Optimal. Leaf size=101 \[ \frac {\sin ^4(c+d x) \sqrt {\cos ^2(c+d x)} \tan (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {b \sin ^2(c+d x)}{a}+1\right )^{-p} F_1\left (\frac {5}{2};\frac {5}{2},-p;\frac {7}{2};\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right )}{5 d} \]
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Rubi [A] time = 0.13, 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 = {3196, 511, 510} \[ \frac {\sin ^4(c+d x) \sqrt {\cos ^2(c+d x)} \tan (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {b \sin ^2(c+d x)}{a}+1\right )^{-p} F_1\left (\frac {5}{2};\frac {5}{2},-p;\frac {7}{2};\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right )}{5 d} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rule 3196
Rubi steps
\begin {align*} \int \left (a+b \sin ^2(c+d x)\right )^p \tan ^4(c+d x) \, dx &=\frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (a+b x^2\right )^p}{\left (1-x^2\right )^{5/2}} \, 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^4 \left (1+\frac {b x^2}{a}\right )^p}{\left (1-x^2\right )^{5/2}} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {F_1\left (\frac {5}{2};\frac {5}{2},-p;\frac {7}{2};\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \sin ^4(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)}{5 d}\\ \end {align*}
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Mathematica [A] time = 4.50, size = 102, normalized size = 1.01 \[ \frac {\sin ^4(c+d x) \sqrt {\cos ^2(c+d x)} \tan (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {a+b \sin ^2(c+d x)}{a}\right )^{-p} F_1\left (\frac {5}{2};\frac {5}{2},-p;\frac {7}{2};\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right )}{5 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-b \cos \left (d x + c\right )^{2} + a + b\right )}^{p} \tan \left (d x + c\right )^{4}, x\right ) \]
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 \[ \int {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.09, size = 0, normalized size = 0.00 \[ \int \left (a +b \left (\sin ^{2}\left (d x +c \right )\right )\right )^{p} \left (\tan ^{4}\left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right )^{4}\,{d x} \]
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 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^4\,{\left (b\,{\sin \left (c+d\,x\right )}^2+a\right )}^p \,d x \]
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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