Optimal. Leaf size=163 \[ \frac {2 A (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-7+2 n);\frac {1}{4} (-3+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (7-2 n) \cos ^{\frac {7}{2}}(c+d x) \sqrt {\sin ^2(c+d x)}}+\frac {2 B (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-5+2 n);\frac {1}{4} (-1+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5-2 n) \cos ^{\frac {5}{2}}(c+d x) \sqrt {\sin ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {20, 2827, 2722}
\begin {gather*} \frac {2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n-7);\frac {1}{4} (2 n-3);\cos ^2(c+d x)\right )}{d (7-2 n) \sqrt {\sin ^2(c+d x)} \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n-5);\frac {1}{4} (2 n-1);\cos ^2(c+d x)\right )}{d (5-2 n) \sqrt {\sin ^2(c+d x)} \cos ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 20
Rule 2722
Rule 2827
Rubi steps
\begin {align*} \int \frac {(b \cos (c+d x))^n (A+B \cos (c+d x))}{\cos ^{\frac {9}{2}}(c+d x)} \, dx &=\left (\cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac {9}{2}+n}(c+d x) (A+B \cos (c+d x)) \, dx\\ &=\left (A \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac {9}{2}+n}(c+d x) \, dx+\left (B \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac {7}{2}+n}(c+d x) \, dx\\ &=\frac {2 A (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-7+2 n);\frac {1}{4} (-3+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (7-2 n) \cos ^{\frac {7}{2}}(c+d x) \sqrt {\sin ^2(c+d x)}}+\frac {2 B (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-5+2 n);\frac {1}{4} (-1+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5-2 n) \cos ^{\frac {5}{2}}(c+d x) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.25, size = 138, normalized size = 0.85 \begin {gather*} -\frac {2 (b \cos (c+d x))^n \csc (c+d x) \left (A (-5+2 n) \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-7+2 n);\frac {1}{4} (-3+2 n);\cos ^2(c+d x)\right )+B (-7+2 n) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-5+2 n);\frac {1}{4} (-1+2 n);\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (-7+2 n) (-5+2 n) \cos ^{\frac {7}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (b \cos \left (d x +c \right )\right )^{n} \left (A +B \cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )^{\frac {9}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^n\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{{\cos \left (c+d\,x\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________