Optimal. Leaf size=230 \[ -\frac {F_1\left (1-n;-\frac {1}{2},\frac {1}{2}-n;2-n;\cos (c+d x),-\cos (c+d x)\right ) (1+\cos (c+d x))^{\frac {1}{2}-n} (n-n \cos (c+d x)) \cot (c+d x) (a+a \sec (c+d x))^n}{d (1-n) \sqrt {1-\cos (c+d x)}}-\frac {\cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d}+\frac {2^{\frac {1}{2}+n} F_1\left (\frac {1}{2};-4+n,\frac {1}{2}-n;\frac {3}{2};1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right ) \cos ^n(c+d x) (1+\cos (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \sin (c+d x)}{d} \]
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Rubi [A]
time = 0.47, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3961, 2960,
2866, 2865, 2864, 138, 3125, 3087, 140} \begin {gather*} \frac {2^{n+\frac {1}{2}} \sin (c+d x) \cos ^n(c+d x) (\cos (c+d x)+1)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n F_1\left (\frac {1}{2};n-4,\frac {1}{2}-n;\frac {3}{2};1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right )}{d}-\frac {\cot (c+d x) (n-n \cos (c+d x)) (\cos (c+d x)+1)^{\frac {1}{2}-n} (a \sec (c+d x)+a)^n F_1\left (1-n;-\frac {1}{2},\frac {1}{2}-n;2-n;\cos (c+d x),-\cos (c+d x)\right )}{d (1-n) \sqrt {1-\cos (c+d x)}}-\frac {\sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 140
Rule 2864
Rule 2865
Rule 2866
Rule 2960
Rule 3087
Rule 3125
Rule 3961
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^n \sin ^4(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 \sin ^4(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))^{4-n} (-a-a \cos (c+d x))^n \, 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 \left (1-2 \cos ^2(c+d x)\right ) \, dx\\ &=-\frac {\cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d}+\left ((-\cos (c+d x))^n (1+\cos (c+d x))^{-n} (a+a \sec (c+d x))^n\right ) \int (-\cos (c+d x))^{4-n} (1+\cos (c+d x))^n \, dx+\frac {\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 (2 a n-2 a n \cos (c+d x)) \, dx}{2 a}\\ &=-\frac {\cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d}+\left (\cos ^n(c+d x) (1+\cos (c+d x))^{-n} (a+a \sec (c+d x))^n\right ) \int \cos ^{4-n}(c+d x) (1+\cos (c+d x))^n \, dx-\frac {\left ((-\cos (c+d x))^n (-a-a \cos (c+d x))^{\frac {1}{2}-n} \sqrt {2 a n-2 a n \cos (c+d x)} \csc (c+d x) (a+a \sec (c+d x))^n\right ) \text {Subst}\left (\int (-x)^{-n} (-a-a x)^{-\frac {1}{2}+n} \sqrt {2 a n-2 a n x} \, dx,x,\cos (c+d x)\right )}{2 a d}\\ &=-\frac {\cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d}-\frac {\left ((-\cos (c+d x))^n (1+\cos (c+d x))^{\frac {1}{2}-n} \sqrt {2 a n-2 a n \cos (c+d x)} \csc (c+d x) (a+a \sec (c+d x))^n\right ) \text {Subst}\left (\int (-x)^{-n} (1+x)^{-\frac {1}{2}+n} \sqrt {2 a n-2 a n x} \, dx,x,\cos (c+d x)\right )}{2 a d}+\frac {\left (\cos ^n(c+d x) (1+\cos (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \sin (c+d x)\right ) \text {Subst}\left (\int \frac {(1-x)^{4-n} (2-x)^{-\frac {1}{2}+n}}{\sqrt {x}} \, dx,x,1-\cos (c+d x)\right )}{d \sqrt {1-\cos (c+d x)}}\\ &=-\frac {\cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d}+\frac {2^{\frac {1}{2}+n} F_1\left (\frac {1}{2};-4+n,\frac {1}{2}-n;\frac {3}{2};1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right ) \cos ^n(c+d x) (1+\cos (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \sin (c+d x)}{d}-\frac {\left ((-\cos (c+d x))^n (1+\cos (c+d x))^{\frac {1}{2}-n} (2 a n-2 a n \cos (c+d x)) \csc (c+d x) (a+a \sec (c+d x))^n\right ) \text {Subst}\left (\int \sqrt {1-x} (-x)^{-n} (1+x)^{-\frac {1}{2}+n} \, dx,x,\cos (c+d x)\right )}{2 a d \sqrt {1-\cos (c+d x)}}\\ &=-\frac {F_1\left (1-n;-\frac {1}{2},\frac {1}{2}-n;2-n;\cos (c+d x),-\cos (c+d x)\right ) (1+\cos (c+d x))^{\frac {1}{2}-n} (n-n \cos (c+d x)) \cot (c+d x) (a+a \sec (c+d x))^n}{d (1-n) \sqrt {1-\cos (c+d x)}}-\frac {\cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d}+\frac {2^{\frac {1}{2}+n} F_1\left (\frac {1}{2};-4+n,\frac {1}{2}-n;\frac {3}{2};1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right ) \cos ^n(c+d x) (1+\cos (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 20.64, size = 7069, normalized size = 30.73 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \left (a +a \sec \left (d x +c \right )\right )^{n} \left (\sin ^{4}\left (d x +c \right )\right )\, dx\]
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 \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int {\sin \left (c+d\,x\right )}^4\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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