Optimal. Leaf size=339 \[ -\frac{a b \left (-16 a^2 b^2+3 a^4-127 b^4\right )}{8 d \left (a^2-b^2\right )^4 \sqrt{a+b \sin (c+d x)}}-\frac{b \left (-81 a^2 b^2+18 a^4-77 b^4\right )}{48 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{3/2}}-\frac{\left (12 a^2-54 a b+77 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right )}{32 d (a-b)^{9/2}}+\frac{\left (12 a^2+54 a b+77 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right )}{32 d (a+b)^{9/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}+\frac{\sec ^2(c+d x) \left (2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)+b \left (3 a^2+11 b^2\right )\right )}{16 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{3/2}} \]
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Rubi [A] time = 0.644966, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2668, 741, 823, 829, 827, 1166, 206} \[ -\frac{a b \left (-16 a^2 b^2+3 a^4-127 b^4\right )}{8 d \left (a^2-b^2\right )^4 \sqrt{a+b \sin (c+d x)}}-\frac{b \left (-81 a^2 b^2+18 a^4-77 b^4\right )}{48 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{3/2}}-\frac{\left (12 a^2-54 a b+77 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right )}{32 d (a-b)^{9/2}}+\frac{\left (12 a^2+54 a b+77 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right )}{32 d (a+b)^{9/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}+\frac{\sec ^2(c+d x) \left (2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)+b \left (3 a^2+11 b^2\right )\right )}{16 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 741
Rule 823
Rule 829
Rule 827
Rule 1166
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx &=\frac{b^5 \operatorname{Subst}\left (\int \frac{1}{(a+x)^{5/2} \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (6 a^2-11 b^2\right )+\frac{9 a x}{2}}{(a+x)^{5/2} \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 \left (a^2-b^2\right ) d}\\ &=-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac{\sec ^2(c+d x) \left (b \left (3 a^2+11 b^2\right )+2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^{3/2}}-\frac{b \operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (-12 a^4+19 a^2 b^2-77 b^4\right )-\frac{5}{2} a \left (3 a^2-10 b^2\right ) x}{(a+x)^{5/2} \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}\\ &=-\frac{b \left (18 a^4-81 a^2 b^2-77 b^4\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^{3/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac{\sec ^2(c+d x) \left (b \left (3 a^2+11 b^2\right )+2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^{3/2}}+\frac{b \operatorname{Subst}\left (\int \frac{\frac{1}{4} a \left (12 a^4-49 a^2 b^2+177 b^4\right )+\frac{1}{4} \left (18 a^4-81 a^2 b^2-77 b^4\right ) x}{(a+x)^{3/2} \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^3 d}\\ &=-\frac{b \left (18 a^4-81 a^2 b^2-77 b^4\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^{3/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}-\frac{a b \left (3 a^4-16 a^2 b^2-127 b^4\right )}{8 \left (a^2-b^2\right )^4 d \sqrt{a+b \sin (c+d x)}}+\frac{\sec ^2(c+d x) \left (b \left (3 a^2+11 b^2\right )+2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^{3/2}}-\frac{b \operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (-12 a^6+67 a^4 b^2-258 a^2 b^4-77 b^6\right )-\frac{1}{2} a \left (3 a^4-16 a^2 b^2-127 b^4\right ) x}{\sqrt{a+x} \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^4 d}\\ &=-\frac{b \left (18 a^4-81 a^2 b^2-77 b^4\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^{3/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}-\frac{a b \left (3 a^4-16 a^2 b^2-127 b^4\right )}{8 \left (a^2-b^2\right )^4 d \sqrt{a+b \sin (c+d x)}}+\frac{\sec ^2(c+d x) \left (b \left (3 a^2+11 b^2\right )+2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^{3/2}}-\frac{b \operatorname{Subst}\left (\int \frac{\frac{1}{2} a^2 \left (3 a^4-16 a^2 b^2-127 b^4\right )+\frac{1}{4} \left (-12 a^6+67 a^4 b^2-258 a^2 b^4-77 b^6\right )-\frac{1}{2} a \left (3 a^4-16 a^2 b^2-127 b^4\right ) x^2}{-a^2+b^2+2 a x^2-x^4} \, dx,x,\sqrt{a+b \sin (c+d x)}\right )}{4 \left (a^2-b^2\right )^4 d}\\ &=-\frac{b \left (18 a^4-81 a^2 b^2-77 b^4\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^{3/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}-\frac{a b \left (3 a^4-16 a^2 b^2-127 b^4\right )}{8 \left (a^2-b^2\right )^4 d \sqrt{a+b \sin (c+d x)}}+\frac{\sec ^2(c+d x) \left (b \left (3 a^2+11 b^2\right )+2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^{3/2}}-\frac{\left (12 a^2-54 a b+77 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-b-x^2} \, dx,x,\sqrt{a+b \sin (c+d x)}\right )}{32 (a-b)^4 d}+\frac{\left (12 a^2+54 a b+77 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-x^2} \, dx,x,\sqrt{a+b \sin (c+d x)}\right )}{32 (a+b)^4 d}\\ &=-\frac{\left (12 a^2-54 a b+77 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right )}{32 (a-b)^{9/2} d}+\frac{\left (12 a^2+54 a b+77 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right )}{32 (a+b)^{9/2} d}-\frac{b \left (18 a^4-81 a^2 b^2-77 b^4\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^{3/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}-\frac{a b \left (3 a^4-16 a^2 b^2-127 b^4\right )}{8 \left (a^2-b^2\right )^4 d \sqrt{a+b \sin (c+d x)}}+\frac{\sec ^2(c+d x) \left (b \left (3 a^2+11 b^2\right )+2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 3.36129, size = 296, normalized size = 0.87 \[ \frac{\frac{1}{2} \left (-81 a^2 b^2+18 a^4-77 b^4\right ) \left ((a+b) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b}\right )+(b-a) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \sin (c+d x)}{a+b}\right )\right )-15 a \left (3 a^2-10 b^2\right ) (a+b \sin (c+d x)) \left ((a+b) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b}\right )+(b-a) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{a+b \sin (c+d x)}{a+b}\right )\right )-3 (a-b) (a+b) \sec ^2(c+d x) \left (\left (6 a^3-20 a b^2\right ) \sin (c+d x)+3 a^2 b+11 b^3\right )-12 (a-b)^2 (a+b)^2 \sec ^4(c+d x) (a \sin (c+d x)-b)}{48 d \left (a^2-b^2\right )^2 \left (b^2-a^2\right ) (a+b \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.795, size = 682, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (-\frac{\sqrt{b \sin \left (d x + c\right ) + a} \sec \left (d x + c\right )^{5}}{3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} +{\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}, 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 [B] time = 1.21919, size = 861, normalized size = 2.54 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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