Optimal. Leaf size=115 \[ -\frac {\, _2F_1\left (1,1+m;2+m;\frac {a+b \sin (c+d x)}{a-b}\right ) (a+b \sin (c+d x))^{1+m}}{2 (a-b) d (1+m)}+\frac {\, _2F_1\left (1,1+m;2+m;\frac {a+b \sin (c+d x)}{a+b}\right ) (a+b \sin (c+d x))^{1+m}}{2 (a+b) d (1+m)} \]
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Rubi [A]
time = 0.08, 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 = {2747, 726, 70}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 726
Rule 2747
Rubi steps
\begin {align*} \int \sec (c+d x) (a+b \sin (c+d x))^m \, dx &=\frac {b \text {Subst}\left (\int \frac {(a+x)^m}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b \text {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 {\text {Subst}\left (\int \frac {(a+x)^m}{b-x} \, dx,x,b \sin (c+d x)\right )}{2 d}+\frac {\text {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 = 0.08, size = 99, normalized size = 0.86 \begin {gather*} -\frac {\left ((a+b) \, _2F_1\left (1,1+m;2+m;\frac {a+b \sin (c+d x)}{a-b}\right )+(-a+b) \, _2F_1\left (1,1+m;2+m;\frac {a+b \sin (c+d x)}{a+b}\right )\right ) (a+b \sin (c+d x))^{1+m}}{2 (a-b) (a+b) d (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a + b \sin {\left (c + d x \right )}\right )^{m} \sec {\left (c + d x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m}{\cos \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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