Optimal. Leaf size=101 \[ \frac{\sin ^2(c+d x) \sqrt{\cos ^2(c+d x)} \tan (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac{b \sin ^2(c+d x)}{a}+1\right )^{-p} F_1\left (\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right )}{3 d} \]
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Rubi [A] time = 0.104974, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3196, 511, 510} \[ \frac{\sin ^2(c+d x) \sqrt{\cos ^2(c+d x)} \tan (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac{b \sin ^2(c+d x)}{a}+1\right )^{-p} F_1\left (\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right )}{3 d} \]
Antiderivative was successfully verified.
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Rule 3196
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \left (a+b \sin ^2(c+d x)\right )^p \tan ^2(c+d x) \, dx &=\frac{\left (\sqrt{\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (a+b x^2\right )^p}{\left (1-x^2\right )^{3/2}} \, 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 ) \operatorname{Subst}\left (\int \frac{x^2 \left (1+\frac{b x^2}{a}\right )^p}{\left (1-x^2\right )^{3/2}} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{F_1\left (\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right ) \sqrt{\cos ^2(c+d x)} \sin ^2(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} \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.288817, size = 102, normalized size = 1.01 \[ \frac{\sin ^2(c+d x) \sqrt{\cos ^2(c+d x)} \tan (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac{a+b \sin ^2(c+d x)}{a}\right )^{-p} F_1\left (\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.664, size = 0, normalized size = 0. \begin{align*} \int \left ( a+ \left ( \sin \left ( dx+c \right ) \right ) ^{2}b \right ) ^{p} \left ( \tan \left ( dx+c \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-b \cos \left (d x + c\right )^{2} + a + b\right )}^{p} \tan \left (d x + c\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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