Optimal. Leaf size=115 \[ \frac {(a+b \sin (c+d x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {a+b \sin (c+d x)}{a+b}\right )}{2 d (m+1) (a+b)}-\frac {(a+b \sin (c+d x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {a+b \sin (c+d x)}{a-b}\right )}{2 d (m+1) (a-b)} \]
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Rubi [A] time = 0.11, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2668, 712, 68} \[ \frac {(a+b \sin (c+d x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {a+b \sin (c+d x)}{a+b}\right )}{2 d (m+1) (a+b)}-\frac {(a+b \sin (c+d x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {a+b \sin (c+d x)}{a-b}\right )}{2 d (m+1) (a-b)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 712
Rule 2668
Rubi steps
\begin {align*} \int \sec (c+d x) (a+b \sin (c+d x))^m \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {(a+x)^m}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \left (\frac {(a+x)^m}{2 b (b-x)}+\frac {(a+x)^m}{2 b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+x)^m}{b-x} \, dx,x,b \sin (c+d x)\right )}{2 d}+\frac {\operatorname {Subst}\left (\int \frac {(a+x)^m}{b+x} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=-\frac {\, _2F_1\left (1,1+m;2+m;\frac {a+b \sin (c+d x)}{a-b}\right ) (a+b \sin (c+d x))^{1+m}}{2 (a-b) d (1+m)}+\frac {\, _2F_1\left (1,1+m;2+m;\frac {a+b \sin (c+d x)}{a+b}\right ) (a+b \sin (c+d x))^{1+m}}{2 (a+b) d (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 99, normalized size = 0.86 \[ -\frac {(a+b \sin (c+d x))^{m+1} \left ((a+b) \, _2F_1\left (1,m+1;m+2;\frac {a+b \sin (c+d x)}{a-b}\right )+(b-a) \, _2F_1\left (1,m+1;m+2;\frac {a+b \sin (c+d x)}{a+b}\right )\right )}{2 d (m+1) (a-b) (a+b)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right ), x\right ) \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.01, size = 0, normalized size = 0.00 \[ \int \sec \left (d x +c \right ) \left (a +b \sin \left (d x +c \right )\right )^{m}\, dx \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \sec \left (d x + c\right )\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m}{\cos \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sin {\left (c + d x \right )}\right )^{m} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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