Optimal. Leaf size=356 \[ \frac {6 \sqrt {6 \pi } b^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {2 \sqrt {2 \pi } b^{5/2} \sin \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {2 \sqrt {2 \pi } b^{5/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {6 \sqrt {6 \pi } b^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {24 b^2 \cos ^3(a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {16 b^2 \cos (a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {4 b \sin (a+b x) \cos ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \cos ^3(a+b x)}{5 d (c+d x)^{5/2}} \]
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Rubi [A] time = 0.83, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3314, 3297, 3306, 3305, 3351, 3304, 3352, 3313} \[ \frac {6 \sqrt {6 \pi } b^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {2 \sqrt {2 \pi } b^{5/2} \sin \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {2 \sqrt {2 \pi } b^{5/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {6 \sqrt {6 \pi } b^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {24 b^2 \cos ^3(a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {16 b^2 \cos (a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {4 b \sin (a+b x) \cos ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \cos ^3(a+b x)}{5 d (c+d x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3304
Rule 3305
Rule 3306
Rule 3313
Rule 3314
Rule 3351
Rule 3352
Rubi steps
\begin {align*} \int \frac {\cos ^3(a+b x)}{(c+d x)^{7/2}} \, dx &=-\frac {2 \cos ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {4 b \cos ^2(a+b x) \sin (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {\left (8 b^2\right ) \int \frac {\cos (a+b x)}{(c+d x)^{3/2}} \, dx}{5 d^2}-\frac {\left (12 b^2\right ) \int \frac {\cos ^3(a+b x)}{(c+d x)^{3/2}} \, dx}{5 d^2}\\ &=-\frac {16 b^2 \cos (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \cos ^3(a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {4 b \cos ^2(a+b x) \sin (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {\left (16 b^3\right ) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (72 b^3\right ) \int \left (-\frac {\sin (a+b x)}{4 \sqrt {c+d x}}-\frac {\sin (3 a+3 b x)}{4 \sqrt {c+d x}}\right ) \, dx}{5 d^3}\\ &=-\frac {16 b^2 \cos (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \cos ^3(a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {4 b \cos ^2(a+b x) \sin (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {\left (18 b^3\right ) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{5 d^3}+\frac {\left (18 b^3\right ) \int \frac {\sin (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (16 b^3 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (16 b^3 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}\\ &=-\frac {16 b^2 \cos (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \cos ^3(a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {4 b \cos ^2(a+b x) \sin (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {\left (18 b^3 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (32 b^3 \cos \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{5 d^4}+\frac {\left (18 b^3 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}+\frac {\left (18 b^3 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (32 b^3 \sin \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{5 d^4}+\frac {\left (18 b^3 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}\\ &=-\frac {16 b^2 \cos (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \cos ^3(a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {16 b^{5/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {16 b^{5/2} \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{5 d^{7/2}}+\frac {4 b \cos ^2(a+b x) \sin (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {\left (36 b^3 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{5 d^4}+\frac {\left (36 b^3 \cos \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{5 d^4}+\frac {\left (36 b^3 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{5 d^4}+\frac {\left (36 b^3 \sin \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{5 d^4}\\ &=-\frac {16 b^2 \cos (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \cos ^3(a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {2 b^{5/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {6 b^{5/2} \sqrt {6 \pi } \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {6 b^{5/2} \sqrt {6 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{5 d^{7/2}}+\frac {2 b^{5/2} \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{5 d^{7/2}}+\frac {4 b \cos ^2(a+b x) \sin (a+b x)}{5 d^2 (c+d x)^{3/2}}\\ \end {align*}
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Mathematica [B] time = 6.31, size = 1429, normalized size = 4.01 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 528, normalized size = 1.48 \[ \frac {2 \, {\left (3 \, \sqrt {6} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + \sqrt {2} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + \sqrt {2} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 3 \, \sqrt {6} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + {\left ({\left (12 \, b^{2} d^{2} x^{2} + 24 \, b^{2} c d x + 12 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )^{3} + 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) - 8 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \cos \left (b x + a\right )\right )} \sqrt {d x + c}\right )}}{5 \, {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )}} \]
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 \[ \int \frac {\cos \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 450, normalized size = 1.26 \[ \frac {-\frac {3 \cos \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{10 \left (d x +c \right )^{\frac {5}{2}}}-\frac {3 b \left (-\frac {\sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 b \left (-\frac {\cos \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{\sqrt {d x +c}}-\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}\right )}{5 d}-\frac {\cos \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{10 \left (d x +c \right )^{\frac {5}{2}}}-\frac {3 b \left (-\frac {\sin \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 b \left (-\frac {\cos \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{\sqrt {d x +c}}-\frac {b \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 d a -3 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}\right )}{5 d}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.09, size = 253, normalized size = 0.71 \[ -\frac {9 \, \sqrt {3} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} + {\left ({\left (-\left (3 i + 3\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + \left (3 i - 3\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left (\left (3 i - 3\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (3 i + 3\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}}}{16 \, {\left (d x + c\right )}^{\frac {5}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{7/2}} \,d x \]
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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