a-b)^3 d}+\frac {\left (3 a^2+9 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {x}{b}\right )^n}{b-x} \, dx,x,b \sin (c+d x)\right )}{16 (a+b)^3 d}\\ &=\frac {\left (3 a^2-9 a b+8 b^2\right ) \, _2F_1(1,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{16 (a-b)^3 d (1+n)}+\frac {\left (3 a^2+9 a b+8 b^2\right ) \, _2F_1(1,1+n;2+n;\sin (c+d x)) \sin ^{1+n}(c+d x)}{16 (a+b)^3 d (1+n)}-\frac {b^6 \, _2F_1\left (1,1+n;2+n;-\frac {b \sin (c+d x)}{a}\right ) \sin ^{1+n}(c+d x)}{a \left (a^2-b^2\right )^3 d (1+n)}+\frac {(3 a-5 b) \, _2F_1(2,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{16 (a-b)^2 d (1+n)}+\frac {(3 a+5 b) \, _2F_1(2,1+n;2+n;\sin (c+d x)) \sin ^{1+n}(c+d x)}{16 (a+b)^2 d (1+n)}+\frac {\, _2F_1(3,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{8 (a-b) d (1+n)}+\frac {\, _2F_1(3,1+n;2+n;\sin (c+d x)) \sin ^{1+n}(c+d x)}{8 (a+b) d (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 241, normalized size = 0.67 \[ \frac {\sin ^{n+1}(c+d x) \left (\frac {\left (3 a^2-9 a b+8 b^2\right ) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{(a-b)^3}+\frac {\left (3 a^2+9 a b+8 b^2\right ) \, _2F_1(1,n+1;n+2;\sin (c+d x))}{(a+b)^3}-\frac {16 b^6 \, _2F_1\left (1,n+1;n+2;-\frac {b \sin (c+d x)}{a}\right )}{a (a-b)^3 (a+b)^3}+\frac {(3 a-5 b) \, _2F_1(2,n+1;n+2;-\sin (c+d x))}{(a-b)^2}+\frac {(3 a+5 b) \, _2F_1(2,n+1;n+2;\sin (c+d x))}{(a+b)^2}+\frac {2 \, _2F_1(3,n+1;n+2;-\sin (c+d x))}{a-b}+\frac {2 \, _2F_1(3,n+1;n+2;\sin (c+d x))}{a+b}\right )}{16 d (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sin \left (d x + c\right )^{n} \sec \left (d x + c\right )^{5}}{b \sin \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )^{n} \sec \left (d x + c\right )^{5}}{b \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.91, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sec ^{5}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )^{n} \sec \left (d x + c\right )^{5}}{b \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\sin \left (c+d\,x\right )}^n}{{\cos \left (c+d\,x\right )}^5\,\left (a+b\,\sin \left (c+d\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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