Optimal. Leaf size=105 \[ -\frac {\sqrt [4]{1-\cos (c+d x)} \cos (c+d x) (\cos (c+d x)+1)^{\frac {1}{4}-n} (a \sec (c+d x)+a)^n F_1\left (1-n;\frac {1}{4},\frac {1}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right )}{d (1-n) \sqrt {\sin (c+d x)}} \]
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Rubi [A] time = 0.27, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3876, 2886, 135, 133} \[ -\frac {\sqrt [4]{1-\cos (c+d x)} \cos (c+d x) (\cos (c+d x)+1)^{\frac {1}{4}-n} (a \sec (c+d x)+a)^n F_1\left (1-n;\frac {1}{4},\frac {1}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right )}{d (1-n) \sqrt {\sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 2886
Rule 3876
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^n \sqrt {\sin (c+d x)} \, dx &=\left ((-\cos (c+d x))^n (-a-a \cos (c+d x))^{-n} (a+a \sec (c+d x))^n\right ) \int (-\cos (c+d x))^{-n} (-a-a \cos (c+d x))^n \sqrt {\sin (c+d x)} \, dx\\ &=-\frac {\left ((-\cos (c+d x))^n (-a-a \cos (c+d x))^{\frac {1}{4}-n} \sqrt [4]{-a+a \cos (c+d x)} (a+a \sec (c+d x))^n\right ) \operatorname {Subst}\left (\int \frac {(-x)^{-n} (-a-a x)^{-\frac {1}{4}+n}}{\sqrt [4]{-a+a x}} \, dx,x,\cos (c+d x)\right )}{d \sqrt {\sin (c+d x)}}\\ &=-\frac {\left ((-\cos (c+d x))^n (1+\cos (c+d x))^{\frac {1}{4}-n} \sqrt [4]{-a+a \cos (c+d x)} (a+a \sec (c+d x))^n\right ) \operatorname {Subst}\left (\int \frac {(-x)^{-n} (1+x)^{-\frac {1}{4}+n}}{\sqrt [4]{-a+a x}} \, dx,x,\cos (c+d x)\right )}{d \sqrt {\sin (c+d x)}}\\ &=-\frac {\left (\sqrt [4]{1-\cos (c+d x)} (-\cos (c+d x))^n (1+\cos (c+d x))^{\frac {1}{4}-n} (a+a \sec (c+d x))^n\right ) \operatorname {Subst}\left (\int \frac {(-x)^{-n} (1+x)^{-\frac {1}{4}+n}}{\sqrt [4]{1-x}} \, dx,x,\cos (c+d x)\right )}{d \sqrt {\sin (c+d x)}}\\ &=-\frac {F_1\left (1-n;\frac {1}{4},\frac {1}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right ) \sqrt [4]{1-\cos (c+d x)} \cos (c+d x) (1+\cos (c+d x))^{\frac {1}{4}-n} (a+a \sec (c+d x))^n}{d (1-n) \sqrt {\sin (c+d x)}}\\ \end {align*}
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Mathematica [B] time = 1.44, size = 214, normalized size = 2.04 \[ \frac {14 \sin ^{\frac {3}{2}}(c+d x) (\cos (c+d x)+1) (a (\sec (c+d x)+1))^n F_1\left (\frac {3}{4};n,\frac {3}{2};\frac {7}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{d \left (6 (\cos (c+d x)-1) \left (3 F_1\left (\frac {7}{4};n,\frac {5}{2};\frac {11}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-2 n F_1\left (\frac {7}{4};n+1,\frac {3}{2};\frac {11}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )+21 (\cos (c+d x)+1) F_1\left (\frac {3}{4};n,\frac {3}{2};\frac {7}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sqrt {\sin \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sqrt {\sin \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.79, size = 0, normalized size = 0.00 \[ \int \left (a +a \sec \left (d x +c \right )\right )^{n} \left (\sqrt {\sin }\left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sqrt {\sin \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {\sin \left (c+d\,x\right )}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \sqrt {\sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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