Optimal. Leaf size=117 \[ -\frac {(a-b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{d}+\frac {(a+b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{d}-\frac {4 a b \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 b (a+b \sin (c+d x))^{3/2}}{3 d} \]
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Rubi [A]
time = 0.15, antiderivative size = 117, 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 = {2747, 718, 839,
841, 1180, 212} \begin {gather*} -\frac {2 b (a+b \sin (c+d x))^{3/2}}{3 d}-\frac {4 a b \sqrt {a+b \sin (c+d x)}}{d}-\frac {(a-b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{d}+\frac {(a+b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 718
Rule 839
Rule 841
Rule 1180
Rule 2747
Rubi steps
\begin {align*} \int \sec (c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=\frac {b \text {Subst}\left (\int \frac {(a+x)^{5/2}}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {2 b (a+b \sin (c+d x))^{3/2}}{3 d}-\frac {b \text {Subst}\left (\int \frac {\sqrt {a+x} \left (-a^2-b^2-2 a x\right )}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {4 a b \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 b (a+b \sin (c+d x))^{3/2}}{3 d}+\frac {b \text {Subst}\left (\int \frac {a \left (a^2+3 b^2\right )+\left (3 a^2+b^2\right ) x}{\sqrt {a+x} \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {4 a b \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 b (a+b \sin (c+d x))^{3/2}}{3 d}+\frac {(2 b) \text {Subst}\left (\int \frac {-a \left (3 a^2+b^2\right )+a \left (a^2+3 b^2\right )+\left (3 a^2+b^2\right ) x^2}{-a^2+b^2+2 a x^2-x^4} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{d}\\ &=-\frac {4 a b \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 b (a+b \sin (c+d x))^{3/2}}{3 d}-\frac {(a-b)^3 \text {Subst}\left (\int \frac {1}{a-b-x^2} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{d}+\frac {(a+b)^3 \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{d}\\ &=-\frac {(a-b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{d}+\frac {(a+b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{d}-\frac {4 a b \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 b (a+b \sin (c+d x))^{3/2}}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 105, normalized size = 0.90 \begin {gather*} \frac {-3 (a-b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )+3 (a+b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )-2 b \sqrt {a+b \sin (c+d x)} (7 a+b \sin (c+d x))}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(285\) vs.
\(2(99)=198\).
time = 1.58, size = 286, normalized size = 2.44
method | result | size |
default | \(\frac {-\frac {2 b \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-4 b a \sqrt {a +b \sin \left (d x +c \right )}+\frac {\arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a^{3}}{\sqrt {a +b}}+\frac {3 b \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a^{2}}{\sqrt {a +b}}+\frac {3 b^{2} \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a}{\sqrt {a +b}}+\frac {b^{3} \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right )}{\sqrt {a +b}}+\frac {\arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a^{3}}{\sqrt {-a +b}}-\frac {3 b \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a^{2}}{\sqrt {-a +b}}+\frac {3 b^{2} \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a}{\sqrt {-a +b}}-\frac {b^{3} \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}}{d}\) | \(286\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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(-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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}}{\cos \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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