Optimal. Leaf size=149 \[ \frac {a d \cos (e+f x) (d \csc (e+f x))^{-1+n} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{f (1-n) \sqrt {\cos ^2(e+f x)}}+\frac {b d^2 \cos (e+f x) (d \csc (e+f x))^{-2+n} \, _2F_1\left (\frac {1}{2},\frac {2-n}{2};\frac {4-n}{2};\sin ^2(e+f x)\right )}{f (2-n) \sqrt {\cos ^2(e+f x)}} \]
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Rubi [A]
time = 0.10, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3317, 3872,
3857, 2722} \begin {gather*} \frac {a d \cos (e+f x) (d \csc (e+f x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{f (1-n) \sqrt {\cos ^2(e+f x)}}+\frac {b d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left (\frac {1}{2},\frac {2-n}{2};\frac {4-n}{2};\sin ^2(e+f x)\right )}{f (2-n) \sqrt {\cos ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3317
Rule 3857
Rule 3872
Rubi steps
\begin {align*} \int (d \csc (e+f x))^n (a+b \sin (e+f x)) \, dx &=d \int (d \csc (e+f x))^{-1+n} (b+a \csc (e+f x)) \, dx\\ &=a \int (d \csc (e+f x))^n \, dx+(b d) \int (d \csc (e+f x))^{-1+n} \, dx\\ &=\left (a (d \csc (e+f x))^n \left (\frac {\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{-n} \, dx+\left (b d (d \csc (e+f x))^n \left (\frac {\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{1-n} \, dx\\ &=\frac {a \cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right ) \sin (e+f x)}{f (1-n) \sqrt {\cos ^2(e+f x)}}+\frac {b \cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {2-n}{2};\frac {4-n}{2};\sin ^2(e+f x)\right ) \sin ^2(e+f x)}{f (2-n) \sqrt {\cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 105, normalized size = 0.70 \begin {gather*} -\frac {d \cos (e+f x) (d \csc (e+f x))^{-1+n} \sin ^2(e+f x)^{\frac {1}{2} (-1+n)} \left (a \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3}{2};\cos ^2(e+f x)\right )+b \csc (e+f x) \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {3}{2};\cos ^2(e+f x)\right ) \sqrt {\sin ^2(e+f x)}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \left (d \csc \left (f x +e \right )\right )^{n} \left (a +b \sin \left (f x +e \right )\right )\, dx\]
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 \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 (d \csc {\left (e + f x \right )}\right )^{n} \left (a + b \sin {\left (e + f x \right )}\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 {\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^n\,\left (a+b\,\sin \left (e+f\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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