Optimal. Leaf size=101 \[ \frac {F_1\left (\frac {3}{2};\frac {3}{2},-p;\frac {5}{2};\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \sin ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac {b \sin ^2(c+d x)}{a}\right )^{-p} \tan (c+d x)}{3 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 101, 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 {\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 3275
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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.22, size = 102, normalized size = 1.01 \begin {gather*} \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 (\frac {a+b \sin ^2(c+d x)}{a}\right )^{-p} \tan (c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.50, size = 0, normalized size = 0.00 \[\int \left (a +\left (\sin ^{2}\left (d x +c \right )\right ) b \right )^{p} \left (\tan ^{2}\left (d x +c \right )\right )\, dx\]
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 \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 = 27, normalized size = 0.27 \begin {gather*} {\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 {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 {tan}\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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