Optimal. Leaf size=26 \[ \text {Int}\left (\frac {(a+b \sec (c+d x))^n}{\sqrt {\sin (c+d x)}},x\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a+b \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx &=\int \frac {(a+b \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx\\ \end {align*}
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Mathematica [A] time = 2.75, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\sin \left (d x + c\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\sin \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.82, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (d x +c \right )\right )^{n}}{\sqrt {\sin \left (d x +c \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\sin \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{\sqrt {\sin \left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \sec {\left (c + d x \right )}\right )^{n}}{\sqrt {\sin {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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