Optimal. Leaf size=105 \[ -\frac {F_1\left (1-n;\frac {5}{4},\frac {5}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right ) (1-\cos (c+d x))^{5/4} \cos (c+d x) (1+\cos (c+d x))^{\frac {5}{4}-n} (a+a \sec (c+d x))^n}{d (1-n) \sin ^{\frac {5}{2}}(c+d x)} \]
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Rubi [A]
time = 0.18, 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 = {3961, 2965,
140, 138} \begin {gather*} -\frac {(1-\cos (c+d x))^{5/4} \cos (c+d x) (\cos (c+d x)+1)^{\frac {5}{4}-n} (a \sec (c+d x)+a)^n F_1\left (1-n;\frac {5}{4},\frac {5}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right )}{d (1-n) \sin ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 140
Rule 2965
Rule 3961
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(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 \frac {(-\cos (c+d x))^{-n} (-a-a \cos (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx\\ &=-\frac {\left ((-\cos (c+d x))^n (-a-a \cos (c+d x))^{\frac {5}{4}-n} (-a+a \cos (c+d x))^{5/4} (a+a \sec (c+d x))^n\right ) \text {Subst}\left (\int \frac {(-x)^{-n} (-a-a x)^{-\frac {5}{4}+n}}{(-a+a x)^{5/4}} \, dx,x,\cos (c+d x)\right )}{d \sin ^{\frac {5}{2}}(c+d x)}\\ &=-\left (-\frac {\left ((-\cos (c+d x))^n (1+\cos (c+d x))^{\frac {1}{4}-n} (-a-a \cos (c+d x)) (-a+a \cos (c+d x))^{5/4} (a+a \sec (c+d x))^n\right ) \text {Subst}\left (\int \frac {(-x)^{-n} (1+x)^{-\frac {5}{4}+n}}{(-a+a x)^{5/4}} \, dx,x,\cos (c+d x)\right )}{a d \sin ^{\frac {5}{2}}(c+d x)}\right )\\ &=-\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 \cos (c+d x)) (-a+a \cos (c+d x)) (a+a \sec (c+d x))^n\right ) \text {Subst}\left (\int \frac {(-x)^{-n} (1+x)^{-\frac {5}{4}+n}}{(1-x)^{5/4}} \, dx,x,\cos (c+d x)\right )}{a^2 d \sin ^{\frac {5}{2}}(c+d x)}\\ &=-\frac {F_1\left (1-n;\frac {5}{4},\frac {5}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right ) (1-\cos (c+d x))^{5/4} \cos (c+d x) (1+\cos (c+d x))^{\frac {5}{4}-n} (a+a \sec (c+d x))^n}{d (1-n) \sin ^{\frac {5}{2}}(c+d x)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(212\) vs. \(2(105)=210\).
time = 2.57, size = 212, normalized size = 2.02 \begin {gather*} -\frac {6 F_1\left (-\frac {1}{4};n,-\frac {1}{2};\frac {3}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+\cos (c+d x)) (a (1+\sec (c+d x)))^n}{d \left (-2 \left (F_1\left (\frac {3}{4};n,\frac {1}{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 )+2 n F_1\left (\frac {3}{4};1+n,-\frac {1}{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 ) (-1+\cos (c+d x))+3 F_1\left (-\frac {1}{4};n,-\frac {1}{2};\frac {3}{4};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+\cos (c+d x))\right ) \sqrt {\sin (c+d x)}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (a +a \sec \left (d x +c \right )\right )^{n}}{\sin \left (d x +c \right )^{\frac {3}{2}}}\, dx\]
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 \frac {\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n}}{\sin ^{\frac {3}{2}}{\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+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n}{{\sin \left (c+d\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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