Optimal. Leaf size=141 \[ \frac {\sin ^2(c+d x) \left (a+b \sin ^4(c+d x)\right )^p \left (\frac {b \sin ^4(c+d x)}{a}+1\right )^{-p} F_1\left (\frac {1}{2};1,-p;\frac {3}{2};\sin ^4(c+d x),-\frac {b \sin ^4(c+d x)}{a}\right )}{2 d}+\frac {\left (a+b \sin ^4(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sin ^4(c+d x)+a}{a+b}\right )}{4 d (p+1) (a+b)} \]
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Rubi [A] time = 0.12, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3229, 757, 430, 429, 444, 68} \[ \frac {\sin ^2(c+d x) \left (a+b \sin ^4(c+d x)\right )^p \left (\frac {b \sin ^4(c+d x)}{a}+1\right )^{-p} F_1\left (\frac {1}{2};1,-p;\frac {3}{2};\sin ^4(c+d x),-\frac {b \sin ^4(c+d x)}{a}\right )}{2 d}+\frac {\left (a+b \sin ^4(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sin ^4(c+d x)+a}{a+b}\right )}{4 d (p+1) (a+b)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 429
Rule 430
Rule 444
Rule 757
Rule 3229
Rubi steps
\begin {align*} \int \left (a+b \sin ^4(c+d x)\right )^p \tan (c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{1-x} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {\left (a+b x^2\right )^p}{1-x^2}-\frac {x \left (a+b x^2\right )^p}{-1+x^2}\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{1-x^2} \, dx,x,\sin ^2(c+d x)\right )}{2 d}-\frac {\operatorname {Subst}\left (\int \frac {x \left (a+b x^2\right )^p}{-1+x^2} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^p}{-1+x} \, dx,x,\sin ^4(c+d x)\right )}{4 d}+\frac {\left (\left (a+b \sin ^4(c+d x)\right )^p \left (1+\frac {b \sin ^4(c+d x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p}{1-x^2} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac {\, _2F_1\left (1,1+p;2+p;\frac {a+b \sin ^4(c+d x)}{a+b}\right ) \left (a+b \sin ^4(c+d x)\right )^{1+p}}{4 (a+b) d (1+p)}+\frac {F_1\left (\frac {1}{2};1,-p;\frac {3}{2};\sin ^4(c+d x),-\frac {b \sin ^4(c+d x)}{a}\right ) \sin ^2(c+d x) \left (a+b \sin ^4(c+d x)\right )^p \left (1+\frac {b \sin ^4(c+d x)}{a}\right )^{-p}}{2 d}\\ \end {align*}
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Mathematica [B] time = 9.90, size = 463, normalized size = 3.28 \[ \frac {2 (2 p-1) \left (\sqrt {-a b}-b\right ) \left (\sqrt {-a b}+b\right ) \sin ^4(c+d x) \cos ^2(c+d x) \left (\left (a-\sqrt {-a b}\right ) \cot ^2(c+d x)+a+b\right ) \left (\left (\sqrt {-a b}+a\right ) \cot ^2(c+d x)+a+b\right ) \left (a+b \sin ^4(c+d x)\right )^p F_1\left (-2 p;-p,-p;1-2 p;-\frac {(a+b) \sec ^2(c+d x)}{\sqrt {-a b}-b},\frac {(a+b) \sec ^2(c+d x)}{b+\sqrt {-a b}}\right )}{d p (a+b)^2 (8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b) \left (p \left (\sqrt {-a b}+b\right ) F_1\left (1-2 p;1-p,-p;2-2 p;-\frac {(a+b) \sec ^2(c+d x)}{\sqrt {-a b}-b},\frac {(a+b) \sec ^2(c+d x)}{b+\sqrt {-a b}}\right )-p \left (\sqrt {-a b}-b\right ) F_1\left (1-2 p;-p,1-p;2-2 p;-\frac {(a+b) \sec ^2(c+d x)}{\sqrt {-a b}-b},\frac {(a+b) \sec ^2(c+d x)}{b+\sqrt {-a b}}\right )+b (2 p-1) \cos ^2(c+d x) F_1\left (-2 p;-p,-p;1-2 p;-\frac {(a+b) \sec ^2(c+d x)}{\sqrt {-a b}-b},\frac {(a+b) \sec ^2(c+d x)}{b+\sqrt {-a b}}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b\right )}^{p} \tan \left (d x + c\right ), x\right ) \]
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 \[ \int {\left (b \sin \left (d x + c\right )^{4} + a\right )}^{p} \tan \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.72, size = 0, normalized size = 0.00 \[ \int \left (a +b \left (\sin ^{4}\left (d x +c \right )\right )\right )^{p} \tan \left (d x +c \right )\, dx \]
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 \[ \int {\left (b \sin \left (d x + c\right )^{4} + a\right )}^{p} \tan \left (d x + c\right )\,{d x} \]
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 \[ \int \mathrm {tan}\left (c+d\,x\right )\,{\left (b\,{\sin \left (c+d\,x\right )}^4+a\right )}^p \,d x \]
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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