Optimal. Leaf size=70 \[ \frac {b^3 \sin (c+d x) (b \cos (c+d x))^{n-3} \, _2F_1\left (\frac {1}{2},\frac {n-3}{2};\frac {n-1}{2};\cos ^2(c+d x)\right )}{d (3-n) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {16, 2643} \[ \frac {b^3 \sin (c+d x) (b \cos (c+d x))^{n-3} \, _2F_1\left (\frac {1}{2},\frac {n-3}{2};\frac {n-1}{2};\cos ^2(c+d x)\right )}{d (3-n) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rubi steps
\begin {align*} \int (b \cos (c+d x))^n \sec ^4(c+d x) \, dx &=b^4 \int (b \cos (c+d x))^{-4+n} \, dx\\ &=\frac {b^3 (b \cos (c+d x))^{-3+n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-3+n);\frac {1}{2} (-1+n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (3-n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 72, normalized size = 1.03 \[ -\frac {\sqrt {\sin ^2(c+d x)} \csc (c+d x) \sec ^3(c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {n-3}{2};\frac {n-1}{2};\cos ^2(c+d x)\right )}{d (n-3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4}, x\right ) \]
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 \[ \int \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \left (b \cos \left (d x +c \right )\right )^{n} \left (\sec ^{4}\left (d x +c \right )\right )\, dx \]
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 \[ \int \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4}\,{d x} \]
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 \[ \int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^n}{{\cos \left (c+d\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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