Optimal. Leaf size=150 \[ \frac {b^5 (a+b \sec (c+d x))^{n+1} \, _2F_1\left (6,n+1;n+2;\frac {b \sec (c+d x)}{a}+1\right )}{a^6 d (n+1)}-\frac {2 b^3 (a+b \sec (c+d x))^{n+1} \, _2F_1\left (4,n+1;n+2;\frac {b \sec (c+d x)}{a}+1\right )}{a^4 d (n+1)}+\frac {b (a+b \sec (c+d x))^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {b \sec (c+d x)}{a}+1\right )}{a^2 d (n+1)} \]
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Rubi [A] time = 0.13, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3874, 180, 65} \[ -\frac {2 b^3 (a+b \sec (c+d x))^{n+1} \, _2F_1\left (4,n+1;n+2;\frac {b \sec (c+d x)}{a}+1\right )}{a^4 d (n+1)}+\frac {b^5 (a+b \sec (c+d x))^{n+1} \, _2F_1\left (6,n+1;n+2;\frac {b \sec (c+d x)}{a}+1\right )}{a^6 d (n+1)}+\frac {b (a+b \sec (c+d x))^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {b \sec (c+d x)}{a}+1\right )}{a^2 d (n+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 180
Rule 3874
Rubi steps
\begin {align*} \int (a+b \sec (c+d x))^n \sin ^5(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(-1+x)^2 (1+x)^2 (a-b x)^n}{x^6} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {(a-b x)^n}{x^6}-\frac {2 (a-b x)^n}{x^4}+\frac {(a-b x)^n}{x^2}\right ) \, dx,x,-\sec (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(a-b x)^n}{x^6} \, dx,x,-\sec (c+d x)\right )}{d}-\frac {\operatorname {Subst}\left (\int \frac {(a-b x)^n}{x^2} \, dx,x,-\sec (c+d x)\right )}{d}+\frac {2 \operatorname {Subst}\left (\int \frac {(a-b x)^n}{x^4} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=\frac {b \, _2F_1\left (2,1+n;2+n;1+\frac {b \sec (c+d x)}{a}\right ) (a+b \sec (c+d x))^{1+n}}{a^2 d (1+n)}-\frac {2 b^3 \, _2F_1\left (4,1+n;2+n;1+\frac {b \sec (c+d x)}{a}\right ) (a+b \sec (c+d x))^{1+n}}{a^4 d (1+n)}+\frac {b^5 \, _2F_1\left (6,1+n;2+n;1+\frac {b \sec (c+d x)}{a}\right ) (a+b \sec (c+d x))^{1+n}}{a^6 d (1+n)}\\ \end {align*}
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Mathematica [B] time = 8.93, size = 562, normalized size = 3.75 \[ -\frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \cos (c+d x) (a+b \sec (c+d x))^n \left (192 a^3 (n-1) (a \cos (c+d x)+b)^2-240 a^3 (n-1) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)^2+a (1-n) \left (96 a^2+4 a b (6-4 n)-4 b^2 \left (n^2-7 n+12\right )\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)^2+40 a^2 (n-1) (2 a-b (n-3)) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)^2-24 a^2 (n-1) (2 a-b (n-4)) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)^2-10 a \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left ((n-1) \left (-14 a^2+2 a b (n-1)+b^2 \left (n^2-5 n+6\right )\right ) (a \cos (c+d x)+b)^2+b \left (24 a^3+12 a^2 b (n-1)-4 a b^2 \left (n^2-3 n+2\right )-b^3 \left (n^3-6 n^2+11 n-6\right )\right ) \, _2F_1\left (2,1-n;2-n;\frac {a \cos (c+d x)}{b+a \cos (c+d x)}\right )\right )+\sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (b \left (120 a^4+120 a^3 b (n-1)-10 a b^3 \left (n^3-6 n^2+11 n-6\right )-b^4 \left (n^4-10 n^3+35 n^2-50 n+24\right )\right ) \, _2F_1\left (2,1-n;2-n;\frac {a \cos (c+d x)}{b+a \cos (c+d x)}\right )+(n-1) \left (-84 a^3+2 a^2 b (18-7 n)+4 a b^2 \left (2 n^2-9 n+9\right )+b^3 \left (n^3-9 n^2+26 n-24\right )\right ) (a \cos (c+d x)+b)^2\right )\right )}{120 a^4 d (n-1) (a \cos (c+d x)+b)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right ), x\right ) \]
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 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.72, size = 0, normalized size = 0.00 \[ \int \left (a +b \sec \left (d x +c \right )\right )^{n} \left (\sin ^{5}\left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (c+d\,x\right )}^5\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
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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