Optimal. Leaf size=97 \[ -\frac {F_1\left (-\frac {1}{2};-\frac {1}{2},-p;\frac {1}{2};\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac {b \sin ^2(c+d x)}{a}\right )^{-p}}{d} \]
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Rubi [A]
time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3275, 525, 524}
\begin {gather*} -\frac {\sqrt {\cos ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {b \sin ^2(c+d x)}{a}+1\right )^{-p} F_1\left (-\frac {1}{2};-\frac {1}{2},-p;\frac {1}{2};\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 3275
Rubi steps
\begin {align*} \int \cot ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx &=\frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {\sqrt {1-x^2} \left (a+b x^2\right )^p}{x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac {b \sin ^2(c+d x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\sqrt {1-x^2} \left (1+\frac {b x^2}{a}\right )^p}{x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {F_1\left (-\frac {1}{2};-\frac {1}{2},-p;\frac {1}{2};\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac {b \sin ^2(c+d x)}{a}\right )^{-p}}{d}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 98, normalized size = 1.01 \begin {gather*} -\frac {F_1\left (-\frac {1}{2};-\frac {1}{2},-p;\frac {1}{2};\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {a+b \sin ^2(c+d x)}{a}\right )^{-p}}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.56, size = 0, normalized size = 0.00 \[\int \left (\cot ^{2}\left (d x +c \right )\right ) \left (a +\left (\sin ^{2}\left (d x +c \right )\right ) b \right )^{p}\, dx\]
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 = 0.41, size = 27, normalized size = 0.28 \begin {gather*} {\rm integral}\left ({\left (-b \cos \left (d x + c\right )^{2} + a + b\right )}^{p} \cot \left (d x + c\right )^{2}, x\right ) \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 = -0.01 \begin {gather*} \int {\mathrm {cot}\left (c+d\,x\right )}^2\,{\left (b\,{\sin \left (c+d\,x\right )}^2+a\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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