Optimal. Leaf size=59 \[ \frac {\, _2F_1\left (1,1+p;2+p;\frac {a+b \sin ^2(c+d x)}{a+b}\right ) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 (a+b) d (1+p)} \]
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Rubi [A]
time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3273, 70}
\begin {gather*} \frac {\left (a+b \sin ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sin ^2(c+d x)+a}{a+b}\right )}{2 d (p+1) (a+b)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 3273
Rubi steps
\begin {align*} \int \left (a+b \sin ^2(c+d x)\right )^p \tan (c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b x)^p}{1-x} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac {\, _2F_1\left (1,1+p;2+p;\frac {a+b \sin ^2(c+d x)}{a+b}\right ) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 (a+b) d (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 61, normalized size = 1.03 \begin {gather*} \frac {\left (a+b-b \cos ^2(c+d x)\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1-\frac {b \cos ^2(c+d x)}{a+b}\right )}{2 (a+b) d (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.52, size = 0, normalized size = 0.00 \[\int \left (a +\left (\sin ^{2}\left (d x +c \right )\right ) b \right )^{p} \tan \left (d x +c \right )\, dx\]
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 = 0.39, size = 25, normalized size = 0.42 \begin {gather*} {\rm integral}\left ({\left (-b \cos \left (d x + c\right )^{2} + a + b\right )}^{p} \tan \left (d x + c\right ), x\right ) \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 \left (a + b \sin ^{2}{\left (c + d x \right )}\right )^{p} \tan {\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 = -0.02 \begin {gather*} \int \mathrm {tan}\left (c+d\,x\right )\,{\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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