Optimal. Leaf size=102 \[ \frac {\sec ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^{p+1}}{2 d (a+b)}-\frac {(a+b p+b) \left (a+b \sin ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sin ^2(c+d x)+a}{a+b}\right )}{2 d (p+1) (a+b)^2} \]
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Rubi [A] time = 0.09, antiderivative size = 102, 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 = {3194, 78, 68} \[ \frac {\sec ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^{p+1}}{2 d (a+b)}-\frac {(a+b p+b) \left (a+b \sin ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sin ^2(c+d x)+a}{a+b}\right )}{2 d (p+1) (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 68
Rule 78
Rule 3194
Rubi steps
\begin {align*} \int \left (a+b \sin ^2(c+d x)\right )^p \tan ^3(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x (a+b x)^p}{(1-x)^2} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac {\sec ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 (a+b) d}-\frac {(a+b+b p) \operatorname {Subst}\left (\int \frac {(a+b x)^p}{1-x} \, dx,x,\sin ^2(c+d x)\right )}{2 (a+b) d}\\ &=-\frac {(a+b+b p) \, _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)^2 d (1+p)}+\frac {\sec ^2(c+d x) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 (a+b) d}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 83, normalized size = 0.81 \[ \frac {\left (a+b \sin ^2(c+d x)\right )^{p+1} \left ((p+1) (a+b) \sec ^2(c+d x)-(a+b p+b) \, _2F_1\left (1,p+1;p+2;\frac {b \sin ^2(c+d x)+a}{a+b}\right )\right )}{2 d (p+1) (a+b)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-b \cos \left (d x + c\right )^{2} + a + b\right )}^{p} \tan \left (d x + c\right )^{3}, x\right ) \]
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 \[ \int {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.01, size = 0, normalized size = 0.00 \[ \int \left (a +b \left (\sin ^{2}\left (d x +c \right )\right )\right )^{p} \left (\tan ^{3}\left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^3\,{\left (b\,{\sin \left (c+d\,x\right )}^2+a\right )}^p \,d x \]
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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